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Orbital Varieties and Unipotent Representations of Classical Semisimple Lie Groups by Thomas Pietraho M.S., University of Chicago, 1996 B.A., University of Chicago, 1996 Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2001 c Thomas Pietraho, MMI. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part and to grant others the right to do so. Author ........................................................................... Department of Mathematics April 25, 2001 Certified by ...................................................................... David A. Vogan Professor of Mathematics Thesis Supervisor Accepted by ...................................................................... Tomasz Mrowka Chairman, Department Committee on Graduate Students
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Orbital Varieties and Unipotent Representations of Classical
Semisimple Lie Groups
by
Thomas Pietraho
M.S., University of Chicago, 1996B.A., University of Chicago, 1996
Submitted to the Department of Mathematicsin partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2001
c© Thomas Pietraho, MMI. All rights reserved.
The author hereby grants to MIT permission to reproduce and to distribute publiclypaper and electronic copies of this thesis document in whole or in part and to grant
others the right to do so.
Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Department of Mathematics
April 25, 2001
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .David A. Vogan
Professor of MathematicsThesis Supervisor
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Tomasz Mrowka
Chairman, Department Committee on Graduate Students
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Orbital Varieties and Unipotent Representations of Classical SemisimpleLie Groups
byThomas Pietraho
Submitted to the Department of Mathematicson April 25, 2001, in partial fulfillment of the
requirements for the degree ofDoctor of Philosophy
Abstract
Let G be a complex semi-simple and classical Lie group. The notion of a Lagrangian coveringcan be used to extend the method of polarizing a nilpotent coadjoint orbit to obtain a unitaryrepresentation of G. W. Graham and D. Vogan propose such a construction, relying on thenotions of orbital varieties and admissible orbit data.
The first part of the thesis seeks to understand the set of orbital varieties contained ina given nipotent orbit. Starting from N. Spaltenstein’s parameterization of the irreduciblecomponents of the variety of flags fixed by a unipotent, we produce a parameterization ofthe orbital varieties lying in the corresponding fiber of the Steinberg map. The parameterset is the family of standard Young or domino tableau of a given shape. The key to the proofis understanding certain closed cycles as defined by D. Garfinkle. This parameterizationis particularly useful; it provides a method of determining the τ -invariant of each orbitalvariety, as well as a way of relating an orbital variety in any classical group to one lying intype A.
The second part of the thesis addresses the representations V (V, π) constructed byGraham and Vogan. A natural question is how well the V (V, π) approximate the set ofunipotent representations that ought to be attached to the nilpotent orbit O. The answeris promising in the setting of spherical orbits. When it is possible to carry out the Graham-Vogan construction, the corresponding infinitesimal character lies in the set of characterssuggested by W. M. McGovern. Furthermore, we show that it is possible to carry out theGraham-Vogan construction for a suffient number of orbital varieties to account for all theinfinitesimal characters attached to O by McGovern.
Thesis Supervisor: David A. VoganTitle: Professor of Mathematics
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Acknowledgments
I would like to thank my advisor, David Vogan, for encouragement and advice. He was thekeystone in my development as a mathematician, and I feel privileged to have had such adistinguished and caring mentor. Ken Gross, Paul Sally, Tony Trono, Neil Tame, and AllanGerry, have all guided me in my pursuit of mathematics. To them, I will forever be in debt.
The representation theory group of students created an active and stimulating atmo-sphere at MIT. My thanks go out to my cofactor turned thesis committee member PeterTrapa, and my mathematical brethren Dana Pascovici, Adam Lucas, Pramod Achar, An-thony Henderson, and Kevin McGerty. The wider community of graduate students at MITis truly exceptional, and I am grateful to them all. I would like to especially thank AlekseyZinger, Lenny Ng, Daniel Chan, and Catalin Zara. Their friendship and support have beenimportant to me during my time at MIT.
I would like to thank my family, for their love, understanding, and for teaching me whatis important in life. Finally, I would like to thank my wife, Jennifer, for showing me howit’s done. She has been a pillar of support, a terrific companion, and a partner in crime.
”The lyf so short, the craft so long to lerne.”- Masthead logo of The Craftsman, ca. 1910.
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Contents
1 Introduction 91.1 Orbital Varieties and Domino Tableaux . . . . . . . . . . . . . . . . . . . . 91.2 Infinitesimal Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 The Graham-Vogan Construction 132.1 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Lagrangian Coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 A Subspace of Sections of LM . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Geometric Left Cells and the Unipotent Variety 193.1 Preliminary Definitions and Basic Facts . . . . . . . . . . . . . . . . . . . . 21
3.1.1 Unipotent Variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.1.2 Nilpotent Orbits and Orbital Varieties . . . . . . . . . . . . . . . . . 223.1.3 Domino Tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1.4 Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Irreducible Components of Fu . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.1 Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.2 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 The Components Irr(O′u ∩ n) . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3.1 A Bijection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Proof of Lemma 3.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4.1 Closed and Nested Clusters . . . . . . . . . . . . . . . . . . . . . . . 303.4.2 Three More Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.5 The τ -Invariant for Orbital Varieties . . . . . . . . . . . . . . . . . . . . . . 353.6 Projection of Orbital Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4 Restriction to Spherical Orbital Varieties 394.1 Model Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Spherical Orbital Varieties and Orbits of S-type . . . . . . . . . . . . . . . . 42
4.2.1 Spherical Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2.2 Smith Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2.3 Rigid and Special Orbits . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 Basepoints in VT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.3.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.3.2 Type A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.3.3 Other Classical Types . . . . . . . . . . . . . . . . . . . . . . . . . . 46
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4.4 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.5 The Trace of the Adjoint Action . . . . . . . . . . . . . . . . . . . . . . . . 53
5 Infinitesimal Characters 575.1 Characters, Weights, and Extensions . . . . . . . . . . . . . . . . . . . . . . 575.2 The Infinitesimal Characters IC1(O) . . . . . . . . . . . . . . . . . . . . . . 60
5.2.1 Infinitesimal Characters of q-unipotent Representations . . . . . . . 605.2.2 The Preimage M−1(O) . . . . . . . . . . . . . . . . . . . . . . . . . 635.2.3 Pruning of IC(O) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.3 Infinitesimal Characters of V (V, π) . . . . . . . . . . . . . . . . . . . . . . . 685.3.1 A Few Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.3.2 Positive Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.3.3 Proof of Theorem 5.3.4 . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.4 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
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Chapter 1
Introduction
The orbit method seeks to classify irreducible unitary representations of a Lie group G byidentifying them with the set of coadjoint orbits g∗/G. A classical theorem of Kostant andKirillov for nilpotent Lie groups provides the motivation for this approach:
Theorem (Kirillov). If G is a connected and simply connected nilpotent Lie group, thenthere is a bijective correspondence
g∗/G −→ Gunitary
between the coadjoint orbits of G and the set of its irreducible unitary representations.
For semisimple Lie groups, however, this approach is far less understood. It is knownwhich unitary representations correspond to semisimple coadjoint orbits. They are therepresentations obtained by parabolic or cohomological induction from a set of unitarycharacters of Levi subgroups of G. General orbits should correspond to representationsthat are parabolically and cohomogically induced from representations attached to nilpotentcoadjoint orbits, which we denote Gfund. We call this set of representations unipotent. Itis not known what this set of unipotent representations ought to be and in what mannersuch a correspondence should be carried out.
The paper [Graham-Vogan] constructs a collection GVO of representation spaces foreach nilpotent coadjoint orbit O of a complex reductive Lie group G. We denote each suchspace as V (V, π) and defer a more precise definition. Very little is known about whichrepresentations of G actually arise in this way, but conjecturally, they should be close tothe set of unipotent representations corresponding to the nilpotent orbit O, or GO
fund. Wefirst address the data used to construct each V (V, π).
1.1 Orbital Varieties and Domino Tableaux
The method of polarization provides a motivated approach to the problem of quantizingnilpotent coadjoint orbits. Unfortunately, it requires a construction of certain Lagrangianfoliations which often do not exist. The construction of Graham and Vogan is designedto mimic polarization, but replaces Lagrangian foliations by somewhat weaker structurescalled Lagrangian coverings. Work of V. Ginsburg imples that Lagrangian coverings alwaysexist in the setting of coadjoint orbits of complex reductive Lie groups [Ginsburg].
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The main ingredients in the construction of Lagrangian coverings of O are the orbitalvarieties V ⊂ O. For each choice of orbital variety V and a choice of an admissible orbitdatum π, the Graham-Vogan construction provides a subspace V (V, π) of sections of abundle LG/QV over G/QV , where QV ⊂ G is the maximal subgroup of G stabilizing V. Thegroup QV contains a Borel subgroup of G and is hence parabolic.
To understand which representations arise as V (V, π), one would like to first parame-terize orbital varieties in a way that also describes the parabolic QV . We will use the termstandard tableau to refer to a standard Young tableau when type G = A, and to a standarddomino tableau in the other classical types. A more complete descripition can be found inSection 3.1.3.
Theorem 3.3.2. Among classical groups, the orbital varieties V ⊂ O are parameterizedby the set of standard tableaux of shape equal to the partition associated to O. The onlyexception occurs in type D when O is very even. In this instance, the number of verticaldominos in the tableau is congruent to 0 or 2 mod 4 depending on the Roman numeralattached to the orbit O.
This mirrors the result obtained in [McGovern2] by examining certain equivalence classeson the Weyl group of G. The proof in this thesis relies on the work of N. Spaltenstein onthe irreducible components of the unipotent variety Fu and the results of D. Garfinkle ondomino tableaux. We can now write VT for the orbital variety that corresponds to thestandard tableau T . The description of the maximal stabilizing parabolic subgroup of VT
is now simple to state.Theorem 3.5.1. Consider an orbital variety V = VT . The Lie algebra q of QV contains
the root space g−αi associated with the simple root −αi iff
(i) the entry i + 1 appears strictly below i in T when type G is A, or
(ii) when type G is not A, either
· i = 1 and the domino with label 1 is vertical, or
· i 6= 1 and the domino with label i lies strictly below i− 1 in T .
The standard tableaux parameterization has two more benefits. Using the work of[Carre-Leclerc], it is possible to define a projection map from orbital varieties of all classicaltypes to ones of type A, both on the level of orbital varieties and their correspondingstandard tableaux. This approach suggests a way of finding a minimal representative ineach VT in the sense of [Melnikov]. We carry this out in the setting of spherical orbits. SeeDefinition 4.3.5.
The standard domino parameterization also admits a means of addressing some relatedcalculations inductively. It sets up a framework for calculating data in the Graham-Voganconstruction.
1.2 Infinitesimal Characters
Let O be a nilpotent coadjoint orbit for a semisimple Lie group G. [McGovern] determines aset IC1(O) of infinitesimal characters that ought to correspond to elements of GO
fund. Thisprovides a convenient way of testing whether the Graham-Vogan construction providesgood candidates for the representations in GO
fund; the set of infinitesimal characters of the
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representations attached to O should contain IC1(O). For spherical O, this is exactly whathappens:
Theorems 5.3.4 and 5.3.5. Let O be a spherical nilpotent orbit of a complex classicalsemisimple Lie group G of rank n and suppose that it is possible to construct the spaceV (V, π). Let χV be the infinitesimal character associated to V (V, π). Then,
(i) If O is rigid, then IC1(O) = χV | V (V, π) ∈ GVO,(ii) If O is a model orbit and n > 2, then IC1(O) ⊂ χV,π| V (V, π) ∈ GVO.
The theorem implies that, at least for spherical nilpotent orbits, the Graham-Voganspaces are strong candiates for unipotent representations. As for non-spherical orbits, it isapparent from Theorem 5.3.5 that the set GVO is too large to be GO
fund. However, additionalconditions on the closure O not considered in [Graham-Vogan] should make it possible torestrict the set of possible infinitesimal characters.
Here is an outline of this thesis. Chapter 2 provides a short description of polarizationand its refinement by Graham and Vogan. Chapter 3 addresses orbital varieties. After listingthe possible approaches to the classification problem, we describe the work of N. Spaltensteinand M. A. van Leeuwen in this direction. A sequence of somewhat technical lemmas provesthe parameterization of orbital varieties described above. We also address the τ -invariantof an orbital variety and describe a method of projecting all orbital varieties onto onesof type A. Chapter 4 begins by examining the Graham-Vogan construstion in a “modelexample.” We restrict our attention to spherical orbits, construct a basepoint within eachorbital variety, and exhibit an inductive construction that we will use to describe V (V, π).Finally, Chapter 5 addresses infinitesimal characters. We describe the work of McGoverndescribing a set IC1(O) of characters, provide a few examples of interesting behavior ofV (V, π), and prove the theorems described above. We finish with some thoughts on furtherwork.
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Chapter 2
The Graham-Vogan Construction
The Graham-Vogan construction of representations associated to a coadjoint orbit O is anextension of the method of polarizing a coadjoint orbit. Polarization is a very effectivetool for attaching representations to coadjoint orbits in the setting of connected nilpotentLie groups, but unfortunately is not as useful among semisimple Lie groups. It relies on aconstruction of a Lagrangian foliation that may not always exist.
To amend this shortfall, [Graham-Vogan] replaces Lagrangian foliations used in polar-ization with the so-called Lagrangian coverings. By a theorem of V. Ginzburg, it is alwayspossible to construct a Lagrangian covering of a coadjoint orbit O. In fact, there is a uniqueone for each orbital variety contained in O. The difficult task now becomes to mimic theconstruction of representations used in polarization in this more complicated setting.
For nilpotent coadjoint orbits, Graham and Vogan suggest a construction of a represen-tation from each pair of the following objects:
· an admissible orbit datum, and
· an orbital variety.
The representation lies in the space of smooth vectors in a degenerate principal seriesrepresentation induced from a representation of a parabolic subgroup of G.
2.1 Polarization
Given a coadjoint orbit O, polarization attempts to find a smooth G-manifold M and aHermitian line bundle LM on M such that O is isomorphic as a symplectic G-space to acertain twisted cotangent bundle T ∗(M,L) such as that defined by [Kostant]. The unitaryrepresentation attached to O is then the space π(M,LM ) described in [Graham-Vogan].
This process provides a nice way of quantizing a coadjoint orbit, and hence we’d like toknow under what circumstances such a manifold M and line bundle LM exist. We note afew properties.
· T ∗(M,LM ) is a symplectic manifold,
· T ∗m(M,LM ) are Lagrangian submanifolds,
· T ∗m(M,LM ) is a Lagrangian foliation of T ∗(M,LM ).
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If O is to be isomorphic as a symplectic G-space to T ∗(M,LM ), it should be possibleto find similar structures on O. To this effect, we need to find a G-invariant Lagrangianfoliation of O. Because O is homogeneous, this can be reduced to finding a subgroup of Gwith certain properties. More precisely, if we fix a basepoint f , then O ∼= G/Gf , where Gf
is the isotropy group. As O is homogeneous, so must be the space of leaves and the entirefoliation is determined by the leaf Λf through f . The question of finding a Lagrangianfoliation is so reduced to finding a subgroup H containing Gf such that:
· H is a closed Lie subgroup,· f |[h,h] = 0,· dimH/Gf = 1
2 dimG/Gf ,
Given such a subgroup H, the manifold M is then isomorphic to the space of leavesG/H, and each leaf is isomorphic to H/Gf . Furthermore, the line bundle LM is induced bya character
· τ ∈ H with dτ = 2πif .
Given a subgroup H and a character τ of H having these properties, the twisted cotan-gent bundle T ∗(M,LM ) has the property that some open set is G-equivariantly symplecto-morphic to a covering space of X.
The representation space π(M,LM ) attached to O in this situation is roughly speakinga set of sections of the line bundle LM . We can pull back LM to a line bundle LO on O.In this way, sections of LM are identified with sections of LO that are constant along theleaves of the Lagrangian foliation, and we can realize π(M,LM ) among them.
G/Gf∼= // O
ρ
²²
LOoo
²²G/H
∼= // M LMoo
As hinted at before, when G is a nilpotent group, the family of subgroups of G is veryrich, and it is always possible to find a group H that makes a given coadjoint orbit a twistedcotangent bundle. That is, polarization is enough to geometrically quantize the coadjointorbits of nilpotent groups.
As G becomes more complicated, however, the pool of subgroups diminishes, and polar-ization becomes more difficult to carry out. Ozeki and Wakimoto [Ozeki-Wakimoto] showthat among reductive Lie groups, subgroups satisfying just the isotropy and dimension re-quirements must be parabolic [Ozeki-Wakimoto]. Furthermore, they prove that polarizationcannot be always carried out:
Corollary 2.1.1. Suppose that G is a split simple group over R or C and is not of type A.If O is a coadjoint orbit of minimal non-zero dimension, then O is not locally isomorphicto a twisted contangent bundle for G.
2.2 Lagrangian Coverings
Polarization, as described above, fails to suggest a representation that one could attachto every coadjoint orbit O. To amend this situation, Graham and Vogan use an idea from
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[Guillemin-Sternberg] and [Ginsburg] and replace the Lagrangian foliation above by a familyof Lagrangian submanifolds that are allowed to overlap, called a Lagrangian covering of O.Following a definition, we will describe a construction of Lagrangian coverings in the caseof nilpotent coadjoint orbits. We then describe how [Graham-Vogan] mimics polarizationin this setting to construct a space of representations attached to a nilpotent O.
Definition 2.2.1. A Lagrangian covering of a symplectic manifold O is a pair (Z, M) ofmanifolds and smooth maps (τ, ρ)
Zτ
~~
ρ
²²O M
such that
· the diagram is a double fibration,· each fiber of ρ is a Lagrangian submanifold of O.
Theorem. [Ginsburg] Let G be a complex reductive Lie group and O be a coadjoint orbit.Then there exists an equivariant Lagrangian covering of O with M a partial flag variety forG.
We relate the construction in the case of a nilpotent coadjoint orbit. Fix a Borel sub-group B of G with unipotent radical N . Write
g = n− ⊕ t⊕ n
for the corresponding triangular decomposition. Let us restrict our attention to nilpotentcoadjoint orbits O, and consider the set O ∩ n. This is a locally closed subset of n and canbe expressed as a union of its irreducible components.
Definition 2.2.2. Consider a nilpotent coadjoint orbit O. Denote the set of irreduciblecomponents of the variety O ∩ n by Irr(O ∩ n). Each element of Irr(O ∩ n) is an orbitalvariety for O.
Proposition 2.2.3. Let V be an orbital variety for O. Then
· |Irr(O ∩ n)| is finite,· dimV = 1
2 dimO,
· Each V is a Lagrangian subvariety of O.
We are ready to construct a Lagrangian covering for the orbit O. In fact, we will con-struct a distinct covering for each orbital variety contained in O. Fix an orbital variety Vand let V0 be its smooth part. Let Q be the subgroup of G that stabilizes V, i.e.
Q = QV = q ∈ G | q · V = V.
This is a parabolic subgroup of G since V is B-stable. Furthermore, we can define themanifold M by
M = g · V | g ∈ G ∼= G/Q.
It is a partial flag variety for G.
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Definition 2.2.4. For a subgroup H ⊂ G and an H-space V , we define G×H V to be theset of equivalence classes in G× V with (gh, v) ∼ (g, h · v) for g ∈ G,h ∈ H, and v ∈ V .
The manifold Z in our Lagrangian covering of O associated to the orbital variety V isnow defined as
Z = G×Q V0.
We define ρ : Z −→ M from the projection of G onto G/Q. The action of G on O givesnatural map G × V → O. It descends to an algebraic map τ : Z −→ O. We now have aLagrangian covering:
G×Q V0
τ
yytttttttttρ
²²G/Gf G/Q
Because the diagram is a double fibration, we can identify fibers of ρ with subsets of O. Infact, each fiber is Lagrangian in O.
We would like to have a construction of representations reminiscent of polarization. Tothat effect, suppose that we have a G-equivariant line bundle LM → M . We can again pullthis bundle back along the fibration ρ, this time to obtain a bundle LZ .
G×Q V0
ρ
²²
τ
wwwwwwwwwLZ
oo
O M LMoo
ρ∗OO
Geometric quantization suggests that the representations attached to O should lie in thespace of sections of LM , or in other words, in the space of sections of LZ that are constant onthe fibers of ρ. This is very similar to the situation arising in the polarization construction,as the fibers of ρ can again be identified with Lagrangian submanifolds of O. This time,however, the full set of sections of LM is too large to quantize O. See [Graham-Vogan]. Tothis effect, it is necessary to pick out a subspace.
2.3 A Subspace of Sections of LM
Choosing an adequate subspace of the sections of LM occupies most of [Graham-Vogan].We relate only a general overview, and direct the reader to [Graham-Vogan] itself for therelevant details. The main idea is to prune the full space of sections of LM , leaving oneswhich also come from an admissible orbit datum of O.
To do this, one must first attach a geometric structure to each orbit datum. This isachieved by mimicking the construction of a Hermitian bundle that often arises in descrip-tions of geometric quantization of integral orbit data. The main difficulty then lies in findinga way of embedding the information from this bundle into the space of sections of LM .
Definition 2.3.1. An admissible orbit datum at f ∈ g∗ is a genuine irreducible unitaryrepresentation π of the metaplectic cover Gf satisfying
π(expY ) = χ(f(Y ))
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for a fixed non-trivial character χ of R.
The process of pruning the set of sections of LM follows the following outline:
(i) Attach some geometric structure to each admissible orbit datum. This will be the setof sections of an infinite-dimensional bundle Seven,∞
π over O.(ii) Represent the sections of Seven,∞
π within the space of sections of a finite-dimensionalbundle Vπ.
(iii) Embed the space of sections of Vπ among sections of LM . More precisely, embed themamong the space of sections of LZ that are constant on fibers of ρ.
(iv) The space of the representation we want is the family of sections of LZ , constant onfibers of ρ, that also come from the sections of Seven,∞
π via this embedding.
We begin by attaching geometric structure to an admissible orbit datum. Let us denotethe metaplectic representation of Gf by τf and form the tensor product representationπ ⊗ τf . While τf and π are genuine representations of Gf , π ⊗ τf in fact descends to arepresentation of Gf itself. This allows us to define a Hilbert bundle over the coadjointorbit O by
Sπ = G×Gf(π ⊗ τf ).
Following [Graham-Vogan], we call it the bundle of twisted symplectic spinors on O. Themetaplectic representation τf of Gf decomposes into two irreducible and inequivalent rep-resentations τ odd
f and τ evenf . Also write τ odd,∞
f and τ even,∞f for the corresponding sets of
smooth vectors. This decomposition passes to the bundle Sπ and the geometric structureattached to the admissible orbit datum π is the subbundle of Sπ defined by
Seven,∞π = G×Gf
(π ⊗ τ even,∞f ).
A similar equivariant Hermitian vector bundle appears in many descriptions of geometricquantization of integral orbit data. [Graham-Vogan] argues that the notion of admissibleorbit data is more natural, and the corresponding bundle in this case is Seven,∞
π . The ideaof [Graham-Vogan] is that the space of the representation attached to O should consist ofthe sections of the bundle LM that somehow also come from sections of Seven,∞
π . Beforethis can be made more precise, a few problems must be overcome. First of all, the bundlesSeven,∞
π are infinite dimensional, while LM may not be. Second, there needs to be a way oftransferring sections of Seven,∞
π to sections of LM .
Seven,∞π
((
G×Q V0
ρ
²²
τ
wwwwwwwwwLZ
oo
O G/Q LM
OO
oo
By enlarging the base space, it is possible to construct a finite-dimensional bundlewhose sections contain the sections of Seven,∞
π . The appropriate base space is the bundle ofinfinitesimal Lagrangians on the coadjoint orbit O.
Definition 2.3.2. Suppose that X is a symplectic manifold. The bundle of infinitesimalLagrangians on X is a fiber bundle B(X) over X. The fiber over each point x ∈ X is theset of Lagrangian subspaces of the tangent space at x of X, denoted by B(TxX).
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Definition 2.3.3. Let O be a coadjoint orbit, and consider V a Lagrangian in the tangentspace g/gf . Write L(V) for the line defined in [Graham-Vogan] (7.4(c)) from the metaplecticrepresentation τf . The admissible orbit datum π defines a G-equivariant vector bundle Vπ
on B(O) by letting the fiber at each V be Hπ ⊗ L(V).
Theorem 2.3.4. [Graham-Vogan] [Kostant] There exists a natural inclusion
i : C∞(O,Seven,∞π ) → C∞(B(O),Vπ).
We would like to incorporate the bundle Vπ over B(O) into our Lagrangian coveringdiagram. Define a map σ : Z → B(O) as follows. Fix z ∈ Z. The definition of Lagrangiancovering forces the fiber of ρ over ρ(z) ∈ M to be a Lagrangian submanifold of O thatcontains τ(z). Hence its tangent space Tτ(z)(ρ−1(ρ(z)) is a Lagrangian subspace of Tτ(z)(O)and thus an element of B(O). Let
σ(z) = Tτ(z)(ρ−1(ρ(z)).
In this way, σ becomes a bundle map over O.
Vπ
²²Seven,∞
π
**
B(O)
²²
Zσoo
ρ
²²
τ
||yyyy
yyyy
yLZ
oo
O M LM
OO
oo
To complete our task, note that we can pull back the bundle Vπ along σ to a bundleσ∗(Vπ) over Z. Smooth sections of Vπ pull back to smooth sections of σ∗(Vπ) and wehave an injective map σ∗ · i : C∞(O,Seven,∞
π ) → C∞(Z, σ∗(Vπ)). Provided that there is aG-equivariant vector bundle isomorphism jπ : σ∗(Vπ) → ρ∗(LM ) we can define a smoothrepresentation of G as:
V (V, π) = ρ∗(C∞(M,LM )) ∩ jπ(σ · i(C∞(O,Seven,∞π )))
If LM is given by a representation γ of the parabolic subgroup Q, then V (V, π) lies inthe space of smooth vectors of the degenerate principal series representation induced fromγ. The entire construction may be summarized by the following diagram.
Vπ
²²
σ∗ // σ∗(Vπ)
²²
jπ
$$IIIIIIIII
Seven,∞π
))
B(O)
²²
G×Q V0σoo
ρ
²²
τ
zztttttttttttLZ
oo
O G/Q LM
ρ∗OO
oo
18
Chapter 3
Geometric Left Cells and theUnipotent Variety
The goal of this section is to obtain a useful parameterization of the orbital varieties con-tained in a given nilpotent coadjoint orbit. We begin by describing the motivation for ourapproach.
Let N denote the set of all complex nilpotent matrixes of size n. Jordan canonicalform dictates that N decomposes into finitely many conjugacy classes, called nilpotentorbits, under the action of SL(n,C). Each is indexed by a partition of n. The set ofpartitions of n, according to Frobenius and Young, also classifies the irreducible complexlinear representations of Sn, the Weyl group of SL(n,C).
The theory of Springer explains this phenomenon [Springer]. Consider a unipotentelement u of G = SL(n,C) and let F denote the full flag variety. We write Fu for thevariety of flags in F that are fixed under the action of u and call it the unipotent variety.Let Ou be the nilpotent orbit through the corresponding nilpotent element. For each Ou,Springer constructs a linear action of Sn on the cohomology H∗(Fu). It commutes with theaction of the isotropy group Gu and yields an irreducible representation πu on Htop(Fu)Gu .The correspondence assigning πu to each Ou is a bijection and gives a geometric explanationof the classical result.
Denote the Springer resolution by π : T ∗F −→ N . A representation of Sn equivalentto πu can also be constructed on the space Vπ′u = H∗(π−1Ou) [Borho]. Both πu and π′u areequipped with canonical bases which respectively correspond to the irreducible componentsof Fu and the set of orbital varieties ofOu. Both of these sets share an identical combinatorialdescription. Let λu denote the partition given by the Jordan canonical form of u.
Theorem. [Spaltenstein] When G = SL(n,C), the set of irreducible components of Fu
corresponds to the set of standard Young tableaux with shape λu.
Each orbital variety of a nilpotent orbit corresponds to a subset of the Weyl groupcalled a geometric left cell [Joseph] . When G = SL(n,C), the notion of a geometric left cellcoincides with that of a Kazhdan-Lusztig left cell and the results of [Joseph] can be usedto classify both. The key is the Robinson-Schensted algorithm which establishes a bijectionbetween the permutations w ∈ Sn and pairs of same-shape standard Young tableaux of size
19
n. We write
RS : Sn −→ (A,B) ∈ SY T (n)× SY T (n)| shape A = shape B
For an element w ∈ Sn, we denote the left tableau of its image by RSL(w) and the righttableau by RSR(w). Under this correspondence, two permutations in Sn belong to the samegeometric left cell iff they share the same right Young tableau. Hence the set of all standardYoung tableaux of size n parameterizes the set of geometric left cells. Joseph’s identificationnow describes the orbital varieties contained in a given nilpotent orbit.
Theorem. [Joseph] The set of orbital varieties contained in the nilpotent orbit Ou corre-sponds to the set of standard Young tableaux of shape λu.
A similar description of orbital varieties for the other complex simple classical groups iscomplicated by two phenomena. First of all, the notions of Kazhdan-Lusztig left cells andgeometric left cells no longer coincide. Second, the ordinary Robinson-Schensted algorithmneeds to be adapted to the new Weyl group. The work of Garfinkle provides the initialsteps [Garfinkle1]. Let Wm be the Weyl group of G. The generalized Robinson-Schenstedalgorithm now takes the form
RS : Wm −→ (A,B) ∈ SDT (m)× SDT (m)| shape A = shape B
where SDT (m) consists of domino tableaux of size m. As in type A, two Weyl groupelements belong to the same Kazhdan-Lusztig left cell if they share the same right dominotableau. However, the converse is no longer true. To remedy this, Garfinkle introduces anequivalence relation on domino tableaux by rearranging, or moving through, certain subsetsof dominos called open cycles. Within each such equivalence class, there is a unique dominotableau of special shape [Lusztig]. This time, two Weyl group elements belong to the sameKazhdan-Lusztig left cell iff their right tableaux are equivalent to the same tableau of specialshape. Because in type A all partitions are of special shape, this is a natural generalizationof the original result.
Garfinkle’s equivalence relation, however, is too strong to describe geometric left cells.By restricting the set of open cycles that one is allowed to move through, McGovern de-fines the appropriate equivalence relation [McGovern2]. In classical groups not of type A,not all partitions of m arise as Jordan block decompositions. Within each of McGovern’sequivalence classes, however, there is a unique one which does.
Theorem. [McGovern2] In types B and C, orbital varieties contained in the orbit Ou areparameterized by standard domino tableaux of shape λu. In type D, the result is the sameunless λu has only even parts, when the number of vertical dominos should be congruent to0 or 2mod 4 according as the Roman numeral attached to Ou is I or II.
The natural question is whether a similar classification can also be obtained by studyingthe unipotent variety. In types B, C, and D, its irreducible components, Irr(Fu), are param-eterized by admissible domino tableaux of shape λu and a choice of sign for certain disjointsubsets of dominos called open and closed clusters [Spaltenstein] [van Leeuwen]. The com-ponent group Au of the centralizer Gu acts on Irr(Fu). Within the above parameterization,this action changes signs of the open clusters. As each orbital variety corresponds uniquely
20
to an Au-orbit on Irr(Fu) [Borho-Brylinski], this parameterizes the orbital varieties in Ou byadmissible domino tableaux of shape λu with a choice of sign for each of its closed clusters.
The key to reconciling the two parameterizations of orbital varieties lies in understandingthe relationship between the cycles and clusters contained in the same tableau. In particular,we prove:
Lemma 3.3.1 Each closed cluster C of a domino tableau contains the closed cycle YCthrough its lowest-numbered domino.
This observation allows us to define a map Φ : ΣDTcl(λ) −→ SDT (λ) from admissibledomino tableaux with signed closed clusters to the set of all standard domino tableaux ofthe same shape. The map Φ moves through the initial cycle of each closed cluster of positivesign while preserving the rest of the tableau.Theorem 3.3.1 The map Φ is a bijection between the set of domino tableaux and the setof admissible domino tableaux of the same shape with signed closed clusters.
We conclude this chapter with two results. For an orbital variety V, we describe theτ -invariant and the maximal stabilizing parabolic QV . Then, we define a map from stan-dard domino tableau to standard Young tableau which can be interpreted in terms of thecorresponding orbital varieties.
3.1 Preliminary Definitions and Basic Facts
3.1.1 Unipotent Variety
For ε = ±1, take <, >ε be a non-degenerate bilinear form on Cm such that
< x, y >ε= ε < y, x >ε ∀x, y ∈ Cm.
The form <,>−1 is symplectic and m = 2n must be even. We call this a form of type C.The form <,>1 is symmetric and m can be even or odd. When m = 2n for some n, we sayit is a form of type D and when m = 2n + 1, we say it is of type B. A full isotropic flag inCm is a sequence
f1 ⊂ f2 ⊂ · · · ⊂ fn
of subspaces of Cm where each fi is isotropic with respect to <,>ε, dim(fi) = i. Then fn
is a maximal isotropic subspace. Denote by F the set of all such flags. It has a naturalstructure of a projective algebraic variety which is irreducible in types B and C and hastwo connected components in type D.
Let Gε be the isometry group of <,>ε and take gε to be its Lie algebra. In types Band D, G1 = O(m) and g1 = so(m), while in type C, G−1 = Sp(2n). Let us fix one ofthese types and simply refer to the isometry group as G. Let u be a unipotent element inG. Define shape u to be the partition whose parts are the sizes of the Jordan blocks of uarranged in decreasing order; it is well-defined and determined by the conjugacy class of u.The converse of this statement is also true: two unipotent elements of the same shape areconjugate in G. Hence to classify all unipotent conjugacy classes in G, one simply needs todetermine which partitions appear as shapes of its unipotent elements.
Set Pε(m) = (λ1, . . . , λj) a partition of m| #k|λk = i is even ∀i with (−1)i = ε.Then the classification takes the form:
21
Theorem. [Gerstenhaber] Conjugacy classes of unipotent elements in Gε are in one-to-onecorrespondence with the set of partitions of Pε(m).
We write λu for the image of a unipotent element u under the above correspondence.The group G has an obvious action on the flag variety F . Denote by Fu the fixed-point setof u on F . In general, Fu is reducible, so let us define by Irr(Fu) the set of its irreduciblecomponents. Let Gu be the centralizer of u in G and Go
u the connected component of theidentity. It acts on Fu and consequently on Irr(Fu). Let Au = Gu/Go
u. We will writeAu = Aλu = Aλ interchangeably when no confusion can arise. Go
u acts trivially on Irr(Fu)and hence Au itself acts on Irr(Fu). We will need to understand this action explicitly. Fornow, we describe Au.
Fact. For a partition λ, let Bλ be the set of its distinct parts λi satisfying (−1)λi = −ε.Then Au is a 2-group with |Bλ| components, i.e.
Au =⊕
|Bλ|Z2.
3.1.2 Nilpotent Orbits and Orbital Varieties
The partition classification of 3.1.1 also can also be used to describe the nilpotent Gε-orbitsin gε.
In cases B and C, nilpotent Gε-orbits in gε are precisely the nilpotent orbits as definedby the group Gad. However, in type D, Gε is larger than Gad whose action defines nilpotentorbits. To each very even partition λu, there correspond two nilpotent orbits whose elementshave Jordan form λu. Nevertheless, there is only one such Gε-orbit. Write Ou (with perhapsa Roman numeral) for a nilpotent orbit corresponding to λu and Ou
′ to the correspondingGε-orbit. We will also write Ou for the nilpotent orbit through u, and Oλ for the nilpotentorbit corresponding to the partition λ.
Let b ⊂ g be a Borel subalgebra, h ⊂ b a Cartan subalgebra, and n the nilradical so thatb = h+n. The irreducible components Irr(Oλu ∩n) are called orbital varieties. By [Joseph],they take the form V (w) = B(n ∩ w−1n) ∩ Oλu for some w ∈ W, the Weyl group. The setof Weyl group elements which map to the same orbital variety under this correspondenceis known as a geometric left cell. McGovern’s parameterization of orbital varieties relies onthis description. Our examination, however, will be based on a parameterization of Irr(Fu)and the action of the component group Au upon it. The key result is:
Fact. [Borho-Brylinski] There is a bijective correspondence between Au-orbits on Irr(Fu)and Irr(O′u ∩ n).
3.1.3 Domino Tableaux
Let T be an integer tableau, or a finite left-justified array of rows of squares, each labeledby an integer. To each tableau T one can assign a partition λT which we call the shapeof T . We view T as a set of ordered pairs (k, Sij) denoting that the square in row i andcolumn j of T is labeled by the integer k. D(k, T ), a domino with label k, is a subset of T ofthe form (k, Sij), (k, Si+1,j) or (k, Sij), (k, Si,j+1). Call these, respectively, vertical andhorizontal dominos. For convenience, we will refer to the set (0, S11) as the 0-domino.
22
When not necessary, we will omit the labels of squares and write Sij for (k, Sij). In thatcase, we define label Sij = k.
Definition 3.1.1. Let T be an integer tableau. If shape T is a partition of 2n + 1 of typeB, T is a domino tableau of type B iff it is partitioned by the dominos (0, S11), D(1, T ),D(2, T ), . . . , and D(n, T ), in a way that the labels increase weakly along rows and columnsof T .
If shape T is a partition of 2n and is of type C, (respectively D), T is a dominotableau of type C (respectively D) iff if it is partitioned by the dominos D(1, T ), D(2, T ),D(3, T ), . . . , D(n, T ) such that the labels increase weakly along rows and columns of T .
For a partition λ, let SDT (λ) denote the set of standard domino tableaux of shape λ.Implicit in this notation are the type of the partition λ and the type of domino tableaux.We will also need to consider domino tableaux all of whose subtableaux are of the sametype.
Definition 3.1.2. For T ∈ SDT (λ) let T (i) denote the tableau formed be the dominos ofT with labels less than or equal to i. Let X= B, C, or D, and take T to be a type X. Thetableau T is admissible iff each T (i) is also a domino tableau of type X for all i.
The dominos that appear within admissible tableaux fall into three categories.
Definition 3.1.3. 1. In types B and D (respectively C), a vertical domino is of type(I+) if it lies in an odd- (respectively even-) numbered column.
2. A vertical domino not of type (I+) is of type (I−).
3. A horizontal domino is of type (N) if its left square lies in an even- (respectively odd-)numbered column.
In fact, we can restate the above definition by noting that a domino tableau is admissibleiff all of its dominos are of type (I+), (I−), or (N), and perhaps the 0-domino. Finally, wewould like to assign plus and minus signs to certain dominos.
Definition 3.1.4. A signed domino tableau is an admissible domino tableau with a signlabel for each domino of type (I+). Denote the set of all signed tableaux of shape λ byΣDT (λ). For a T ∈ ΣDT (λ), let |T | denote the underlying domino tableau.
Example 3.1.5. Consider the partition [5,3] of type D and the three tableaux below. T is anadmissible D-tableau; D(1, T ) and D(3, T ) are dominos of type (I+), D(2, T ) is a dominoof type (I−), and D(4, T ) is of type (N).
4321 3
12
42
41 3+ -
(a) T (b) T ′ (c) T ′′
T ′ on the other hand is not admissible, as shape T (1) is not a D-partition. Here, D(1, T ′)and D(2, T ′) are horizontal dominos not of type (N). Finally, T ′′ is a signed domino tableau;its dominos of type (I+) are signed. Also, |T ′′| = T.
23
3.1.4 Cycles
Starting with a domino tableau T , Garfinkle defines a way to form a new domino tableau,not necessarily of the same type, that preserves the labels of certain alternate squares.There is a distinguished way to define this new domino tableau MT (k, T ) by requiring thata particular domino D(k, T ) be not be preserved by the process, but as much as possible ofthe rest of T and the labels of certain squares remain the same. The map is called movingthrough the domino D(k, T ).
We specify four choices of the squares of T whose labels should be fixed under movingthrough. For X = B or C, the square Sij is said to be X-fixed iff i + j is odd. For X=D orD’, Sij is X-fixed iff i + j is even. Squares of T that are not X-fixed are called X-variable.When our choice of X is clear, we will refer simply to fixed and variable squares. Under themoving through map, the labels of fixed squares will be preserved while those of variableones may change.
Recall Garfinkle’s definition of a cycle [Garfinkle1]. We will think of cycles as both,subsets of dominos of T , as well as just the sets of their labels. We will call a cycle whosefixed squares are X-fixed an X-cycle. Because the definitions of B and C-fixed as well as Dand D’-fixed are the same, the B and C-cycles as well as D and D’-cycles for a tableau Tcoincide.
For reference and to establish notation, we recall Garfinkle’s definition of the image ofa single domino under the moving through map.
Definition 3.1.6. Let MT (D(k, T ), T ) be the image of the domino D(k, T ) under movingthrough. If D(k, T ) = (k, Si,j−1), (k, Sij) or (k, Sij), (k, Si+1,j) and Sij is fixed, let r bethe label of Si−1,j+1. Then
1. If r > k, let MT (D(k, T ), T ) = (k, Sij), (k, Si−1,j),
2. If r < k, let MT (D(k, T ), T ) = (k, Sij), (k, Si,j+1)
On the other hand, If D(k, T ) = (k, Sij), (k, Si,j+1) or (k, Si−1,j), (k, Sij) and Sij isfixed, let r be the label of Si+1,j−1. Then
1. If r > k, let MT (D(k, T ), T ) = (k, Si,j−1), (k, Sij),
2. If r < k, let MT (D(k, T ), T ) = (k, Sij), (k, Si+1,j)
While MT (D(k, T ), T ) denotes the image of the domino D(k, T ) under moving through,we let MT (k, T ) be the new tableau obtained by moving through the cycle containingD(k, T ). While this notation does not explicitly indicate the type of the cycle that is movedthrough, this will always be clear from the context.
Example 3.1.7. Consider again the D-tableau T from Example 3.1.5.
1 2 34 1
23
41 2 3
41
23
4
(a) T (b) MT (2, T ) (c) T (d) MT (2, T )
24
The shaded squares in (a) are D-fixed. With this choice, MT (D(1, T ), T ) = S11, S12and the set D(1, T ), D(2, T ) constitutes a D-cycle in T . The set D(3, T ), D(4, T ) isT ’s other D-cycle. In the diagrams, the dark lines outline each cycle. Moving throughD(1, T ), D(2, T ) yields the inadmissible tableau in (b).
The shaded squares in diagram (b) are B-fixed. The B-cycles are the sets D(1, T ),D(2, T ), D(3, T ), and D(4, T ). This time, moving through the cycle containing D(2, T )gives the admissible tableau in (d).
3.2 Irreducible Components of Fu
The irreducible components of the unipotent variety Fu were described by N. Spaltensteinin [Spaltenstein]. We present this parameterization as interpreted by M.A. van Leeuwen[van Leeuwen]. Its advantage lies in a particularly translucent realization of the action ofAu on these components that allows us to parameterize the orbital varieties Irr(O′u ∩ n).
Let us fix a unipotent u and a flag F of the appropriate type. Spaltenstein’s parameter-ization of Irr(Fu) begins by associating to F a signed domino tableau. Let F ∈ Fu be a flagof type X= B, C, or D, and recall the definition of the flags F (i). Let λ(i) be the shape ofthe Jordan form of the unipotent operator induced by u upon F (i). It turns out that for alli, the difference between λ(i) and λ(i+1) is precisely a domino [Spaltenstein]. By assigningthis domino a label i + 1, we obtain a domino tableau T of shape λu from any flag in Fu.In fact, the construction implies that this domino tableau will be admissible.
Admissible tableaux, however, do not fully separate the components of Fu. If two flagsgive rise to different domino tableaux in this way, they lie in different components of Fu.However, the converse is not true. The inverse image Fu,T of a given admissible tableau Tunder this identification is in general not connected. Nevertheless, the irreducible compo-nents of Fu,T are precisely its connected components [van Leeuwen](3.2.3). Accounting forthis disconnectedness yields a parameterization of Irr(Fu). To this effect, we consider the setof signed admissible tableaux ΣDT (λu). Itself, it is too large to parameterize Irr(Fu), butwith an appropriate equivalence relation, it will give us the parameterization. The mannerin which this equivalence is defined separates [van Leeuwen] from [Spaltenstein]. We follow[van Leeuwen] and define the concept of clusters. Our definition differs from [van Leeuwen],but it is equivalent.
3.2.1 Clusters
Clusters partition the set of dominos of T ∈ ΣDT (λ) into subsets. They are definedinductively and depend only on the underlying domino tableau |T |. Hence suppose wealready know the clusters of T (k−1) and would like to know how D(k, T ), the domino withlabel k inside T , fits into the clusters of T (k). Here is a summary:
Definition 3.2.1. In type B and C, let cl(0) be the cluster containing T (1).
1. If D(k, T ) = Sij , Si+1,j and type D = (I−), then D(k, T ) joins the cluster of thedomino containing Si,j−1. If j = 1, then D(k, T ) joins cl(0).
2. If D(k, T ) = Sij , Si+1,j and type D = (I+) then D(k, T ) forms a singleton clusterin T (k), unless i ≥ 2 and Si−1,j+1 is not in T . In the latter case, D(k, T ) joins thecluster of the domino containing Si−1,j .
25
3. Take D(k, T ) = Sij , Si,j+1, so that type D = (N). Let C1 be the cluster of thedomino containing Si,j−1 but if j = 1, let C1 = cl(0). If i ≥ 2 and Si−1,j+2 is notin T , let C2 be the cluster of the domino that containing Si−1,j+1. If C1 = C2 orC2 does not exist, the new cluster is C1 ∪ D(k, T ). If C1 6= C2, the new cluster isC1 ∪ C2 ∪D(k, T ).
4. The clusters of T (k−1) left unaffected by the above simply become clusters of T (k).
Definition 3.2.2. A cluster is open if it contains an (I+) or (N) domino along its rightedge and is not cl(0). A cluster that is neither cl(0) nor open is closed. Denote the set ofopen clusters of T by OC(T ) and the set of closed clusters as CC(T ).
This definition differs from [van Leeuwen] as we do not call cl(0) an open cluster. Theopen clusters of T correspond to the parts of λ contained in Bλ. As the latter set param-eterizes the Z2 factors of Aλ, we will ultimately use open clusters to describe the action ofAλ on the irreducible components of Fu. To be more precise, define a map
bT : Bλ −→ OC(T ) ∪ cl(0).
For r ∈ Bλ, let bT (r) be the cluster that contains a domino ending a row of length r in T .This map is well-defined: any two dominos that end two rows of the same length belong tothe same cluster; furthermore, such a cluster is always open or it is cl(0). The map bT isalso onto OC(T ), but it is not one-to-one as T may have fewer open clusters than |Bλ|. Forfuture reference, we will need this definition.
Definition 3.2.3. For a cluster C, let IC be the domino in C with the smallest label andtake Sij as its left and uppermost square. For X = B or C, we say C is an X-cluster iff i + jis odd. For X = D or D’, C is an X-cluster iff i + j is even.
Example 3.2.4. Consider this admissible C-tableau for the partition [8, 6, 5, 5].Its clusters are the sets1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11, 12. The set 1, 2 is preciselycl(0), the second set is a closed C-cluster, while the third is open and also of type C.
2
1 5
6
9
8
10
1211+
3
4+
+
-7
The clusters are outlined with darker lines. The C-cycles of Sare 1, 2, 3, 5, 8, 10, 9, 4, 6, 7, and 11, 12, showing that thecluster and cycle structures for a given tableau may not coincide.Note, however, that each cluster contains the cycle of the sametype through its smallest-numbered domino. This is true in gen-
eral.
3.2.2 Equivalence Relations
Armed with the notion of open and closed clusters, we can now define two equivalencerelations on ΣDT (λ).
Definition 3.2.5. If T, T ′ ∈ ΣDT (λ), let T ∼op,cl T ′ iff |T | = |T ′| and the products of signsin all corresponding open and closed clusters of T and T ′ agree. Denote the equivalenceclasses by ΣDTop,cl(λ) and write [T ] for the equivalence class of T ∈ ΣDT (λ). We thinkof ΣDTop,cl(λ) as the set of admissible domino tableaux of shape λ with signed open andclosed clusters.
26
Similarly, if T, T ′ ∈ ΣDT (λ), let T ∼cl T ′ iff |T | = |T ′| and the products of signs inall corresponding closed clusters of T and T ′ agree. Denote the set of these equivalenceclasses by ΣDTcl(λ) and write [[T ]] for the equivalence class of T . Elements of ΣDTcl(λ)are represented by admissible domino tableaux of shape λ with signed closed clusters.
The set ΣDTop,cl(λu) will parameterize the irreducible components of Fu. There is aconsiderable amount of freedom in how a flag of Fu,T can be assigned an equivalence class ofsigned admissible domino tableaux. A particular choice is presented in [van Leeuwen](3.4),and we call this map Γu.
To understand the action of Au on the components of the unipotent variety in terms ofthis parameterization, we first describe an action of Au on ΣDTop,cl(λu). Let
ξr : ΣDTop,cl(λ) −→ ΣDTop,cl(λ)
act trivially if bT (r) = cl(0) and by changing the sign of the open cluster bT (r) otherwise.For each r ∈ Bλ, let gr denote the generator of the corresponding Z2 factor of Au. Wedefine the action of gr on ΣDTop,cl(λu) by
gr[T ] = ξr[T ].
This action accurately reflects the action of Au on Irr(Fu). We summarize this result below.
Theorem. The map Γu establishes an Au-equivariant bijection between the componentsIrr(Fu) and ΣDTop,cl(λu), the set of admissible domino tableaux of shape λu with signedopen and closed clusters.
Example 3.2.6. Consider λu = [5,3], a D-partition. The set Irr(F[5,3]) is parameterized byΣDTop,cl([5, 3]), or the following eight tableaux with signed open and closed clusters:
1 2 3 4+ +
1 2 3 4+-
1 2 3 4+ -
1 2 3 4--
1 2 4+ 3
1 2 43-
1 2 3+ 4
1 2-
34
Tableaux (a)-(d) each have an open cluster 3, 4 and a closed cluster 1, 2, while (e)-(f) all have one open cluster 1, 2, 3, 4. A[5,3] = Z2 × Z2 and each of its factors acts in thesame way by changing the sign of the sole open cluster in each of these tableaux.
According to Fact 3.1.2, the set of irreducible components of O′u∩ n, denoted Irr(O′u∩ n),corresponds to the set of Au-orbits of Irr(Fu). The above characterization of Irr(Fu) alongwith the description of the Au action yield the following:
Corollary. The orbital varieties Irr(O′u ∩ n) are parametrized by ΣDTcl(λ), the set ofadmissible domino tableaux of shape λ with signed closed clusters.
27
Example 3.2.7. With λu as in Example 3.2.6, Irr(O[5,3]∩n) is parameterized by ΣDTcl([5, 3]),or the following four domino tableaux with signed closed clusters. They are derived fromthe dominos of Example 3.2.6 by identifying the A[5,3]-orbits.
1 24
3+
1 24
3-
21
34 2
14
3
3.3 The Components Irr(O′u ∩ n)
We aim to reconcile Corollary 3.2.2 with McGovern’s original parameterization of Irr(O′u ∩n). To this effect, we define a bijection between ΣDTcl(λ) and SDT (λ) by applying Garfin-kle’s moving through map to certain distinguished cycles.
3.3.1 A Bijection
Consider an X-cluster C and let IC be the domino in C with the smallest label. Let YC bethe X-cycle through IC . We call it the distinguished cycle of C. We will use it to define amap between admissible domino tableaux with signed closed clusters and the set of dominotableaux of the same shape. Our construction relies on:
Lemma. A closed cluster of an admissible domino tableau T contains its distinguishedcycle, i.e. if C ∈ CC(T ), then YC ⊂ C.
We defer the proof to its own section. Armed with this fact, we can now propose a map
Φ : ΣDTcl(λ) −→ SDT (λ)
by moving through the distinguished cycles of all closed clusters with positive sign. Moreexplicitly, for a tableau T ∈ ΣDTcl(λ), let CC+(T ) denote the set of closed clusters of Tlabeled by a (+) and let σ(T ) = YC | C ∈ CC+(T ) be the set of their distinguished cycles.Then we define
Φ(T ) = MT (|T |, σ(T )).
Theorem. Φ : ΣDTcl(λ) −→ SDT (λ) is a bijection.
Proof. We check that this map is well-defined, that its image lies in ΣDT (λ), and thenconstruct its inverse. For this map to be well-defined, we need to know that the definitionof Φ does not depend on which order we move through the cycles in σ(T ). It is enoughto check that if YC and Y ′C ∈ σ(T ), then Y ′C is also lies in σ(MT (|T |,YC)). While thisstatement is not true for arbitrary cycles, in our setting, this is Lemma 3.4.7.
The image of Φ indeed lies in ΣDT (λ). That Φ(T ) is itself a domino tableau followsfrom the fact that moving through any cycle of T yields a domino tableau. That it is ofthe same shape as T follows as well because Φ moves through only closed cycles. HenceΦ(T ) ∈ SDT (λ).
The definition of a cluster, and in particular 3.2.1(2), forces the initial domino IC ofevery closed cluster to be of type (I+). By the Definition 3.1.6, the image of MT (IC , T )
28
in MT (YC , T ) is inadmissible, i.e. it is a horizontal domino not of type (N). In general,all the inadmissible dominos in Φ(T ) appear within the image of distinguished cycles undermoving through. Furthermore, the lowest- numbered domino within each cycle is the imageof the initial domino of some distinguished cycle. With this observation, we can constructthe inverse of Φ. We define a map
Ψ : Φ(ΣDTcl(λ)) −→ ΣDTcl(λ)
that satisfies Ψ Φ = Identity. Let ι(Φ(T )) be the set of cycles in Φ(T ) that containinadmissible dominos. We define Ψ(Φ(T )) = MT (Φ(T ), ι(Φ(T ))). By the above discussion,ι(Φ(T )) contains precisely the images of cycles in σ(T ). Hence
Ψ(Φ(T ) = MT (Φ(T ), ι(Φ(T ))) = MT (MT (|T |, σ(T ))) = T
as desired. Thus Φ is a bijection onto its image in SDT (λ). As we already know that thesets ΣDTcl(λ) and SDT (λ) both parameterize the same set of orbital varieties, Φ must bea bijection between them.
Corollary 3.3.1. The orbital varieties Irr (O′u∩ n) are parameterized by the set SDT (λu).
We can translate this to:
Theorem 3.3.2. In types B and C, orbital varieties contained in the orbit Ou are param-eterized by standard domino tableaux of shape λu. In type D, the result is the same unlessλu has only even parts, when the number of vertical dominos should be congruent to 0 or2 mod 4 according as the Roman numeral attached to Ou is I or II.
Example 3.3.3. Recall our parameterization of Irr(O′[5,3] ∩ n) in Example 3.2.7 and firstconsider the tableau 3.2.7 (a). The set of its closed clusters labeled with (+) consists of1, 2 and hence Y1,2 = 1, 2. Its image under Φ is then the tableau of 3.1.5 (b). Thetableaux 3.2.7 (b),(c), and (d), have no closed clusters labeled with (+) and their underlyingdomino tableaux are unaffected by Φ. Hence the image under Φ of the set ΣDTcl([5, 3]) isthe following set of domino tableaux:
4321 3
12
4 2 43
12
14
3
or precisely the set SDT ([5, 3]), as described by McGovern’s original result.
3.4 Proof of Lemma 3.3.1
We aim to show that a closed cluster C contains its distinguished cycle YC . The proof hastwo parts. First, we show that YC is contained in a slightly larger set of clusters C, definedas the union of C with all of its nested clusters. Then, we show that YC intersects each ofthese nested clusters trivially.
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3.4.1 Closed and Nested Clusters
First, we define and describe the set C. Let C be a closed cluster of a tableau T and denoteby rowk T = Sk,j | j ≥ 0 the kth row of T . Define colk T similarly. If rowk T ∩ C 6= ∅, letinfk C = infj| Sk,j ∈ rowk T ∩ C and supk C = supj | Sk,j ∈ rowkT ∩ C.
Consider the following tableau of type D. It has two closed clusters given by the sets C1 =1, 2, 3, 4, 5, 8, 9, 10, 11, 12 and C2 = 6, 7. C1 is a D-cluster while C2 is a B-cluster. YC11
2
3
4
5
6 7 8
9 1012
11is then a D-cycle and equals 1, 3, 5, 11, 12, 10, 9, 2. T has two other D-cycles, 4, 6 and 7, 8. Both intersect C1, but are not contained withinit. The B-cluster YC2 equals 6, 7 and is contained in C2. This exampleshows that an X-cluster may not contain all the X-cycles through itsdominos. However, it always contains its initial cycle. Also notice that
C1 completely surrounds C2. We call such clusters nested.Nested clusters complicate the description of closed clusters. To simplify our initial results,we would like to consider the set formed by a cluster together with all of its nested clusters.To be more precise:
Definition 3.4.1. Let C′ be a cluster of T . C′ is nested in C if all of the following aresatisfied:
• infk|rowkT ∩ C′ 6= ∅ > infk|rowkT ∩ C 6= ∅
• supk|rowkT ∩ C′ 6= ∅ < supk|rowkT ∩ C 6= ∅
• infk|colkT ∩ C′ 6= ∅ > infk|colkT ∩ C 6= ∅
• supk|colkT ∩ C′ 6= ∅ < supk|colkT ∩ C 6= ∅
We define C = C ∪ C′ ∈ CC(T ) | C′ nested in C and, finally, let periphery(C) =D(k, T ) ∈ C | D(k, T ) is adjacent to a square not in C. Note that periphery(C) ⊂ C.
Example 3.4.2. In the above tableau, C2 is nested in C1. C1 ∪ C2 = C = T, periphery(C) =YC1 ⊂ C1.
The next two propositions describe properties of dominos that occur along the leftand right edges of C. They are essential to our goal of relating closed clusters to theirdistinguished cycles. Recall the our definition of the cycle YC endows C as well as C witha choice of fixed and variable squares by defining the left and uppermost square of IC asfixed.
Proposition 3.4.3. Take C a closed cluster of a domino tableau T and consider k suchthat rowkT ∩ C 6= ∅. Then:
(i) type D(label(Tk,infk C), T ) = type D(label(Tk,infk C), T ) = (I+)
(ii) type D(label(Tk,supk C), T ) = typeD(label(Tk,supk C), T ) = (I−).
Proof. Part (i) is true for all non-0 clusters by 3.2.1(2) while part (ii) is the defining propertyof closed clusters.
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Proposition 3.4.4. For a closed cycle C, suppose that IC = Sij , Si+1,j and that Sij fixed.If D = Spq, Sp+1,q lies on periphery(C), then
1. Spq is fixed if type D = (I+).
2. Sp+1,q is fixed if type D = (I−)
Proof. (i) Assume the contrary, i.e. that there is a D′ ∈ periphery(C) of type (I+) whoseuppermost square is not fixed. By examining such a domino that lies closest to IC , wefind that periphery(C) must then contain two type (I+) dominos E = Skl, Sk+1,land E′ = Sk+1,l, Sk+2,m with the squares Skl and Sk+2,m fixed and |m− l| minimal.
Assume m < l. The opposite case can be proved by a similar argument. Because E′
is of type (I+), there is an integer t such that m < t < l, Sk+1,t ∈ periphery(C), andt is maximal with these properties. Let F be the domino containing Sk+1,t. F has tobe Sk+1,t, Sk+2,t and of type (I−). If its type was (I−) or (N), 3.2.1(2) would forceSk+1,t+1 to be in periphery(C) as well. If F on the other hand was Sk+1,t, Sk,t, thiswould contradict the minimality of |m− l|. We now consider two cases.
(a) Assume t = l − 1. Because E and F belong to periphery(C) and hence to C,C must contain a domino of type (N) of the form Su,l−1, Su,l with u > k + 2and u minimal with this property. The set of squares Sp,l−1|k + 2 < p <u ∪ Spl|k + 1 < p < u must be tiled by dominos, which is impossible, as itscardinality is odd.
(b) Assume t < l−1. We will contradict the maximality of t. Because E and F bothbelong to C, C must contain a sequence Hα of dominos of type (N) satisfying
Hα = Sk+1+f(α),t+2α, Sk+1+f(α),t+2α+1
where 0 ≤ α ≤ l−t+12 . We choose each Hα such that for all α, f(α) is minimal
with this property. Because the sets Sk+p,l|k + 1 < p < k + 1 + f( l−t+12 ) and
Sk+p,t|k + 2 < p < k + 1 + f(0) have to be tiled by dominos of type (I+) and(I−) respectively, f(0) has to be even and f( l−t+1
2 ) has to be odd. Hence thereis a β such that f(β) is even and f(β + 1) is odd.Assume f(β) < f(β + 1), but the argument in the other case is symmetric. LetG be the domino containing the square Sk+1+f(β),t+2β+2. G must belong to C, asHβ and G is either of type (I−) or (N). The type of G cannot be (N), however,as this would contradict the condition on f . Hence G must be of type (I−). IfG = Sk+1+f(β),t+2β+2, Sk+f(β),t+2β+2. Then by successive applications of 3.2.1(1) and (2), the set of dominos
Sk+f(β)−γε,t+2β+ε, Sk+1+f(β)−γ−ε,t+2β+ε| ε = 1 or 2 and 0 ≤ γ ≤ f(β)− 2
is contained in C as well. But this means that t + 2β + ε for ε = 1 or 2 satisfiesthe defining property of t, contradicting its maximality. This shows (i).
(ii) We would like to show that for every (I−) domino in peripheryC, the bottom square isfixed. It is enough to show that this is true for one such domino, as an argument similar
31
to that in (i) can be repeated for the others. Let l = infk|rowkT ∩ C = ∅. Then by3.4.3 and the definition of fixed, we know that Sl,infl C is fixed. As Sl,supl C , Sl+1,supl Cis a domino of type (I−) in periphery(C), we have found the desired domino.
3.4.2 Three More Lemmas
Lemma. periphery (C) ⊂ YC ⊂ C.
Proof. We prove the second inclusion. The first is apparent from the argument. We showthat for each D in the periphery(C), as sets of squares, MT (D, T ) ⊂ C. We differentiatecases accounting for different domino positions along periphery(C).
(i) Take D = (k, Sij), (k, Si+1,j) and suppose typeD = (I+). Because D lies onperiphery(C), Proposition 3.4.4 implies that Sij is fixed. Due to Definitions 3.2.1(2)and 3.2.2, Si,j+1 ∈ C.(a) Suppose Si−1,j+1 in not in C. Then r = label(Si−1,j+1) < k. Otherwise Si−1,j
and Sij would both belong to the same cluster by Definition 3.2.1(2); since Si−1,j
and Si−1,j+1 are in the same cluster by Definition 3.2.1(1) or (3), this contradictsour assumption. Now Definition 3.1.6(2) says
MT (D, T ) = (k, Sij), (k, Si,j+1),
and since Sij and Si,j+1 both belong to C, so must MT (D, T ).
(b) Suppose now that Si−1,j+1 ∈ C. Then Si−1,j ∈ C as well since by Definition3.2.1(1) or (3), they both belong to the same cluster. Now Definition 3.1.6implies MT (D, T ) ⊂ Sij , Si−1,j , Si,j+1. As all of these squares lie in C, wemust also have MT (D, T ) ⊂ C.
(ii) Suppose D = (k, Sij), (k, Si,j+1) and that Si,j+1 is fixed. By Definitions 3.2.1(2)and 3.2.2 Si,j+2 ∈ C.(a) Suppose Si−1,j+1 is not in C. Then Si−1,j+2 is not in C, as by Definition
3.2.1(1) or (3) they both belong to the same cluster. By the definition of acluster, r = label(Si−1,j+2) < k and by Definition 3.1.6(2) forces MT (D, T ) =Si,j+1, Si,j+2. Since the squares Si,j+1 as well as Si,j+2 are both contained inC, so is MT (D, T ).
(b) Suppose Si−1,j+1 ∈ C. Then because MT (D, T ) ⊂ Si,j+1, Si,j+2, Si−1,j+2,MT (D, T ) ⊂ C.
(iii) Take D = (k, Sij), (k, Si,j+1) and suppose Sij is fixed. Then Si,j−1 ∈ C by Definition3.2.1(3).
(a) Suppose Si+1,j−1 is not in C. Then r = label(Si+1,j−1) > k by Definition 3.2.1(2)or (3). Now Definition 3.1.6(1) implies that MT (D,T ) = Sij , Si,j−1 ⊂ C.
(b) If Si+1,j−1 ∈ C, then Si+1,j ∈ C as well, since by Definition 3.2.1(1) or (3)they belong to the same cluster. But by Definition 3.1.6(1) or(2), MT (D,T ) ⊂Sij , Si+1,j , Si,j−1 ⊂ C.
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(iv) Let D = (k, Sij), (k, Si+1,j) and take type D = (I−). The square Si+1,j is then fixedand Si+1,j−1 ∈ C.(a) Assume that Si+1,j−1 ∈ C. Then Si+2,j ∈ C as well. Since MT (D, T ) equals
Si+1,j , Si+1,j−1 or Si+1,j , Si+2,j, MT (D, T ) ∈ C as both possibilities are con-tained in C.
(b) Assume Si+1,j−1 is not in C. We have r = label(Si+1,j−1) > k, for otherwiseD(r, T ) and hence Si+1,j−1 ∈ C. But then MT (D,T ) = Si+1,j , Si+1,j−1, so itis contained in C.
Because of Proposition 3.4.4, these are all the cases we need to consider. For instance,dominos of type D = (I+) whose bottom squares are fixed cannot occur along periphery(C).
What remains is to see that YC is contained within the closed cluster C itself. It isenough to show that its intersection with any closed cluster nested in C is trivial. Our proofrelies on the notion of X-boxing. For the following definition, recall Garfinkle’s notion ofφX -box [Garfinkle1](1.5.2)
Definition 3.4.5. For X = B, C, D, or D’, we say a domino D(k, T ) is X-boxed iff it iscontained in some φX -box.
The importance of this concept lies in its behavior with respect to cycles and movingthrough. The following proposition is a restatement of [Garfinkle1](1.5.9 and 1.5.22).
Proposition 3.4.6. Suppose D(k, T ) and D(k′, T ) both belong to the same X-cycle.
(i) D(k, T ) is X-boxed iff MT (D(k, T ), T ) is not X-boxed.
(ii) D(k, T ) and D(k′, T ) are both simultaneously X-boxed or not X-boxed.
Armed with this notion, we can now address:
Lemma. If C′ ⊂ C is a closed cluster nested in C, then YC ∩ C′ = ∅.Proof. It is enough to show that periphery C′∩YC = ∅, as this forces C′∩YC = ∅. We dividethe problem into a few cases.
(i) Suppose type YC , type YC′ = C, D′. We investigate periphery C′ ∩ YC . It cannotcontain dominos of types (I+) and (I−); because the boxing property is constanton cycles according to Proposition 3.4.6(ii), such dominos would have to be simul-taneously C and D-boxed, which is impossible. If D(k, T ) ∈ periphery C′ ∩ YC′ isof type (N), D(k, T ) and MT (D(k, T ), T ) are both C and D’-boxed. This contra-dicts Proposition 3.4.6(i), forcing periphery C′ ∩ YC = ∅. The proof is identical whentype YC , type YC′ = B, D.
(ii) Suppose type YC , type YC′ = C,D. The proof is similar to the first case, exceptthis time, dominos of type (N) cannot be simultaneously C and D-boxed. Again, theproof is identical when type YC , type YC′ = B, D′.
(iii) Suppose type YC , type YC′ ⊂ B, C or D,D′. Then by by the definition of cycles,YC ∩ YC′ = ∅. We know periphery(C′) ⊂ YC′ ⊂ C′ by Lemma 3.4.2, implying againthat periphery(C′) ∩ YC = ∅.
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Lemma 3.4.7. Consider closed clusters C and C′ and their initial cycles YC and YC′ . ThenYC is again a cycle in MT (|T |,YC′).Proof. If C and C′ are clusters of the same type, then so are their initial cycles and thelemma is [Garfinkle 1.5.29]. Otherwise, without loss of generality, take C to be a C-clusterand C′ to be a D-cluster. As the other cases are similar, we can also assume that YC isC-boxed while YC′ is D-boxed.
Suppose that the dominos D(r) ∈ YC and D(s) ∈ YC′ lie in relative positions describedby the following diagram:
s
r
with the box labeled by r fixed. The same squares in MT (|T |,YC′) have the labels
s′
r
To prove the lemma, we need to show that s < r implies s′ < r and s > r implies s′ > r.There are two possibilities for the domino D(s). It is either horizontal or vertical and mustoccupy the following squares:
Case (i) s s
ror Case (ii)
s
s
r
• Case (i). In this case, s < r always. Garfinkle’s rules for moving through imply thatMT (|T |, D(r)) ∩ C′ 6= ∅. This is a contradiction since we know by hypothesis thatYC 6= YC′ . Hence this case does not occur.
• Case (ii). First suppose s > r. Then the our squares within MT (|T |,YC′) must looklike
s
s′
r
for some s′ 6= s. Since the tableau MT (YC′ , T ) is standard, this requires that s′ > simplying s′ > r which is what we desired. Now suppose s < r and suppose the squaresin our diagram look like
s
t s
r u
As in case (i), we find that D(t) /∈ C′. Since D(t) ∈ C, type D(s) = I+ implies typeD(t) = I−, type D(r) = I−, and type D(u) = I+. Otherwise, the rules definingclusters would force s to lie in the cluster C. Now D(u) lies in the initial cycle ofa closed cluster of same type as C′. Since it lies on the periphery and its type is
34
I+, then its top square must be fixed. In particular, D(u) /∈ C. But s < r impliesMT (D(r))∩D(u) 6= ∅. This is a contradiction, implying that this case does not arise.
To finish the proof, we must examine the possibility that D(s) and D(r) lie in the relativepositions described by
r
s .
This case is completely analogous and we omit the proof.
3.5 The τ-Invariant for Orbital Varieties
A natural question is whether the two methods of parameterizing orbital varieites by stan-dard tableaux indeed yields the same parameterization. A partial answer is that the orbitalvarieties attached to the same standard tableau share the same generalized τ -invariant.This is a useful in our setting, as the construction of the Graham-Vogan space associatedto an orbital variety V requires us to be able to explicity identify its τ -invariant. Aftera few definitions, we describe how to calculate the τ -invariant for an orbital variety VT
corresponding to a standard tableau T . We then show that this result is independent of themethod of the parameterization.
Let ∆ be the set of roots in g, ∆+ the set of positive roots and Π the set of simpleroots. Write g =
⊕α∈∆+ g−α ⊕ t ⊕ ⊕
α∈∆+ gα for the triangular decomposition of g andlet b = t⊕⊕
α∈∆+ gα. Write W for the Weyl group, and let Pα be the standard parabolicsubgroup with Lie algebra pα = b ⊕ g−α. Following [Joseph], for an element w ∈ W , anorbital variety V, and a standard parabolic subgroup we define
τ(w) = Π ∩ w(−∆+), (3.1)τ(P ) = α ∈ Π |Pα ⊂ P, (3.2)τ(V) = α ∈ Π |Pα(V) = V. (3.3)
We would like to be able to read off τ(V) from the standard tableau parameterizing V as themaximal parabolic subgroup Q stabilizing V is precisely the standard parabolic subgroupsatisfying τ(Q) = τ(V). A result of Joseph suggests one approach to this problem.
Theorem. [Joseph] Let Vw be the orbital variety of type A that corresponds to the Weylgroup element w ∈ Sn. Then τ(w) = τ(Vw).
The Robinson-Schensted algorithm now suggests that one should be able to describe theτ -invariant of Vw in terms of the left and right tableau RSL(w) and RSR(w). For a standardYoung tableau T , write rT (i) for the number of the row that contains the square with labeli.
Theorem. [Joseph] Consider an element w ∈ Sn and let T = RSL(w). The simple rootαi ∈ Π lies in τ(w) iff the square labelled i lies higher in T than the square with label i + 1,or more precisely, iff
rT (i + 1) > rT (i).
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This gives a quick way of describing the τ -invariant in type A. Garfinkle’s generalizationof the Robinson-Schensted algorithm to hyperoctahedral groups provides a means of usingthis approach in the setting of the other classical groups. For a standard domino tableauT , we say that a domino D lies higher in T than another domino D′ iff there is an integerj ≥ 1 such that D lies entirely within the union of rows 1, . . . , j of T while D′ lies entirelywithin the union of the rows j + 1, . . . , l of T .
Theorem. [Garfinkle2] Let w be an element of the Weyl group of a classical Lie group notof type A. Write T = RSL(w). The simple root αi ∈ Π lies in τ(w) iff one of the followingis satisfied:
(i) i = 1 and the domino D(1, T ) is vertical,(ii) i > 1 and D(i− 1, T ) lies higher that D(i) in T .
The above theorems find the τ invariant using the parameterization obtained from theWeyl group. To find the τ -invariant using the other approach, we rely on Spaltenstein’soriginal work on the unipotent variety Fu. For an irreducible component C ∈ Irr Fu,τ(C) = α |PαC = C. Then
Theorem. [Spaltenstein] Let X = B, C, or D. Consider C ∈ Irr Fu,|T |, that is, anirreducible component whose classifying tableau T in SDTop,cl has underlying domino tableau|T |. Then αi ∈ τ(C) iff one of the following is satisfied:
(i) i = 1, D(1, T ) is vertical, and X 6= D,
(ii) i > 1 and D(i− 1, |T |) lies higher that D(i, |T |) in |T |,(iii) i > 1 and D(i− 1, T ), D(i, T ) ∈ CC+(T ),(iv) If X = D, then α1 ∈ τ(C) iff 1, 2 ∈ CC−(T ) and α2 ∈ τ(C) iff 1, 2 ∈ CC+(T ).
This provides us with a means of finding the τ -invariant of an orbital variety independent ofthe above results on the Weyl group. Both approaches rely on a description of the relativepositions of squares of dominoes within the standard tableau describing the orbital variety.It is not a priori apparent that the two parameterizations of orbital varieties by standardtableaux are the same, or have the same τ -invariant for that matter. However, decipheringSpaltenstein’s description makes it possible to decide this question.
Theorem 3.5.1. Consider an orbital variety VT that corresponds to the standard dominotableau T under the correspondence of this chapter. The simple root αi lies in τ(VT ) iff oneof the following conditions is satisfied:
(i) i = 1 and the domino D(1, T ) is vertical,(ii) i > 1 and D(i− 1, T ) lies higher that D(i) in T .
In particular, this means that τ(w) = τ(Vw) = τ(VT ) = τ(QVT).
Proof. Let π be the projection from SDTop,cl onto SDTcl by identifying the A-group action.Also define a map Φ : SDTop,cl → SDT as the composition Φ π. We prove that ifT ∈ SDTop,cl parameterizes KT ∈ Irr Fu, then τ(KT ) = τ(VΦ(T )).
That α1 ∈ τ(KT ) iff α1 ∈ τ(VΦ(T )) is clear in types B and C since D(1, T ) never lies
within a closed cluster and hence remains unchanged under Φ. For i > 1, suppose that
36
D(i, T ) or D(i − 1, T ) lies in some C ∈ CC+(T ). If the number of dominos in C is greaterthan 2, then the fact that αi ∈ τ(KT ) iff αi ∈ τ(VΦ(T )) follows from [Garfinkle III.1.4]. Nowsuppose that C = D(i), D(i− 1). Then αi ∈ τ(KT ). But D(i− 1) is higher than D(i) inMT (C, T ) , implying by the definition of Φ that αi ∈ τ(VΦ(T )). The remaining possibility isthat only one of D(i) and D(i−1) lies in C. The fact that αi ∈ τ(KT ) iff αi ∈ (VΦ(T )) againfollows by inspection. Finally, when X = D, the conditions for αi, i ≤ 2 to lie in τ(KT )desribed by Spaltenstein translate exacly to our conditions for τ(VΦ(T )).
These results; however, do not resolve the question of whether the two parameterizationsof orbital varieties by standard tableaux are the same. Neverthelss, it is possible to provea somewhat stronger statement than what we have. It turns out that the two parameter-izations share the same generalized τ -invariant. This is readily verified by inspecting theactions of the Tαβ operators in both settings. For the purposes of this work, however, onlythe results on the τ -invariant itself are necessary.
3.6 Projection of Orbital Varieties
A natural question to ask is how the orbital varieties in classical types not of type A relateto those of type A, and furthermore, can this relationship be easily interpreted through thecorresponding standard tableaux? A result in this direction would be useful in subsequentwork on Graham-Vogan spaces, as it may be used to break down the calculations requiredto understand these spaces.
Let g be a classical complex Lie algebra of type Xn = Bn, Cn, or Dn and let n be theunipotent part of b. There is a natural projection map πA from n to nA, the correspondingunipotent part in type An. Let O be a nilpotent orbit of type Xn. It turns out that theimage of an orbital variety for O under πA is always an orbital variety for some nilpotentorbit P of type A. In fact, if P arises in this way, then all of its orbital varieties lie inthe image of πA for O. We will use this idea to motivate our approach to describing theGraham-Vogan spaces for classical groups not of type A.
We interpret this approach in terms of the standard domino tableaux used to parame-terize the relevant orbital varieties. In an effort to find certain Littlewood-Richardson coeffi-cients, Carre and Leclerc define a bijection between the set of semistandard domino tableauxof a given shape and a set of pairs of Yamanouchi and semistandard Young tableaux. Be-cause this map is very useful in answering the above questions, we state this more precisely.
Definition 3.6.1. A Yamanouchi domino tableau is a semistandard domino tableau whosecolumn reading is a Yamanouchi word. We denote the set of Yamanouchi domino tableauxof shape λ and evaluation µ as Y am2(λ, µ). A Yamanouchi word is a word w = w1w1 . . . wl
such that each right factor wi . . . wl contains at least as many letters j than j + 1. Finally,a column reading of a domino tableau T is the word obtained by reading the successivecolumn of T from bottom to top and left to right, counting each horizontal domino onlythe first time that it arises.
Theorem 3.6.2 (Carre-Leclerc). There is a bijection
DT (λ, µ) −−−−→(π1,π2)
qν∈LY am2(λ, ν)× Y T (ν, µ),
37
where the L contains all partitions ν such that the Y am2(λ, ν) is not empty. Furthermore,its restriction to standard domino tableaux induces a bijection
SDT (λ) −−−−→(π1,π2)
qνY am2(λ, ν)× SY T (ν).
The bijection is an algorithm that takes a tableau T and modifies it succesively until itscolumn reading becomes a Yamanouchi word. Simultaneously, it builds a standard Youngtableau to record the sequence of moves.
Definition 3.6.3. Define a map
πA : SDT (n) −→ SY T (n)
by πA(T ) = π2(T ) where π2 is the second component of the Carre-Leclerc map. We alsodenote by πA the map induced on orbital varieties obtained by identifying T with VT .
38
Chapter 4
Restriction to Spherical OrbitalVarieties
Armed with a description of the orbital varieties contained in a given nilpotent orbit aswell as the corresponding τ -invariants, we can attempt to describe the Graham-Voganrepresentations attached to a nilpotent orbit in the setting of classical groups.
The Graham-Vogan construction takes on a particularly nice form for orbital varietiesof S-type, to de defined in the following sections. We would like to restrict our attention tonilpotent orbits all of whose orbital varieties are S-type, and are therefore led to considerto the set of spherical nilpotent orbits.
After a brief description of S-type orbital varieties and how spherical orbits fit withinthe framework of all nilpotent coadjoint orbits, we introduce an inductive procedure thatwe will use in the next chapter while calculating the infinitesimal characters correspondingto V (V, π). It is suggested by a construction of a minimal representative of each orbitalvariety that we detail in this chapter.
We begin by illustrating our method with a “Model Example,” which is sufficiently naıveto quickly describe our approach.
4.1 Model Example
We will calculate the infinitesimal character associated to V (V, π) constructed from a par-ticular orbital variety in type C. Although the example we choose is a bit naıve from severalperspectives, we use it to motivate our approach to the more general calculation. We willaddress its shortcomings within the following chapter.
Suppose G = Sp(2 ·4) and realize the Lie algebra g as a set of 8×8 matrices of the form
sp(8) =
m(A,B,C) =(
A BC −At
)|A, B,C ∈ M(4,C) and B, C ∈ SY M(4,C)
.
Let O be the nilpotent coadjoint orbit in g∗ corresponding to the partition [23, 12]. It hasdimension 18. There are three orbital varieties contained in O, corresponding to the dominotableaux
39
1
2 3
4
1 2
3
4
1
2 4
3
123
4
Let V be the orbital variety corresponding to the first domino tableau. Then dimV =12 dimO = 9. As a representative, we take f = m(A,B, 0) with
A =
0 0 0 00 0 1 00 0 0 00 0 0 0
and B =
0 0 0 00 0 0 00 0 0 00 0 0 1
.
To calculate the infinitesimal character of the Graham-Vogan space for V, we need todescribe the parameters
(i) an admissible orbit datum (π,Hπ),
(ii) Vo, the smooth part of V,
(iii) the stabilizing parabolic QV ⊂ G,
(iv) a smooth representation (γ, Wγ) of QV ,
(v) and a G-equivariant isomorphism of vector bundles jπ.
where notation is as in Chapter 2.Write Gf for the isotropy subgroup of f and gf for its Lie algebra. As G is complex, the
metaplectic cover Gf is isomorphic to Gf × Z/2Z. We choose one admissible orbit datum;it is trivial on G
f and acts by the non-trivial character on Z/2Z.The orbital variety V is smooth so that in the notation of the first chapter, Vo = V.
From Theorem 3.5.1 , we find that the stabilizer of V is the standard parabolic subgroupQV with Levi factor isomorphic to GL(2)×GL(2). One can quickly check that, in this case,both QV and the standard Borel subgroup B act with dense orbit on V.
This observation simplifies calculations, as it allows us to replace the Lagrangian coveringG×QV V by G/Qf , where Qf = QV ∩Gf and V contains QV/Qf as a dense subset. We notethat B/Bf is also dense in V. The equivariant line bundle τ∗Vπ is induced by a characterα of Bf . It is given by the square root of the absolute value of the real determinant of Bf
acting on the tangent space b/bf of V at f . This is
α
(A ∗0 At−1
)= | t31 t63 |−1 , where A =
t1 ∗ ∗ ∗0 t3 ∗ ∗0 0 t3 ∗0 0 0 1
.
Because we are looking for a map jπ, we would like to find a character γ of B whose
40
restriction to Bf is α. Such a character is given by
γ
(A ∗0 At−1
)= | t1 t2 t3 t4 |−3 , where A =
t1 ∗ ∗ ∗0 t2 ∗ ∗0 0 t3 ∗0 0 0 t4
.
The character γ extends uniquely from B to QV . Let the half-density bundle on G/QV begiven by the character ρQV and define another character γ′ on QV to equal γ ⊗ ρ−1
QV . Then
V (O,V, π, γ, jγ,π) ⊂ IndGQ(γ′).
Hence the infinitesimal character that we associated to the representation space V (V, π)equals −(3
2 , 32 , 3
2 , 32) + ρ = (5
2 , 32 , 1
2 , 12), where ρ = (4, 3, 2, 1) and equality is up to Weyl group
action. This is precisely the unique infinitesimal character attached to the orbit O[23,12] by[McGovern]. Similar calculations for the other orbital varieties in this orbit yield the sameinfinitesimal character.
A significant simplification in this example came from the fact that the parabolic sub-group QV acted with dense orbit on V. It made it easy to find the isomorphism jπ. Unfor-tunately, this is not always the case.
Example 4.1.1. [Melnikov] Let G = SL9 and let
T=1 2 3 6 94 5 87
Then VT , the orbital variety in O[5,3,1] corresponding to T has dimension 31. However,dimQV · f ≤ 30 for all f ∈ V.
This counterexample can be extended to produce others in larger groups of type A.Counterexamples in other classical types arise among orbital varieties VT where VπA(T )
produces a counterexample in type A. The reasons for this will become a bit more apparentin the next few sections.
Although we cannot always assume that QV acts with dense orbit on V, fortunately,there are many instances where we can apply this simplifying assumption.
· There are important classes of nilpotent orbits all of whose orbital varieties do admita dense orbit of their stabilizing parabolic.
· Furthermore, even among nilpotent orbits containing orbital varieties without thisproperty, there are always some orbital varieties that admit the above simplification.
We start our investigation of the V (V, π) spaces with these. There are other complicationsthat do not appear in this model example, but in order to spare the reader, we will addressthem when the need arises.
41
4.2 Spherical Orbital Varieties and Orbits of S-type
We would like to use the methods of our model example to calculate the infinitesimal char-acter associated to V (V, π) for as many nilpotent orbits as feasible. The main assumptionrequired is that the stabilizer of an orbital variety has a dense orbit in that variety. Suchorbital varieties are called of S-type, as are the nilpotent orbits all of whose orbital varietiessatisfy this condition. Among classical groups, there is a class of small nilpotent orbits thatare of S-type. We first describe this set and then place it among other important nilpotentcoadjoint orbits.
4.2.1 Spherical Orbits
Let G be a complex simple Lie group and B a Borel subgroup.
Definition 4.2.1. A nilpotent coadjoint orbit O ⊂ g∗ is spherical iff it contains an openB-orbit.
The work of Panyushev provides a concise description of spherical nilpotent orbits con-tained in classical groups.
Theorem 4.2.2. [Panyushev] Spherical nilpotent orbits in complex classical Lie groups areprecisely those that correspond to the following partitions:
(i) [2b, 1c] in type A,
(ii) [3a, 22b, 1c] with a ≤ 1 in type B,
(iii) [2b, 12c] in type C, and
(iv) [3a, 22b, 1c] with a ≤ 1 in type D.
These orbits arise in [McGovern3] for another reason. In the setting of complex simplyconnected semisimple Lie groups, there is an orbit whose G-module structure of the coor-dinate ring of regular functions on a nilpotent orbit O, denoted R(O), is more transparentthan in general; all multiplicities in R(O) are either 0 or 1. The last property characterizesspherical orbits. The largest such orbit is called the model orbit.
Theorem 4.2.3. [McGovern3] In each of the classical types, the model orbit is the largestspherical nilpotent orbit and corresponds to the following partition. Let ε = 0 or 1.
(i) [2n, 1ε] in type A2n+ε−1,
(ii) [3, 24m−2ε, 12ε] in type B2(2m−ε)+1,
(iii) [2n] in type C,
(iv) [3, 22m−2, 11+2ε] in type D2(2m+ε).
Spherical orbits may be characterized alternately as those nilpotent coadjoint orbitscontained in the closure of the model orbit. Yet another characterization of spherical orbitsis that they are precisely the orbits that contain a representative which is a sum a rootvectors corresponding to orthogonal simple roots [Panyushev2] [McGovern3].
42
4.2.2 Smith Orbits
Let V be an orbital variety and let QV be its stabilizer in G. To proceed as in our modelexample, we would like the action of QV on V to admit a dense orbit. As an example aboveshowed, this is unfortunately not always true, but instead, leads us to a definition.
Definition 4.2.4. [Melnikov] An orbital variety V ⊂ O is of S-type iff it admits a denseQV orbit. A nilpotent coadjoint orbit is a S-type iff all its orbital varieties are of S-type.
Fact 4.2.5. If O is a spherical nilpotent orbit of a classical Lie group, then O is of S-type.
Proof. This is a consequence of the dimension argument in Corollary 4.4.2.
From now on, we will restrict our attention to the setting of spherical nilpotent orbits.For all spherical nilpotent orbits, we can use the methods of our model example to calculatethe infinitesimal character associated to V (V, π). For completeness, we give a descriptionof the S-type orbits in groups of type A.
Proposition 4.2.6. [Melnikov] Suppose that n ≥ 13, the partition λ has λ2 > 2, and
(5, 3, 1, . . . ) ≤ λ ≤ (n− 4, 4)
in the usual partial order on partitions. Then the orbit Oλ in type An−1 is not of S-type.
In light of this, one should also expect orbits in other classical types not to be of S-typewhenever a type A orbit in πA(λ) is not S-type. Nevertheless, in addition to sphericalnilpotent orbits, Melnikov finds other classes of nilpotent orbits in type A that are S-type.Note, however, that her results are incomplete, as they fail to resolve the status of a numberof nilpotent orbits in type A.
Proposition 4.2.7. [Melnikov] A nilpotent orbit Oλ in type A is of S-type whenever λsatisfies one of the following:
(i) λ > (n− 4, 4),(ii) λ = (λ1, λ2, 1, . . . , 1) with λ2 ≤ 2,
(iii) λ = (2, . . . ) where λi ≤ 2 for all i.
The question arises as to why we do not address this larger class of S-type orbits insteadof just the spherical orbits. We will address this issue later, but the short answer is thatit may be necessary to again modify the Graham-Vogan construction for such results to beuseful. We finish this section by listing how spherical orbits fit among two other importantclasses of nilpotent orbits.
4.2.3 Rigid and Special Orbits
There is an order reversing map d on the set of nilpotent orbits in g. The map d is involutivewhen it is restricted to its range.
Definition 4.2.8. An orbit in the range of d is called special.
A general characterization of special orbits appears in [Collingwood-McGovern](6.3.7).We interpret it among small orbits.
43
Fact 4.2.9. The following is a complete list of special spherical orbits in classical Liegroups:
(i) All orbits are special in type A,
(ii) the orbits corresponding to partitions of the form [3, 22b, 1c] and [1c] in type B,
(iii) the orbits corresponding to the partitions of the form [22b, 12c] and [2b] in type C,
(iv) the orbits corresponding to the partitions of the form [22b, 1c] and [3, 1c] in type D.
Definition 4.2.10. A nilpotent orbit is rigid in g if it is not induced from any properparabolic subalgebra.
Proposition 4.2.11. [Collingwood-McGovern] An nilpotent orbit corresponding to the par-tition [p1, p2, . . . ] is rigid iff
· 0 ≤ pi+1 ≤ pi ≤ pi+1 + 1 for all i, and· |j | pj = i| 6= 2 if ε(−1)i = −1,
where ε = ±1 and is precisely defined as in Chapter 3.
Fact 4.2.12. The following is a complete list of non-rigid spherical orbits in classical Liegroups:
(i) All non-zero orbits are not rigid in type A,
(ii) the orbits corresponding to partitions of the form [3, 12c] and [22b, 12] in type B,
(iii) the orbits corresponding to the partitions of the form [22, 12c] and [22c] in type C, and(iv) the orbits corresponding to the partitions of the form [3, 1c] and [22c] in type D.
4.3 Basepoints in VT
From the previous sections, we know that each spherical orbital variety V contains a pointwhose orbit under the Borel subgroup is dense in V. For our purposes, we would like ashort expression for some such point to simplify the forthcoming calculations. For orbitalvarieties within classical nilpotent orbits, such an expression can be easily read off from thestandard tableau corresponding to V.
In type A, such a basepoint is essentially defined in [Melnikov]. We provide a slightlymore general construction and extend the result to other classical types. The main tool forthe latter is the surjection from domino tableaux onto standard Young tableaux defined in[Carre-Leclerc]. It induces a map on the level of orbital varieties that helps us define thebasepoint in the “type A component” of each V.
4.3.1 Notation
Let g be a Lie algebra of classical type. For a fixed type X, we will write gn for the Liealgebra of rank n. Let ∆+ be the set of its positive roots, and Π for its set of simple roots.To fix notation, we let Π equal
ei − ei+1 (type A)ei − ei+1, en (type B)ei − ei+1, 2en (type C)ei − ei+1, en−1 + en (type D).
44
Let gα denote the root space corresponding to α ∈ ∆ and choose Eα ∈ gα. Also chooseTi such that t =
⊕CTi. The triangular decomposition can then be written as
g =∑
−α∈∆+
gα ⊕ t⊕∑
α∈∆+
gα.
Lie algebras of type A appear naturally inside Lie algebras of the other classical types, andinspired by the results on orbital varieties of the previous chapter, we would like be ableto interpret results in type A within the context of the other classical types. To this effect,define two linear maps that relate the Lie algebras of type A to those of other classicaltypes. Let g be an algebra of type Xn and define the projection
πA : g → gln
by πA(Eei−ej ) = Eei−ej , πA(Ti) = Ti, zero on the other root spaces, and extending linearly.Here we interpret Eei−ej and Ti differently on the domain and range of the map. Also define
ιA : gln → g
by requiring that ιA πA = I and Image ιA =⊕
gei−ej ⊕ t.
To facilitate induction, we also define a linear map ι that embeds each Lie algebra oftype Xn−1 into the Lie algebra of type Xn. Somewhat informally
ι : gn−1 → gn
is defined by adding one to each index that appears in the root space decomposition. Forexample, in type A, ι(gei−ej ) = gei+1−ej+1 and ι(ti) = ti+1. Finally, for an integer i, definei = n + 1 − i, where n is the size of the underlying algebra. In particular, this meansι(gei−ej
) = gei−ej. While this notation is somewhat confusing, as it means different things
for Lie algebras of different rank, it does simplify expressions in later calculations. We hopethe reader forgives us, as the rank should always be clear from context.
4.3.2 Type A
Consider a spherical nilpotent orbit O and let VT ⊂ O be the orbital variety associated tothe standard Young tableau T ∈ Y T (n). Let T i denote the set of labels contained in thei-th column of T, so that in our case T i = ∅ if i > 2. We will define a point fT containedin VT whose orbit under the Borel subgroup is dense in VT .
Fact 4.3.1. Let φ : T 2 −→ T 1 be an injection with the property that φ(k) < k for allk ∈ T 2. Such a map always exists, and furthermore, the point
fT =∑
k∈T 2
Eek−e gφ(k)
is contained in the variety VT .
Proof. The fact that a map φ always exists is clear by inspection. A spherical nilpotentorbit in type A is uniquely determined by the rank of its elements. For each fT defined
45
above, f2T = 0, so it lies in some spherical orbital variety. That it lies precisely in VT follows
from induction and the above rank condition.
This definition includes Melnikov’s construction as a special case. More precisely, it isalways possible to choose φ in such a way so that φ(k) = k − 1 whenever α]k−1
/∈ τ(T ). Inthis incarnation, fT is a minimal representative of VT in the sense described below. Letf ∈ n and for its root space decomposition, let us write
f =∑
ε∈∆+
cε(f)Eε.
Definition 4.3.2. An element f ∈ V is a representative of V if f does not belong to anyother orbital varieties. A representative f of V is minimal if
1. each cε(f) ∈ Z,2. for every αi /∈ τ(V), cαi(f) 6= 0,3. If g is another representative of V satisfying the above, the the number of non-zero
cε(g) will be greater than or equal to the number of non-zero cε(f).
We would like the basepoints we choose to be minimal representatives, as these con-ditions will simplify the ultimate infinitesimal character calculations. In type A, we havealready seen that this is always possible and in further work we would like fT to be closeto satisfying this condition.
Example 4.3.3. Consider the orbital variety VT associated with the standard Young tableau
T =
1 3245
The points
f1 =
0 1 00 0 00 0 0
and f2 =
0 0 10 0 00 0 0
both lie in VT and are both defined as fT by Fact 4.3.1, each by different choice of φ. For thefirst, φ(3) = 2 while for the second, φ(3) = 1. However, only f1 is a minimal representativeof VT .
4.3.3 Other Classical Types
Let X = B, C, or D, and let VT be the orbital variety in a spherical nilpotent orbit of typeX associated with the standard tableau T. The basepoint that we define in VT is the sumof two points,
· fπA(T ), a version the basepoint of the type A orbital variety associated to VT by thetableau projection map and
· MXT , a point defined from the set of horizontal dominos in T .
46
To define the second term, we first need to distinguish among the different types of horizontaldominos that appear in the domino tableaux of spherical orbital varieties. For each suchtableau T , define the set of NT = k ∈ Labels(T ) |D(k) is horizontal to be the set of itshorizontal dominos. Let ST be the set of dominos in NT that intersect the first column ofT , or more formally, ST = k ∈ NT | k ∈ T 1. If M is a family of sets of integers, let M
denote the union of all integers contained in elements of M. We now inductively define aset N1
T of pairs of labels in T by N?1 = ∅ and
NT1 =
NT (n−1)1 ∪ k, n if D(n) ∈ ST \ (NT (n−1)
1 )
and if X = C, k = n− 1,
NT (n−1)1 otherwise.
Finally, let NT2 = ST \(NT
1 ) and NT3 = NT \((NT
1 )∪NT2 ). Note that the set NT
3 is alwaysempty in type C while NT
2 is always empty in types B and D.
Example 4.3.4. Suppose T and U are the following domino tableau:
T =
0 12
3 4
5
U =
1
2 3
4
For the tableau T of type B, ST = 2, 5, NT1 = 2, 5, NT
2 = ∅, and NT3 = 1. If we
consider U as a domino of type C, then SU = 1, 4, NU1 = ∅, NU
2 = 1, 4, and NU3 = ∅.
If we consider U as a domino of type D, then SU = 1, 4, NU1 = 1, 4, NU
2 = ∅, andNU
3 = ∅.
We are now ready to define the point MXT . Let MX
T =∑i,j∈NT
1Eei+ej
+∑
α∈UXT
Eα,
where for a standard tableau T in type X, UXT is the set of roots described by
UXT =
e]k−1
+ ek | k ∈ NT3 X = D
2ek |NT2 X = C
ek | k ∈ NT3 ∪ e3 | 2 ∈ T 3 and 3 ∈ T 2 X = B
Definition 4.3.5. For X= B, C, or D, and a domino tableau T , let
fXT = fπA(T ) + MX
T
where fπA(T ) is interpreted as lying inside the Lie algebra of type X.
Lemma 4.3.6. The point fT is a minimal representative of VT .
Proof. In type A, this is [Melnikov]. For other classical types, we define T ′ = Φ−1(T ) ∈ΣDTcl(shape T ). We first show that fT ∈ VS , where S = Φ(T ∗) and T ∗ ∈ ΣDTcl(shape T )has the same underlying domino tableau as T ′. We then show that T ′ and T ∗ must sharethe set of closed clusters with positive sign, which implies that S = T by the definition of
47
Φ. This verifies that fT is a representative of VT . Minimality of fT may then be checkedby inspection.
We would like to show that for all k ≤ n, fT (k) ∈ Oshape T ′(k). By induction, it isenough to verify this for k = n− 1. Note that for spherical orbits, the partition of the orbitcontaining a nilpotent element f is completely determined by rank f and rank f2. Theabove statement can be now verified by inspecting the definition of fT and comparing rankfT (n−1) and rank f2
T (n−1) with rank fT and rank f2T . In this way, we have fT ∈ VS , where
S = Φ(T ∗) and T ∗ is some tableau in ΣDTcl(shape T ) sharing its underlying tableau withT ′.
Now note that if C is a closed cluster of T ′ or T ∗, then because the orbit Oshape T isspherical, the initial cycle IC through C must have the form IC = i, i + 1, . . . , j. Theorem3.5.1 implies that the simple root αi ∈ τ(T ) iff there is a closed cluster C ∈ C+ withIC = i, i + 1, . . . , j for some j. Further note that if C ∈ C+, then Eei+ej appears inthe expansion of fT with non-zero coefficient while Eei−ej has coefficient zero. Similarly, ifC ∈ C−, then Eei−ej appears in the expansion of fT with non-zero coefficient while Eei−ej
has coefficient zero. But this forces Φ(T ∗) to have the same τ -invariant as Φ(T ′), whichimplies that Φ(T ∗) = Φ(T ′). Hence fT is a representative of VT .
Lemma 4.3.7. Consider an orbital variety VT in a spherical nilpotent orbit of classicaltype that corresponds to the standard tableau T , and let Q = QVT
be the maximal parabolicstabilizing it. Then the orbits
B · fT and Q · fT
are dense in VT .
Proof. For the result in in type A, see [Melnikov](Proposition 4.13). In general, densenessfollows by induction from Corollary 4.4.2 below.
Example 4.3.8. Let X = C and consider the orbital variety VT associated with the dominotableau
T=
1
2 3
45
. Then the Young tableau πA(T ) =
1 3245
.
We have N = 1, 4, 5, N1 = 1 and N2 = N \ N2. Finally, the basepoint fT =(A M0 −At
), where
A = fπA(T ) =
0 0 0 0 00 0 0 0 00 0 0 1 00 0 0 0 00 0 0 0 0
and M =
0 1 0 0 01 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 1
.
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4.4 Induction
Our calculation of infinitesimal characters of Graham-Vogan representations attached tothe orbital variety VT will proceed by a type of induction on the standard tableau T. As inour model example, we would like to describe the action of Qf on the space q/qf. Becausewe consider only spherical orbits, it is equivalent to describe this action on the isomorphicspace b/bf. In this section, we describe this space inductively, verifying Lemma 4.3.7 in theprocess.
Fix a standard tableau T of a given classical type and write VT for the orbital varietycorresponding to it. Ideally, we would like to be able extract information about the orbitalvariety VT from the orbital variety VT (n−1) and in this manner set up a type of induction.Based on our construction of minimal representatives, there are two instances where aninductive step is not completely apparent. First, in type C, when both D(n) and D(n− 1)are horizontal dominos, while D(n− 2) is not, it appears that one should let the inductiontake place from the orbital variety VT (n−2) to VT (n). A somewhat more serious problem isthat in types B and D, the standard domino tableau T (n−1) does not always correspond toan orbital variety of the same classical type as VT , so in order for induction to make sense,we have to be careful defining the appropriate orbital variety. To this effect, we define astandard tableau T ↓ by
T ↓ =
MT (C, T (n− 1)) X = B or D, type VT (n−1) 6= X andC the cycle in T (n− 1) through n− 1,
T (n− 2) X = C, D(n) and D(n− 1) are horizontal dominos,while D(n− 2) is not,
T (n− 1) otherwise.
With this definition, shape T ↓ and shape T are partitions of the same classical type. There-fore, we are able to associate an orbital variety VT ↓ of the same type as VT to the standardtableau T ↓. We will write f↓ for fT ↓ , and b↓, q↓, and g↓ for the Lie algebras correspondingto VT ↓ .
As in our model example, we would like to describe the action of Qf on the spaceU = b/bfT
. If we think inductively, however, we can break this task down into a study ofthe quotients
Un = (b/bfT)/(b↓/b
↓f↓).
In fact, ⊕
k≤n
Uk ' U.
It will be often convenient to divide our work into cases that arise from an inductiveconstruction of of the representative fT . The cases are distinguished by the possible formsof the difference fT − ι(fT ↓). We describe the possiblities along with what they imply onthe level of tableaux.
(C1) When fT = ι(fT ↓), the domino T \ T ↓ lies entirely in the first column of T .(C2) When fT = ι(fT ↓) + Ee1−eφ(n)
, the domino T \ T ↓ lies entirely in the second
49
column of T .(N1) When fT = ι(fT ↓) + Ee1+ek
and X = B or D, then this is the case whenT ↓ 6= T (n − 1) and k, k + 1, . . . , n − 1 is a cycle in T (n − 1). If X = Cand k 6= 2, then k = n− 1 and T ↓ = T (n− 2).
(N2) When fT = ι(fT ↓) + E2e1 , then X = C and T ↓ = T (n− 1).(N3) When fT = ι(fT ↓) + Ee1−e2 + Ee1 , we have X = B and T \ T ↓ is a
horizontal domino that intersects the third column of T . When fT =ι(fT ↓) + Ee1−e2 + Ee1+e2 , we have X = D and again T \ T ↓ is a horizontaldomino that intersects the third column of T .
(*) When fT = ι(fT ↓) + Ee1 , we have X = B and T \ T ↓ = D(3) ∈ T 2 whileD(2) ∈ NT
3 .
We can now attack the description of the space Un. Because we would like our descriptionto reflect the original action of QfT
, we infact describe the quotient
Un = (b/bfT)/
(ι(b↓)/ι(b↓
ι(f↓)))
and embed it in b. There is a certain amount of choice possible in the parametrization ofUn; however, the determinant of the QfT
action is independent of these choices.
Lemma 4.4.1. Consider a standard tableau T and recall our construction of fT , a repre-sentative of the orbital variety VT . The group QfT
acts on the space b/bfTand this action
restricts to the quotientUn = (b/bfT
)/(ι(b↓)/ι(b↓
ι(f↓)))
.
In each of the above cases, we describe a space U ′n ⊂ b with the property that the determinant
of the natural action of QfTequals the determinant of the QfT
action on the quotient Un.Recall that for a standard tableau S, we denote the set of labels contained in column j of Sby Sj. When the square or domino with label n lies entirely in the first column of T , thatis, case (C1),
Un =
⊕T 2 ge1−ei
in type A
⊕(πAT )2
NT∪∗ge1−ei
⊕⊕(πAT )2 ge1+egφ(i)
other types.(4.1)
When the square or domino with label n lies entirely in the second column of T , that is,case (C2), we define two spaces V and W by
V =⊕
j>gφ(n)
j /∈N∪(gπAT )2
geφ(n)−ej ⊕⊕
(πAT (n−1))2
φ(i)>φ(n)
ge1−ei⊕ t1
W =⊕
j 6=φ(n)∪1
j 6=φ(i)
i∈ ^(πAT ↓)2
geφ(n)+ej .
50
We can then say
Un =
V in type AV ⊕W ⊕ ge1 ⊕NT
3 =? geφ(n)type B
V ⊕W ⊕ g2eφ(n)⊕ ge1+eφ(n)
type CV ⊕W ⊕NT
3(ge1−ei
⊕ ge1−e]i−1) type D.
(4.2)
Now suppose that we are in case (N1) and k, n = NT1 . Then in each of the classical
types not equal to A,
Un =⊕
(πAT ↓)2ge1+eφ(i)
⊕⊕
(NT↓1 )
ge1−ej⊕
⊕
(πAT ↓)2ge1−ei
⊕ t1 (4.3)
Case (N2) appears only in type C and there
Un =⊕
(πAT ↓)2(ge1−ei
⊕ ge2−ei)⊕
⊕
NT↓not ge2−e1
(ge1−ei⊕ ge2−ei
)⊕
⊕
(πAT ↓)2(ge1+φ(i) ⊕ ge2+φ(i))⊕ t1
(4.4)
Case (N3) appears only in types B and D.
Un = ⊕
j>2(ge2−ej ⊕ ge2+ej )⊕ ge2 ⊕ t1 ⊕ t2, type B⊕j>2(ge2−ej ⊕ ge2+ej )⊕ t1 ⊕ t2 type D
(4.5)
Finally, while in the special case of (*),
U3 = ge1−e2 ⊕ ge1+e3 ⊕ ge2 . (4.6)
Proof. First form a decomposition b = b1 ⊕ ι(b↓) that is compatible with the root spacedecomposition. For B ∈ b, write B = B1 + B2 with B1 ∈ b1 and B2 ∈ ι(b↓). Note thatB ∈ bfT
if and only if
[B, fT ] = 0. (4.7)
To describe Un, we assume that B2 lies in ι(bfT↓ ), i.e. that
[B2, ι(fT ↓)] = 0. (4.8)
We would like to know what additional conditions on B are necessary to make sure that itsatisfies (4.7). If we write
B =∑
α∈∆+
cαEα +∑
i≤n
ciTi,
then (4.7) imposes linear conditions on the coefficients c in the expansion of B. If we choose
51
a representative α or i within each linear condition and denote the set of representatives byP , then
b/bf '⊕
α∈P
gα ⊕⊕
i∈P
ti.
The natural action of Qf has the same determinant on both spaces. To describe U ′n, we only
need to include representatives for linear conditions that do not already arise as conditionsfor (4.8). We carry out this plan in each of the cases by describing the set of representativesin each of the cases.
Case (C1). In this case, fT = ι(fT ↓). Condition (4.7) boils down to
[B1, ι(fT ↓)] = 0. (4.9)
Write B1 =∑
S cαEα + c1T1. If we expand the left hand side of (4.9) in terms of root spacecoordinates, the resulting linear conditions imposed by (4.9) all take the form cγ = 0 forγ in some set Ω. The quotient U ′
n then takes the form⊕
Ω gα. Deciphering (4.9) explicitlyleads to the description in the statement of the lemma.
Case (C2). In this case, fT = ι(fT ↓) + Ee1−eφ(n). Equation (4.7) reduces to
[B1, ι(fT ↓)] + [B1, Ee1−eφ(n)] + [B2, Ee1−eφ(n)
] = 0. (4.10)
We can again write B1 =∑
S cαEα + c1T1 and expand (4.10) in terms of root space coordi-nates. This time, (4.10) imposes more complicated conditions on the coefficients. For eachlinear condition on the coefficient c obtained from (4.10), we select as representative thelargest root γ such that cγ appears within the linear equation. If , however, ci also appearswithin a linear condition, we select the coefficient i instead. When we account for linearconditions that already appear in (4.8), we obtain the description of U ′
n in the statement ofthe lemma.
Case (N1). In this case, fT = ι(fT ↓) + Ee1+ek. Equation (4.7) reduces to
[B1, ι(fT ↓)] + [B1, Ee1+ek] + [B2, Ee1+ek
] = 0. (4.11)
In types B and D, the method of case (C2) can be used verbatim, we only have to accountfor the different linear conditions imposed by (4.11). When X = C, we merely have toaccount for the different definition of T ↓ in this case by letting B1 =
∑S cαEα +c1T1 +c2T2
for the appropriate set S.
Case (N2). In this case, fT = ι(fT ↓) + Ee1+e2 . Equation (4.7) reduces to
[B1, ι(fT ↓)] + [B1, Ee1+e2 ] + [B2, Ee1+e2 ] = 0
and the method of case (C2) can again be used verbatim to describe U ′n.
Case (N3). In type B, fT = ι(fT ↓) + Ee1−e2 + Ee1 , while in type D, fT = ι(fT ↓) +Ee1−e2 + Ee1+e2 . In both cases, fT ↓ = 0 and bf
T↓ = b↓. Hence equation (4.7) reduces to
[B, Ee1−e2 ] + [B, Ee1+e2 ] = 0 (4.12)
52
in type B and
[B,Ee1−e2 ] + [B, Ee1+e2 ] = 0 (4.13)
in type D.Case (*). In this case, fT = ι(fT ↓) + Ee1 .Equation (4.7) reduces to
[B1, ι(fT ↓)] + [B1, Ee1 ] + [B2, Ee1 ] = 0
and the method of case (C2) can again be used verbatim to describe U ′n.
Corollary 4.4.2. For a standard tableau T, dimUn = dimVT − dimVT ↓ .
Proof. We can compute dimVT −dimVT ↓ from the formula for the dimension of a nilpotentorbit. Let [λ1, · · ·λp] be the dual partition to shape T . In each of the cases, dimVT−dimVT ↓
equals
12(dimOshape T − dimOshape T ↓) =
λ2 + λ3 Case (C1)λ1 Case (C2) and X = Aλ1 − 1 + λ3 Case (C2) and X = B or Dλ1 + 1 Case (C2) and X = Cλ1 Case (N1) and X = B or Cλ1 − 1 Case (N1) and X = D2λ1 − 1 Case (N2)λ1 Cases (N3) and (*)
One can now check these are exactly the dimensions of the corresponding spaces Un. Wedetail the calculation in the inductive case (C2) when the group is of classical type C. Theother cases are not dissimilar. Recall the two types of vertical dominos that arise within adomino tableau, and denote by I− and I+ the set of labels of the dominos of that type thatare contained in the tableau T . Reading off the parameterization of Un in this case impliesthat dimUn equals
#j < φ(n) | j /∈ N ∪ (πAT )2
+ #
j j 6= φ(i) for i ∈ (πAT )2, j 6= n
+ #i ∈ (πAT )2 |φ(i) > φ(n)
+ 3
=j ∈ I− | j 6= φ(n), and if j > φ(n), then j ∈ Im φ
+j ∈ I− ∪N | j 6= n
= (#I− − 1) + (#I− + #N − 1) + 3= 2#I− + #N + 1 = λ1 + 1,
as desired.
4.5 The Trace of the Adjoint Action
Let tf be a maximal torus inside the Lie algebra qf . It can be verified that the basepointfT was chosen so that qfT
∩ t is a maximal torus in qfT. The inductive procedure of the
previous sections provides a quick description of the coordinates of tf . The trace of the
53
adjoint action of tf on q/qf can then be calculated as a sum of the traces of the actions ofthe quotient spaces Ui. In keeping with the inductive philosphy of this chapter, we computethis trace on the space Un, separating each of the inductive cases.
Fact 4.5.1. Let f = fT and write an element a ∈ t as a =∑
1≤i≤n ai ti. Then a lies in tfiff
∑2≤i≤n ai ti lies in the torus ι(t)ι(f↓) and additionally
· a1 = aφ(n) in case (C2),
· a1 = −ak in case (N1), where k, n is a pair in NT1 ,
· a1 = 0 in cases (N2), (N3), as well as (*).
Proof. This follows immediately from an inductive description of the basepoint fT in eachof the above cases.
Proposition 4.5.2. Let [p1, p2, . . . , pl] be the partition that corresponds to the nilpotentorbit O through fT . Write [λ1, λ2, . . . , λm] for its dual partition. Finally, write an arbitraryelement of t as
∑i≤n ai ti. The trace of the adjoint action of tf on the quotient Un is listed
below, sorted according to the inductive cases.
(C1) In type A, the trace is −λ2 a1 +∑
i∈T 2 aı, while it is −(λ2 + λ3)a1 in the otherclassical types.
(C2) In type A, the trace is −λ1 a1+∑
i∈T 1 aı. In the other classical types with NT3 = ∅
while in type D, it is(−λ1 − c)a1,
where
· c = 2 in type C,· c = −2 + λ3 in types B and D.
(N1) Let k, n = NT1 . In types B and D, the trace is
−(λ1 − c)a1
with c defined as in case (C2). In type C, the trace is 0.
(N2) This case occurs only in type C where the trace is 0.
(N3) This case occurs only in types B and D where the trace is −λ1a1.
(*) Here , the trace is −2a1 − a3.
Proof. We use the description of the quotient Un in Lemma 4.4.1 together with Fact 4.5.1.In type A, determining the trace is simply a matter of reading off the coordinates. Weprovide the calculations for the other classical types which are only a little more subtle.Write I+ and I− for the set of labels of the vertical dominos in T of the corresponding type,and write N for the set of horizontal dominos in T . We will use | · | to denote the order ofeach of these sets.
54
Case (C1). By reading off the coordinates, we find that the trace is∑
πA(T )2
(−a1 + aı) +∑
πA(T )2
(−a1 − aφ(i)) +∑
i∈NT
(−a1 − aı) (−a1 + a3)
where the final parenthetical expression appears iff some sub-tableau of T lies in case (*).Applying (4.5.1) reduces the above to
−2 |(πAT )2| a1 − |NT | a1 (−1) = −(λ2 + λ3)a1.
Case (C2). We begin with type B. The trace is
∑
j>φ(n)
j /∈N∪ ^(πAT )2
(−a1 + aj) +∑
(πAT ↓)2φ(i)>φ(n)
(−a1 + aı) +∑
j /∈φ(n)∪1
j 6=φ(i)
i∈(πAT ↓)2
(−a1 − aj)
=∑
i<φ(n)i/∈N∪(πAT )2
(−a1 + aı) +∑
i∈(πAT ↓)2φ(i)>φ(n)
i>φ(n)
(−a1 + aı) +∑
i/∈φ(n)∪ni 6=Im(φ)
i∈(πAT ↓)2
(−a1 − aı)
=∑
i<φ(n)i/∈N∪(πAT )2
(−a1 + aj) +∑
i>φ(n)i/∈N∪(πAT )2
(−a1 + aı) +∑
i 6=φ(n)
(−a1 − aı)+? + a1 + (a1)
=− (2 (|I+| − 1) + |N |+ 1 (+1))a1
This is (−λ1 − λ3 + 2)a1. In type C , the calculation is similar, with final line equal to
−(2 (|I−| − 1) + |N |+ 2 + 1 + 1)a1 = −(λ1 + 2)a1.
In type D, the final line is
−(2 (|I+| − 1) + |N |)a1 − (a1 − ak + a1 − ak) = −(2 (|I+| − 1) + |N |)a1 − (2a1 − 2ak)
where the parenthetical expression appears iff k = NT1 . This reduces to the statement in
the Proposition.Case (N1). The proof is similar to case (C2).Case (N2). We find that the trace equals:
∑
i∈I+
(a1 − aı) +∑
i∈N
(a1 − aı) +∑
i∈φ(I+)
(a1 + aı).
After applying Fact 4.5.1, this expression reduces to 0.The remaining two cases are simple.
For future use, let us define the vector (cn, cn−1, . . . , c1) by letting ci equal the numberof times the term ai appears in the expression for the trace of the adjoint action on
⊕i≤n Ui
described by Proposition 4.5.2.
55
56
Chapter 5
Infinitesimal Characters
Armed with the constructions of the previous chapter, one is ready to examine the repre-sentations that arise from the Graham-Vogan construction. As in the model example, werestrict our examination to those representations that arise from spherical orbital varieties,as the corresponding Lagrangian coverings are then just quotients of the group G.
5.1 Characters, Weights, and Extensions
Let O be a spherical nilpotent orbit of a classical Lie group G, fix a Borel subgroup B, andconsider the orbital variety V ⊂ O that corresponds to the standard tableau T by Corollary3.3.1. Write Q for its stabilizer in G; it is determined explicitly by Theorem 3.5.1. Recallthe basepoint f = fT ∈ V and its stablizer Qf ⊂ Q. Lemma 4.3.7 implies that both, theparabolic Q, and in fact, the Borel B act on f with dense orbit in V.
The Graham-Vogan construction examines the Lagrangian covering:
G×Q V0
π
yytttttttttρ
²²G/Gf G/Q
Because Q/Qf and B/Bf are dense in V, Graham and Vogan’s construction suggestslooking at the character α of Qf that is given by the square root of the absolute value of thereal determinant of Qf acting on the tangent space q/qf of V at f . Utilizing the notationof Chapter 1, one would like to know when a homomorphism jπ exists. This conditiontranslates to the existence of an existence of a representation γ of Q that contains α on Qf
as a subrepresentation. That is,γ|Qf
⊃ α.
Ideally, one would like γ itself to be a character. In this case, jπ is an isomorphism, asrequired by [Graham-Vogan], rather than just an injection. We examine the weight wα
of the character α. First note that α is a real character. Recall the vector (cn, . . . , c1)defined at the end of the previous chapter. If we split the weight of α into holomophic andanti-holomorphic parts, we obtain:
wα = ( cn2 , cn−1
2 , . . . , c12 )( cn
2 , cn−1
2 , . . . , c12 ).
57
Here, we interpret a weight of Qf as an equivalence class of weights of Q and the above isjust a representative of such an equivalence class.
We would like to answer the existence question for γ by examining its correspondingweights. To this effect, first suppose that γ is a real character. Write the Levi subalgebra l
as a sum of reductive parts as⊕
i≤s g(li). A real character γ of L takes the form
γ(A) =∏
i≤s
|det Ai|αi (5.1)
where A ∈ L, αi ∈ R and Ai is the restriction of A to the ith reductive part of L. We canrewrite 5.1 as
γ(A) =∏
i≤s
(detAi)αi2 (detAi)
αi2 (5.2)
splitting it into holomorphic and anti-holomorphic parts. In this manner, we associate theweight
wγ = (αn2 , αn−1
2 , . . . , α12 )(αn
2 , αn−1
2 , . . . , α12 ).
We would like to know conditions under which wγ lies in the same equivalence class ofweights of Q as wα. In the case of spherical orbits, Fact 4.5.1 imples that this occurs iff
· αi + αj = ci + cj whenever i = φ(j),
· αi − αj = ci − cj whenever i, j ∈ NT1 , and
· αi = ci for all i /∈ NT ∪ T 2 ∪ φ(T 2).
If we write wγ = wα + ε for some weight ε, then these conditions translate to
· εi + εj = 0 whenever i = φ(j),
· εi − εj = 0 whenever i, j ∈ NT1 , and
· εi = 0 for all i /∈ NT ∪ T 2 ∪ φ(T 2).
Denote the set of weights wγ that satisfy the above conditions by HWr(wα). We wouldalso like to know which weights in HWr(wα) correspond to a real character γ of Q. WriteHW 1
r (wα) for this set. First, let us define some notation. For a parabolic subgroup Q ofG, we group the coordinates that correspond to the same reductive part of its Levi L bysetting them off with an additional set of parentheses. If
l =⊕
i≤s
g(li) and g(lj) ∩ t =⊕
ci≤j≤di
CTj ,
then we will write a weight a as
a = ((an an−1 . . . ad1)(ac2 . . . ad2) . . . (ack. . . adk
) . . . (adl. . . a1)).
For instance, if l = gl3⊕gl2⊕gl1, then we will write a weight as a = ((a6 a5 a4) (a3 a2) (a1)) .This notation provides us with a convenient way of testing which weights correspond tohighest weights of one-dimensional representations of the parabolic Q.
58
Fact 5.1.1. A weight a ∈ HWr(w) lies in HW 1r (w) iff all coefficients corresponding to a
given reductive part of the Levi of Q are the same. That is, iff
ack= ack+1 = . . . adk
for all 1 ≤ k ≤ l.
Proof. Suppose that a satifies the above hypothesis. Then a character of Q with weight ais given by a product of exponents of absolute values of determinants of the reductive partsof L. The exponent of the determinant of the part corresponding to ck, ck + 1, . . . , dk isgiven by twice their common value, as per the descripition of real characters of 5.2.
If γ is a real character of Q that restricts to α on Qf , then wγ ∈ HW 1r (wα). Conversely,
given a weight w in HW 1r (wα), we can construct a real character γw according to the
procedure described by 5.2.Now suppose that γ is an arbitrary character of Q that restricts to the real character α
on Qf . Then γ takes the formγ = χ · γ′
where γ′ is a real character such that γ′|Qf= α, and χ is a unitary character such that
χ|Qf= 1. In particular, this means that χ|Tf
= 1. If we write A ∈ T as∑
aiTi, then
χ(A) =∏
i≤n
(ai|ai|
)βi
=∏
i≤n
aβi
2i (ai)−
βi
2 .
In this manner, we define the weight
wχ = (βn
2 , βn−1
2 , . . . , β1
2 )(−βn
2 ,−βn−1
2 , . . . ,−β1
2 ).
The character χ restricts to the identity on Qf iff wχ lies in the equivalence class of 0 ofweights of Q. This occurs iff
· βi + βj = 0 whenever i = φ(j),· βi − βj = 0 whenever i, j ∈ NT
1 , and· βi = 0 for all i /∈ NT ∪ T 2 ∪ φ(T 2).
Furthermore, because χ should be a unitary character of L, its entries also need to satisfythe conditions of Fact 5.1.1. We can generalize the definitions of HWr(wα) and HW 1
r (wα)in the obvious way.
For a weight wγ , write whγ and wa
γ for its holomorphic and anti-holomorphic parts. Notethat wγ ∈ HW (wα) iff wh
γ = (dn, dn−1, . . . , d1) satisfies:
· di + dj = ci + cj whenever i = φ(j),· di − dj = ci − cj whenever i, j ∈ NT
1 , and· di = ci for all i /∈ NT ∪ T 2 ∪ φ(T 2).
Furthermore, wγ ∈ HW 1(wα) iff wγ ∈ HW (wα) and the coefficients di of whγ satisfy the
conditions of Fact 5.1.1. This analysis simplifies notations by allowing us to refer to weightswith only one part. With this characterization in mind, we redefine the sets HW and HW 1.
59
Definition 5.1.2. Let w be the weight of a one-dimensional representation of Qf and defineHW (w) to be the set of weights of representations of Q that restrict to w on the torus tf .Futhermore, let HW 1(w) be the set of weights in HW (w) that correspond to weights ofcharacters of Q.
We would like to answer the following questions:
· As suggested by the Graham-Vogan construction, can α always be extended to acharacter γ of Q? What about a finite-dimensional representation of Q?
· What are the infinitesimal characters of the representations constructed in this way,and how do they fit into the set of infinitesimal characters that ought to be attachedby the orbit method to the nilpotent orbit O?
The arguments of this section reduce an answer to the first question to a descriptionof the set HW 1(wα). Proposition 4.5.2 together with the conditions of 5.1.1 calculate theweight wα of the character α. A character γ that restricts to α on Qf exists wheneverHW 1(wα) is non-empty.
To answer the second question, we turn to the work of W. M. McGovern, who describesa procedure for constructing a set of infintesimal characters that ought to be attached toan arbitrary nilpotent orbit among classical groups.
5.2 The Infinitesimal Characters IC1(O)
A classification of unitary representations of complex reductive Lie groups can be obtainedfrom a construction that begins with a set of the so-called special unipotent representationsfirst suggested by Arthur (see [Barbasch]). This classification, however, is unsatisfactoryfrom the point of view of the orbit method: special unipotent representations can onlyhave as associated varieties the closures of special nilpotent orbits. To remedy this short-fall, [McGovern] suggests enlarging the set of special unipotent representations to a set ofrepresentations, called q-unipotent, with all possible associated varieties.
We recall McGovern’s construction of the infinitesimal characters of q-unipotent rep-resentations for classical groups. However, not all q-unipotent infinitesimal characters ob-tained by his method can reasonably correspond to representations attached to nilpotentorbits. After describing this phenomenon more closely, we prune the set of q-unipotent in-finitesimal characters to a set that should be attached to nilpotent orbits. We describe theseinfinitesimal characters explicitly for spherical nilpotent orbits. It should be noted that thiscan nevertheless serve only as an approximation to the set of infinitesimal characters thatought to be attached to a nilpotent orbit. For instance, there are unipotent representationsthat ought to be attached to certain nilpotent orbits that do not have half-integer coordi-nates (see for instance [McGovern4])(Section 5). Unfortunately, the methods of this sectiondo not account for such infinitesimal characters.
5.2.1 Infinitesimal Characters of q-unipotent Representations
We reproduce the procedure for attaching infinitesimal characters to nilpotent orbits. Givena nilpotent orbitO, we first decribe a way of producing an element hO in a Cartan subalgebraof g.
60
Proposition 5.2.1. For each nilpotent element f ∈ g, there is a homomorphism φ : sl2 −→g that maps the element
(0 10 0
)onto f. If the nilpotent orbit O = Of through f corre-
sponds to the partition [p1, . . . , pl], then the matrix hO = φ
(1 00 −1
)has the following
eigenvalues:
p1 − 1, p1 − 3, . . . ,−(p1 − 1), p2 − 1, . . . , pl − 1, . . . ,−(pl − 1).
We can describe the element hO more precisely in terms of its coordinates.
Proposition 5.2.2. 1. If g is of type A, then the coordinates of hO, regarded as anelement of a Cartan subalgebra of g, are its eigenvalues in non-increasing order.
2. If g is of type B, C, or D, embed it in some sl(n) via the standard representation.
(a) Suppose the partition of Of has the numeral I or none at all. Also supposethat 0 occurs as an eigenvalue of the matrix hO with multiplicity k. Then thecoordinates of hO are its positive eigenvalues together with [k/2] zeros, arrangedin non-increasing order.
(b) If the numeral of Of is II, then the coordinates of hO are obtained in a similarmanner, except that the final coordinate is replaced by its negative.
The element hO lies in h and not in its dual h∗. Therefore, it can be regarded as aninfinitesimal character of the dual algebra Lg. We summarize Arthur’s original constructionof special unipotent representations. Consider a map
φ : SL(2) −→ LG.
Each such map gives a representation πφ of G via a transfer of the parameters of the trivialrepresentation of PGL(2) = LSL(2) to G. Representations obtained in this manner arecalled spherical special unipotent. For each πφ, Arthur conjectured that there exist:
1. a finite set Πφ of representations that contain πφ providing an analogue of a LanglandsL-packet, and
2. a linear combination of distribution characters of representations in Πφ that is a stabledistribution.
The representations contained in Πφ are called special unipotent. Barbasch and Vogan gavea more precise definition of special unipotent representations.
Definition 5.2.3. [Barbasch-Vogan] An irreducible representation of G is special unipotentif its annihilators equal Jmax(λO) for λO = 1
2hO.
A classification of unitary representations can be produced starting from this set ofrepresentations, see [Barbasch]. However, from the point of view of the orbit method, thisapproach is unsatisfactory. The associated variety of a special unipotent representation canonly be the closure of a special nilpotent orbit. By enlarging the group LG, McGovern
61
suggests a way of enlarging the set of special unipotent representations to a set whoserepresentations that have all possible associated varieties.
We follow [McGovern] in the definition of the enlargement of LG. In each of the classicaltypes except for type B, let n be the dimension d of the standard representation of LG. Intype B, however, let n = d + 1. More precisely, the values of n are as in the following table:
Type n
Am m + 1Bm 2m + 1Cm 2m + 1Dm 2m
The enlargement of LG that we seek is SL(n). There is a natural injection LG −→SL(n), of course with some choice in type B. Let T be the diagonal subgroup of SL(2).We can consider all the possible maps SL(2) −→ SL(n) and T −→ LG which make thefollowing diagram commute:
T
''
// SL(2) // SL(n)
uuLG
Mimicking the Joseph construction, each such choice allows us to transfer the trivialrepresentation from LT to a representation of G. The number n was chosen in such away as to obtain as many representations as possible while making sure that the resultingrepresentations of G depend only on a nilpotent orbit in sl(n).
Definition 5.2.4. [McGovern] Let U be a nilpotent orbit in sl(n), and recall the elementλU = 1
2hU . Let λ′U be any SL(n)-conjugate of λU lying inside a Cartan subalgebra of Lg.When regarded as an infinitesimal character of g, λ′U is called q-unipotent.
We would like to know which nilpotent orbit in g∗ should be attached to each of theq-unipotent infinitesimal characters. The philosophy of the orbit method dictates that thisis the open orbit O contained in the associated variety of U(g)/Jmax(λ′U ).
Theorem 5.2.5. [McGovern] Suppose that the orbit U ⊂ sl(n)∗ corresponds to the partitionp. The open orbit O in the associated variety of U(g)/Jmax(λ′U ) has partition:
1. pt in type A,
2. (pt)B in type B,
3. (l(pt))C in type C,
4. (pt)D in type D, except when p is very even, in which case O depends on the choiceof λU and can be either (pt, I) or (pt, II).
Here the maps pX are the X-collapses of the partition p and l(p) is the partition obtainedfrom p by subtracting 1 from its smallest term. For g of a specified type, we can thereforedefine a map
M : nilpotent orbits in sl(n) −→ nilpotent orbits in g
by letting M(U) = O, the orbit in the associated variety of U(g)/Jmax(λ′U ). With the helpof the above theorem, we can also think of M as a map on partitions.
62
5.2.2 The Preimage M−1(O)
According to the philosophy of the previous section, the q-unipotent infinitesimal charactersthat are attached to the nilpotent coadjoint orbit O of a classical group g is the set
IC(O) = λ′U | U ∈ M−1(O).
This is at least a first approximation of the infinitesimal characters of the representationsthat should be attached to O by the orbit method. We describe this set explicitly for thespherical nilpotent orbits in classical Lie algebras.
Proposition 5.2.6. Let O = Op be a spherical nilpotent orbit in a classical Lie algebra g.Then in each of the classical types, the set M−1(p) is as follows:
Type A pt,Type BWhen p = [22k, 12n−4k+1] k 6= n
2 , [2n− 2k + 1, 2k], [2n− 2k, 2k + 1]p = [22k, 12n−4k+1] n even, [n + 1, n]p = [3, 12n−2], [2n− 1, 12], [2n− 2, 2, 1], [2n− 2, 13]p = [3, 22k, 12n−4k−2] k 6= n−1
2 , 0, [2(n− k)− 1− ε, 2k + 1 + ε, 1] | ε = 0, 1p = [3, 2n−1] n odd; [n2, 1]
Type CWhen p = [12n], [2n + 1]
p = [2, 12n−2], [2n, 1]p = [2k, 12n−2k], k 6= 1 or n, [2n− k + 1, k], [2n− k + 1, k − 1, 1]p = [2n]; [n + 1, n], [n2, 1], [n + 1, n− 1, 1]
Type DWhen p = [22k, 12n−4k] k 6= n
2 [2n− 2k, 2k], [2n− 2k − 1, 2k + 1]p = [2n] n even, [n2]p = [3, 12n−3], [2n− 2, 12], [2n− 3, 2, 1], [2n− 3, 13]p = [3, 22k, 12n−4k−3] k 6= n−2
2 , [2(n− k − 1)− ε, 2k + 1 + ε, 1] | ε = 0, 1p = [3, 2n−2, 1] n even, [n, n− 1, 1].
Proof. The proof is much simpler than the statement. It consists of understanding the abovemap and analyzing all the possibilities. The details are left to the interested reader.
Unfortunately, even among this list, there already appear orbits U whose associatedq-unipotent infinitesimal characters λ′U cannot be attached to the nilpotent orbit O ⊂ g∗
in any reasonable way. To say this more precisely, we need to define characteristic cycles.Let X be a finitely-generated Harish-Chandra U(g) bimodule. Take X0 to be a finite-dimensional generating set for X that is stable under the adjoint action of U(g), and letXn = Un(g× g)X0. Then Xn is a filtration of X.
The associated graded module grX is isomorphic to S(g× g) as a grU(g× g)-module,and is annihilated by S(g∆). Hence grX can be regarded as a finitely-generated, one-sidedmodule over S(g).
Let V1, . . . ,Vs be the irreducible components of the associated variety V(grX). Further-more, let P1, . . . Ps be the corresponding minimal primes over AnngrX. Then grX admitsa finite filtration in which every graded su bquotient has the form S(g)/Q for Pi ⊂ Q for
63
some i. Let ni be the number of graded subquotients isomorphic to S(g)/Pi. This is inde-pendent of X0, and we can define the characteristic cycle Ch(X) to be
∑niVi. When X is
irreducible, s = 1, and the sum contains only one term.
Definition 5.2.7. Let U be a nilpotent orbit in sl(n) as before and recall the infinitesimalcharacter λ′U . Write U for the spherical q-unipotent bimodule U(g)/Jmax(λ′U ). We definethe number mλ′U to be the multiplicity of V(U) in Ch(U).
The orbit method dictates that in order for U to correspond to a cover of a nilpotentcoadjoint orbit O , mλ′U cannot exceed the order of the fundemental group of O. That is, Ushould not be too large to meaningful ly correspond to O. It turns out that for certain U ,this unfortunately does occur. Examples of this phenomenon arise already among sphericalnilpotent orbits.
Example 5.2.8. Let U be the nilpotent orbit corresponding to the partition [6, 3] in sl(9).Fix the type of the Lie algebra g to be C. Then M(U) = O[23,12] ⊂ sp(8)∗. Furthermore,λ′U = (5
2 , 32 , 1, 1
2). However, mλ′U = 4 while |π1(O[23,12])| = 2. According to the above philos-ophy, mλ′U should not be the infinitesimal character of a unipotent representation attachedto O[23,12]. In fact, this is also true for any U with partition of the form [2n−k+1, k]. Thereare similar examples in the other classical groups not of type A. Based on this example, weknow that in order to find the set of the infinitesimal characters of representations attachedto the nilpotent orbit O, we have to prune the set IC(O).
5.2.3 Pruning of IC(O)
As in the above example, in some cases, mλ′U > |π1(O)|. We would like to exclude theinfinitesimal characters that arise from these orbits, so our goal is to determine for whichU ∈ M−1(O) this occurs. First, we n eed to know the order of the fundamental group ofeach nilpotent orbit. Write a partition p as [p1, p2, ..., pl] and for each p define the numbers
a = number of distinct odd pi,b = number of distinct even nonzero pi,c = gcd(pi).
Proposition 5.2.9. Let O = Op be an orbit in a classical Lie algebra g. The order of thefundamental group is independent of the Roman numeral assigned to the orbit (if any) and|π1(Op)| =1. c in type A,2. in type B,
- 2a if p is rather odd,
- 2a−1 otherwise,
3. 2b in type C,4. in type D,
- 2 · 2max(0,a−1) if p is rather odd,
- 2max(0,a−1) otherwise.
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We follow [McGovern] in determining the multiplicity mλ′U . The process is a bit complexand requires notation incompatible with some used here, so rather than referring the readerto [McGovern], we replicate the relevant parts here usi ng new notation.
Definition 5.2.10. Let M(U) = O, and suppose that U corresponds to the partition p. Ineach of the classical types X = B,C, and D, we define two numbers µ and ν.
When X = D, let q = podd = (qλ11 , . . . , qλt
t ) and break it up into chunks as follows.Starting from the left, each chunk takes on one of the forms: (qλi
i , qλi+1
i+1 ) with both λi
and λi+1 odd; (qλii ) with λi even; or (qλi
i ) with λi odd and λi+1 even. Let ν be thenumber of chunks of the first two types. The number µ is defined the same way butwith q = [(peven)D]odd.
When X = B, break up podd into chunks as in type D. Let ν1 be the number ofchunks of the first type. Let c be the leftmost chunk of the third type and let ν2 bethe number of chunks of the second type to the right of c, plus one. If no c exists,let ν2 = 0. Finally, let ν = ν1 + ν2. The number µ is defined the same way but with([r(peven)]B)odd.
When X = C, define ν in the same way as in type B. To define µ, replicate itsdefinition in type D but with the partition [(peven)D]odd.
Finally, in each of the cases let ν∗ = max(0, ν − 1) and µ∗ = max(0, µ− 1).
Definition 5.2.11. Write the infintesimal character λ′U in coordinates as
(( i2)ri , . . . , (1
2)r1 , 0r0).
If λ′U contains a coordinate of −12 , simply write this as an additional 1
2 .
In type B, define the following numbers:
κ = number of even positive i with ri odd and ri−1 even,
κ1 = number of even positive i with ri odd, ri−1 even, and either ri−2 > ri withi > 2, or r0 > 1
2r2,
κ2 = number of even positive i with ri odd, ri−1 even positive, and the largestinteger j with the following property is even: for even m, i ≤ m ≤ j, rm is odd,while for odd m in the same range, rm is positive even.
In type D, first let i0 be the smallest odd integer i with ri odd if one exists. Otherwise,let i0 = ∞. Then define:
κ = number odd i with ri odd and either ri−1 even or i = i0,
κ1 = number of odd i > i0 with ri odd, ri−1 even, and either ri−2 > ri,
κ2 = number of odd i.i0 with ri odd, ri−1 even positive, and the largest integerj with the following property is odd: for even m, i ≤ m ≤ j, rm is positive even,while for odd m in the same range , rm is odd.
In type C, the definition is a bit longer. Define a string of integers i, . . . j to be relevantif
65
j > i ≥ 0,
for i < m ≤ j, rm is odd,
either i > 0 and ri is odd, or i = 0 and ri = 12(ri+2 − 1),
the string is maximal subject to the above.
Now let
ES = positive even integers i in S such that ri > 1 and i > 2 or ri−1 6= 1,FS = odd integers i in S with ri > 1,κ′S = max(#(ES ∪ FS)− (length(S)− 2), 0).
We can now list the relevant strings as S1, . . . , Sr in such a way that the ones withκ′S = 2 come first, followed by the ones with κ′S = 1, and then the ones with κ′S = 0.Enumerate the integers in ∪SES as i1, . . . is in such a way that the ones in S1 comefirst, etc. Now let κ(ia) = 1 iff a ≤ ν∗ and 0 otherwise. Also let κ(jb) = 1 iff b ≤ µ∗
and 0 otherwise. Finally, for each relevant string S, we can define
κS =∑
ia∈ES
κ(ia) +∑
jb∈FS
κ(jb).
We are now ready to decribe the multiplicity mλ′U . Let
nB = 2κ−min(ν∗, κ1)−min(µ∗, κ2),
nC =∑
S max(length(S)− 2− κS , 0),
nD = 2κ−min(µ∗, κ1)−min(ν∗, κ2) + κ3.
Proposition 5.2.12 (McGovern). Consdier the type A nilpotent orbit U = Uq. Withnotation as above, mλ′U
equals
1. 1 in type A,2. 2nB in type B,3. 2nC in type C,4. 2max(nD−2,0)in type D.
Corollary 5.2.13. Consider a spherical nilpotent orbit O and let M(Up) = O. Then
1. nB = 2κ except when p = [2n − 1, 12], or [2(n − k) − 1, 2k + 1, 1], in which case itequals 2κ− 1
2. nC = κ− 1 when q has the form [2n− k + 1, k],and is 0 otherwise,3. nD = 2κ.
Proof. In type B, both µ and ν are less than 2, except when p = [2n−1, 12], [2(n−k)−1, 2k+1, 1], [2n− 2, 2, 1], or [2n− 2k− 2, 2k + 2, 1]. In the case of the former two, min(ν∗, κ1) = 1,and in the case of all four, min(µ∗, κ2) = 0. For spherical orbits of type D, both µ and νare less than 2. Finally, in type C, relevant strings of length greater than 2 arise only whenp = [2n− k + 1, k].
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We are now ready to state a second approximation to the set of infintesimal charac-ters that should appear as infinitesimal characters of representations attached to sphericalnilpotent coadjoint orbits. As mentioned before, for a given nilpotent orbit O of a givenclassical type, this is the set of characters of the form λ′U with M(U) = O that also satisfythe condition
mλ′U ≤ |π1(O)|.We will denote this set by IC1(O), and compute it in the next proposition. We also adoptnew notation for a q-unipotent infinitesimal character by associating it with the partitionof the type A orbit that is used to compute it. For example, the orbit U[42,1] lies in thepreimage M−1(O[24] of the type C orbit with partition [24]. Then
λ′U = (32 , 3
2 , 12 , 1
2)
which we write asλ′U = [42, 1].
This expression is unique as long as the type of the orbit O is specified. In the case of veryeven orbit in type D, we take this to mean that the infinitesimal character with all termsnonnegative at tached to the orbit with numeral I and the infinitesimal character with onenegative terms attached to the orbit with numeral II.
Proposition 5.2.14. Let Op be a spherical nilpotent orbit in a classical Lie algebra g thatcorresponds to the partition p. The set IC1(Op) of infinitesimal characters attached to Op
by the above procedure is as follows:
Type A ptType BWhen p = [22k, 12n−4k+1] [2n− 2k, 2k + 1],
p = [3, 12n−2], n 6= 2 [2n− 2, 2, 1], [2n− 2, 13]p = [3, 12], [22, 1], [2, 13], [3, 12]p = [3, 22k, 12n−4k−2] k 6= n−1
2 , 0, [2n− 2k − 2, 2k + 2, 1]p = [3, 2n−1] [n2, 1]
Type CWhen p = [12n], [2n + 1]
p = [2k, 12n−2k]k 6= 2 [2n− k + 1, k − 1, 1]p = [22, 12n−4] [2n− 1, 12], [2n− 1, 2]p = [2n]n 6= 2 [n2, 1], [n + 1, n− 1, 1];p = [22] [22, 1], [3, 12], [3, 2];
Type DWhen p = [22k, 12n−4k] k 6= n
2 [2n− 2k − 1, 2k + 1],p = [2n] [n2]p = [3, 12n−3], [2n− 3, 2, 1], [2n− 3, 13]p = [3, 22k, 12n−4k−3] [2n− 2k − 3, 2k + 2, 1]
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5.3 Infinitesimal Characters of V (V , π)
Recall the character α, defined as the square root of the absolute value of the real determi-nant of the Qf action on q/qf used to define V (V, π). Suppose that α extends to a characterγ on Q. According to Section 5.1, such an extension exists whenever the set HW 1(wα) isnot empty. The first goal of this section is to decide whether and when this occurs. Thisis important as the construction of V (V, π) relies on the existence of a bundle isomorphismjγ,π defined in Chapter 2. In the setting of spherical nilpotent orbits, jγ,π exists preciselywhen there is a character γ of the parabolic Q stabilizing V that restricts to α on Qf .
The second goal of the section is to decide how well the infinitesimal characters ofV (V, π) fit within those that ought to be attached to the nilpotent orbit O. Suppose thatthe half-density bundle on G/Q is given by the character ρG/Q, and define γ′ = γ ⊗ ρ−1
G/Q.
The space V (V, π) is then a subset of IndGQ(γ′). If we write wγ for the character of γ and
ρ for the half-sum of the positive roots of G, then the associated infinitesimal character isχγ = wγ + ρ. One expects that χγ should be a character attached to O by the previoussection, that is, it should lie in the set IC1(O).
We begin with a short litany of examples of what is not true. It turns out that thesimplest solutions one would like to have for both of the above questions are not possible.
5.3.1 A Few Examples
First, we show that it is not always possible to find a character γ of Q that restricts to αon Qf . This occurs already in type A for the minimal orbit in rank 5.
Example 5.3.1. Let g = gl5 and consider the orbital variety VT associated to the standardYoung tableau
T =
1 4235
The basepoint f = fT = Ee2−e3 constructed in 4.3.5 has dense B orbit by Lemma 4.3.7.The τ -invariant of T and hence that of VT can be gleaned from Theorem 3.5.1 and equalse1 − e2, e3 − e4, e4 − e5. If Q is the parabolic stabilizing VT and L is its Levi subgroup,the τ -invariant forces l = gl2 × gl3. We can now compute the weight of the square root ofthe absolute value of the determinant of the Qf action on q/qf . According to the inductiveprocedure of Proposition 4.5.2, the weight of α is wα = −t1− t3 + t4 + t5 which we write as
wα = ((−1, 0), (−1, 1, 1))
by grouping terms that correspond to the same reductive part of the Levi. The set ofweights that restrict to wα consists of the one-parameter family
HW (wα) = wα + ε = ((−1, ε1), (−1− ε1, 1, 1)).
The weight wα + ε corresponds to a one-dimensional representation of Q iff the conditionsdescribed after Fact 5.1.1 are satisfied, that is, iff −1 = ε and − 1 − ε = 1. This is notpossible, implying that HW 1(wα) = ∅ and that there does not exists a character γ of Q
68
that restricts to α on Qf !
One can reasonably expect that this counterexample can be used to construct others inlarger groups by considering orbital varieties associated to standard Young tableaux thatcontain the above T as a subtableau. In other words, the property that HW 1(wα) = ∅should preserved by induction on tableau. Surprisingly, this too is false.
Example 5.3.2. Let g = gl6 and consider
S=
1 42 635
Then S contains T as a subtableau. Following the procedure of the previous example, wefind that wα = ((−2), (0, 0), (0, 1, 1)) which extends to a two-parameter set of weights of theform
HW (wα) = wα + ε = ((−2− ε1), (ε1, ε2), (−ε2, 1, 1)).The weight wα + ε corresonds to a one-dimensional representation γ of Q whenever ε1 =ε2 = −1. The associated infinitesimal character is then
wγ = wα + (1,−1,−1, 1, 0, 0) =(−1,−1,−1, 1, 1, 1) + (52 , 3
2 , 12 ,−1
2 ,−32 ,−5
2)=(3
2 , 12 , 1
2 ,−12 ,−1
2 ,−32).
This is precisely the infinitesimal character attached to O[22,12] by the procedure outlinedin the previous section.
One also hopes that if there does exists a character γ that restricts to α, then theassociated infinitesimal character is among the ones that ought to tbe associated to theoriginal orbit O, that is, χγ ∈ IC1(O). Unfortunately, this also fails.
Example 5.3.3. Let g = gl6 and consider the tableau
T= 1 32 54 6
The Levi of the parabolic stablizing VT is l = gl2 ⊕ gl2 ⊕ gl2. Proposition 4.5.2 impliesthat wα = ((−3
2 ,−32), (1
2 ,−12), (3
2 , 32)). The set of weights that restrict to α on tf is the
three-parameter family
HW (wα) =w(ε1, ε2, ε3) = ((−3+ε1
2 ,−3+ε22 ), (1+ε2
2 ,−1+ε32 ), (3+ε3
2 , 3+ε12 ))
.
For w(ε1, ε2, ε3) to lie in HW 1(wα), we must have ε1 = ε2 = ε3 and 1 + ε2 = −1− ε3. Thisforces ε1 = ε2 = ε3 = −1. Hence
HW 1(wα) = w(−1,−1,−1) = ((−1,−1), (0, 0), (1, 1))
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which corresponds to the character of the parabolic Q given by
γ
A1 ∗ ∗0 A2 ∗0 0 A3
=
(|A1|−2|A3|2) 1
2
The infinitesimal character of IndGQ(γ ⊗ ρ−1
G/Q) is then
χγ =(−1,−1, 0, 0,−1,−1) + (52 , 3
2 , 12 ,−1
2 ,−32 ,−5
2)=(3
2 , 12 , 1
2 ,−12 ,−1
2 ,−32).
But χγ does not lie in IC1(O[3,3]), which consists of the single infinitesimal character(1 1 0 0 − 1 − 1). In fact, χγ lies in IC1(O[4,2])!
5.3.2 Positive Results
The phenomenon of the final example above occurs only for certain model spherical orbits.For all other spherical orbits, the infinitesimal character χγ , if defined, is indeed attachedto O by the method of the previous section.
Theorem 5.3.4. Let O be a rigid, non-model spherical nilpotent orbit and consider anorbital variety V with stabilizer Q. Suppose that there exists a character γ of Q that restrictsto the character α on Qf defined as the absolute value on the real determinant of its actionon q/qf . Then the infinitesimal character χγ lies in IC1(O).
We defer the proof to its own section. We have not yet answered the questions of whenit is possible to extend the character α of Qf to a character of Q, and whether the set ofsuch extensions for a given orbit provides enough candidates whose associated infinitesimalcharacters exhaust IC1(O). Example 5.3.2 shows that it is certainly not always possiblefind an extension γ of α for every orbital variety V ⊂ O. However, there exists at leastone orbital variety within each orbit whose associated α does admit such an extension.Futhermore, there exists a sufficient number of such orbital varieties in O to account for allinfinitesimal characters in IC1(O).
Theorem 5.3.5. Let O be a rigid spherical orbit or a model orbit with n > 2. For everyχ ∈ IC1(O), there exists an orbital variety V ⊂ O satisfying
· αV extends to a character γ of Q, and· χγ = χ.
Proof. For every nilpotent orbit, we construct a set of standard tableaux. Given a nilpotentorbit O, it is always possible to construct a unique domino tableau satisfying the following:
· There exists an integer k such that ∀i ≤ k, i ∈ T 1 and i /∈ T 2,· k is maximal among all standard tableaux of shape equal to the partition correspond-
ing to O.
Write TO for the above tableau. When OT is a very even orbit in type D with Romannumeral II, define a tableau TII satisfying:
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· n− 1, n = NT1 ,
· 1, 3, 5, . . . ∈ T 1, and
· 2, 4, 6, . . . ∈ T 2.
The Levi of the stabilizing subgroup of VTO has exactly two reductive components. Wefirst examine the case where the largest element of the partition p corresponding to O is 2and the Roman numeral associated to O, if any, is I. Let [λ1, λ2] be the partition dual to p.The weight wα has form:
wα = ((c1, c1, . . . c1), (c2, c2, . . . c2))
where
(c1, c2) =
(−λ1,−λ2) in type A,(−λ1 + 2, 0) in types B and D,(−λ1 − 2, 0) in type C.
The elements of HW (wα) have the general forms w(ε1, ε2, . . . , εs) =
((c1 − ε1, c1 − ε2, . . . c1 − εs), (c2, . . . , c2 + ε2, c2 + ε1)) if O is rigid,((ε1, c1 − ε2, . . . c1 − εs), (c2 + εs, . . . , c2 + ε2, c2 + ε1)) p = [2n], n is odd in type C,((c1 − ε1, c1 − ε2, . . . c1 − εs), (c2 + εs, . . . , c2 + ε2, c2 + ε1)) otherwise.
In the first case, w(ε1, ε2, . . . , εs) ∈ HW 1(wα) iff εi = 0 for all i. In the third case,w(ε1, ε2, . . . , εs) ∈ HW 1(wα) iff εi = εj for all i and j. This produces a one-parameterfamily of weights that depends on the common value of the εi = ε. In the second case,w(ε1, ε2, . . . , εs) ∈ HW 1(wα) iff εi = εj for i, j ≥ 2 and ε1 = c1 − ε2. This again yieldsa one-parameter family of weights that depends on the common value of the εi = ε withi ≥ 2.
Whenever O is rigid, |IC1(O)| = 1 and an easy comparison with Proposition 5.2.14shows that w(0, 0, . . . 0) + ρ = IC1(O). In the third case above in types B and D,|IC1(O)| = 1 again and with ε = 0, w(0, 0, . . . 0) + ρ = IC1(O). In type C, whenn > 2, |IC1(O)| = 2. Note that w(−1,−1, . . . − 1) 6= w(0, 0, . . . 0) and it is an easy checkthat w(−1,−1, . . .− 1) + ρ,w(0, 0, . . . 0) + ρ = IC1(O). The second case is similar. Nowconsider the case when O is very even in type D with numeral II, and consider the orbitalvariety VTII
. We find that
wα = 12((−2n + 2,−2n + 2,−2n + 4), (−2n + 4,−2n + 6), . . . , (−4,−2), (−2, 0), (0)).
The elements in HW (wα) have the form w(β, ε1, . . . , εs) = 12(−2n + 2 + β,−2n + 2 +
β,−2n + 4− ε1), (−2n + 4 + ε1, . . . (−2 + εs−1,−εs), (εs)). The set of elements in HW 1(wα)is a one-parameter family, consisting of w(εs) = 1
2((. . . (−4 + εs,−4 + εs), (−εs,−εs), (εs)).When n is odd, let εs = −1 and when n is even, let εs = 3. Inductively, it is now easy toshow that w(εs) + ρ = (n−1
2 , n−12 , n−3
2 , n−32 , . . . , 3, 3, 1,−1).
Now assume that p has largest part 3 and contains parts of size 2. Then according toProposition 4.5.2,
wα = ((c1, c1, . . . , c1 − 1), (c2, c2, . . . c2))
where (c1, c2) = (−λ1 + 1, 0) in both tyles B and D. The elements in HW (wα) have the
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general form
w(ε1, . . . εs+2) = ((c1 − ε1, . . . c1 − εs, c1 − εs+1), (c2 + εs+2, c2 + εs, . . . c2 + ε1))
According to Fact 5.1.1, w(ε1, . . . εs+2) ∈ HW 1(wα) iff εi = 0 for all i ≤ s and s + 2, andεs+1 = −1. Furthermore, w(0, 0, . . . , 0,−1, 0) + ρ) = IC1(O).
Now if p has no parts of size two and n > 2, then O is neither rigid nor model, but thesame result holds. We find that
wα = ((c1)(c2, . . . c2))
and the general form of the elements in HW (wα) is
w(ε1, ε2) = ((c1 − ε1)(ε2, 0, 0 . . . 0))
where c1 = −λ1. Now w(ε1, ε2) lies in HW 1(O) iff ε2 = 0. It is an easy check thatw(1, 0) + ρ,w(0, 0) + ρ = IC1(O). This finishes the proof of the theorem.
5.3.3 Proof of Theorem 5.3.4
We begin with an example.
Example 5.3.6. Let g = gl7 and let O[4,3] be the nilpotent orbit corresponding to the parti-tion [4, 3]. Consider the standard Young tableau
T=
1 42 53 67
The orbital variety VT has stabilizer Q with Levi L whose Lie algebra is l = gl4 ⊕ gl3.We would like to know that if γ is a character of Q that restricts to α on Qf , then wγ + ρlies in IC1(O). By Proposition 4.5.2 and the analysis of Section 5.1,
wα = ((−32 ,−1,−1,−1), (3
2 , 32 , 3
2)),
and
HW (wα) = w(ε1, ε2, ε3) = ((−32 ,−2+ε1
2 ,−2+ε22 ,−2+ε3
2 ), (3+ε32 , 3+ε2
2 , 3+ε12 )).
The conditions following Fact 5.1.1 now imply that w(ε1, ε2, ε3) ∈ HW 1(wα) iff εi = 1 forall i. Hence wγ must equal w(1, 1, 1) and
wγ + ρ = (−32 ,−3
2 ,−32 ,−3
2 , 2, 2, 2) + (3, 2, 1, 0,−1,−2,−3)= (3
2 , 12 ,−1
2 ,−32 , 1, 0,−1) ∈ IC1(O[4,3]),
as desired. Now note that
wα + ρ = w(0, 0, 0) + ρ = (32 , 1, 0,−1, 1
2 ,−12 ,−3
2).
72
While wα does not correspond to a character of Q, this is nevertheless a permutation ofwγ + ρ and also lies in IC1(O[4,3]). This observation suggests an approach to our problem.We will prove:
Lemma 5.3.7. Suppose that we are in the setting of Theorem 5.3.4. Then there exists aweight wβ such that
· wβ + ρ ∈ IC1(O), and· wβ + ρ is in the same Weyl group orbit as wγ + ρ.
The lemma implies that wγ + ρ ∈ IC1(O), proving Theorem 5.3.4.We will first shed some light on the method of our proof. As in our examples,the form
of a general element of HW (wα) depends on a number of independent variables εii≤s
and can be written as w(ε1, ε2, . . . εs). The general form can also be expressed as b =(bn bn−1 . . . b2 b1). Each entry bi can be a constant or it may depend on a single independentvariable, as prescribed by the conditions following Fact 5.1.1. At most two entries candepend on the same independent variable. We can divide the entries of b into disjointmaximal strings of entries of the form bl, bl−1, . . . bk which satisfy:
· bl and bk both depend on the same independent variable, and· there is no pair (l′, k′) such that bl′ and bk′ both depend on the same independent
variable and additionally l′ > l and k′ < k.
For such a maximal string, call I = (k, k + 1, . . . , l) a dependent interval of b. It is an easyconsequence of Fact 4.5.1 that if i lies in a dependent interval, the entry bi is not constant.If i ∈ I and bi depends on the variable εNi , we will say that εNi corresponds to I. Note thateach εi for i ≤ s corresponds to one and only one dependent interval I. Dependent intervalscome in two flavors:
· ∀i ≤ l−k+12 , the entries bl−i and bk−i depend on εNi for some integer Ni ≤ s,
· there exists an non-negative integer i ≤ l−k+12 such that bl−i and bki depend on
different variables.
We will call dependent intervals of the first type simple. For each simple dependent intervalI = (k . . . l), we define a permutation σI as a product of transpositions by
σI =∏
i<l−k2
(l − i k+l2 + i).
The permutation σI simply interchanges the first (l−k+1)/2 entries of I with the second setof (l−k +1)/2 entries, preserving the relative order of elements in each set. By hypothesis,we know that there exists a character γ of Q that restricts to α on Qf . Hence there exists aconstant ci for each variable εi such that w(c1, c2, . . . ) ∈ HW (wα) that equals wγ . If thereexists a ci 6= 0 that corresponds to the dependent interval I, we say that I is non-zero.
Example. 5.3.6 (Again) The above example contains only one dependent interval. It equalsI = (1, . . . , 6) and corresponds to the entries
(b6 b5 b4 b3 b2 b1) = (−2+ε12 ,−2+ε2
2 ,−2+ε32 , 3+ε3
2 , 3+ε22 , 3+ε1
2 ).
73
In fact, I is a simple dependent interval, and
σI = (3 6) (5 2) (4 1)
Now note that if we write wα +ρ as (c7 c6 . . . c1), then wγ +ρ = (cσ(7) cσ(6) . . . cσ(1)). Hence,at least in this case, we have produced a method of describing the permutation relatingwγ + ρ and wα + ρ.
In general, after describing the weight wβ, we will show that if all non-zero dependentintervals in HW (wα) are simple, then
Lemma 1. wβ + ρ ∈ IC1(O), andLemma 2. wγ +ρ = σ(wβ +ρ), where σ is the product of the σI taken over all non-zero
simple dependent intervals I and acts by permuting the order of the entriesof the weights.
Furthermore, we will show that
Lemma 3. a non-zero non-simple dependent interval cannot exist under the hypothesesof the Theorem.
We first describe wβ.
Definition 5.3.8. We define wβ inductively. Let vδ = wδ − ι(w↓δ ), for δ = α or β. Then let
vβ = vα +
−(n + 2)T1 Case (N1), X = C−(n + 1)T1 Case (N2), X = CT1 Case (N3), X = B, DT3 Case (*), X = B.
Fact 4.5.1 implies that wβ ∈ HW (wα). Because of the hypotheses of the Theorem, we knowthat there is a weight wγ ∈ HW 1(wα). If we write a general element of HW (α) as
w(ε1, . . . εs) = (bn, bn−1, . . . , b1),
then there exists constants c1, . . . cs such that w(c1, . . . cs) = wγ . Because O is rigid, thereexists at least one entry bp that is constant. Note that it does not belong to any dependentinterval. We will prove:
Fact A. If bp is adjacent to a non-zero non-simple dependent interval, then there areno constants c1, . . . cs such that w(c1, . . . cs) ∈ HW 1(wα).
Fact B. If I1 = (k1, . . . l1) is a non-zero non-simple dependent interval that is adjacentto a simple dependent interval I2 = (k2, . . . l2), then there are no constantsc1, . . . cs such that w(c1, . . . cs) ∈ HW 1(wα).
If w(ε1, . . . εs) contains a non-zero non-simple dependent interval, it must contain at leastone that is adjacent to either a simple dependent interval or a constant. Facts A and Bthen provide a contradiction, proving Lemma 3.
74
Fact C. If bp is adjacent to a non-zero simple dependent interval I = (k, . . . l), then
σI ((wβ + ρ)l . . . (wβ + ρ)k) = ((wγ + ρ)l . . . (wγ + ρ)k)
Fact D. If I1 = (k1, . . . l1) is a non-zero simple dependent interval that is adjacent toeither a simple dependent interval or a zero non-simple dependent intervalI2 = (k2, . . . l2), then
σI ((wβ + ρ)l1 . . . (wβ + ρ)k1) = ((wγ + ρ)l1 . . . (wγ + ρ)k1)
Lemma 3 shows that w(ε1, . . . εs) consists solely of simple dependent intervals and con-stants. For an integer i that either lies in a zero dependent interval or whose correspondingentry is a constant, we know that (wγ + ρ)i = (wβ + ρ)i. If, however, i lies in a non-zerodependent interval, Facts C and D show that (wγ +ρ)i = (wβ +ρ)σ(i), which implies Lemma2. It remains to prove Lemma 1 and the four Facts.
Proof of Lemma 1. We would like to show that wβ ∈ IC1(O). Let S = IC1(O) −ι(IC1(O↓)) and define w = wβ + ρ − ι(w↓β − ρ↓). It is easy to verify the lemma for smalln. By induction, it is enough to show that w ∈ S. The proof in Type C includes all theessential elements of the general proof, and is particularly easy to state. We detail eachinductive case.
(C1) Proposition 4.5.2 implies that w = (λ1, 0, . . . , 0). Recall the character notation ofSection 5.2. Note that k = λ2 and that the difference w = [2n−k +1, k−1, 1]− [2n−2− k + 1, k − 1, 1] always lies in the one or two element set S.
(C2) This time, w = (λ2, 0, . . . , 0). Again using the notation of Section 5.2, we find thatw = [2n− k + 1, k − 1, 1]− [2n− k + 1, k − 3, 1] always lies in S.
(N1) Here, w = (n− 2, 0, . . . , 0). Using the notation of Section 5.2, w = [n + 1, n− 1, 1]−[n + 1, n− 3, 1], which lies in S by 5.2.14.
(N2) Here, w = (n−1, 0, . . . , 0). Using the notation of Section 5.2, w = [n2, 1]− [n, n−2, 1],which lies in S by 5.2.14.
This accounts for all the cases that arise in type C. For the other classical types, the proofrequires the same inductive verification and follows inductively, except in one instance.When the partition corresponding to O has no parts of size 1, then wβ /∈ IC1(O). Thisdoes not contradict the Lemma, as O is not rigid, but it does complicate the induction step.If W is an orbital variety such that W↓ ⊂ O, then the associated wβ again lies in IC1(O).This proves Lemma 1.
Proof of Fact A. Write I = [k, k + 1, . . . l] for the non-zero non-simple dependent intervaladjacent to bp, and further assume that p = k − 1. The proof for the other possibility issymmetric. Utilizing the notation suggested by Fact 5.1.2, the entries of I must have theform
bl, bl−1, . . . bm1), (bm1−1, . . . , bm2) . . . (bmq−1, . . . bk.
We examine two possiblities. Either
75
· l, k − 1 ∈ T 2 \ T 1 and k ∈ T 1 \ T 2, or
· l, k ∈ NT1 .
The precise statement of the first case follows from the non-zero assumption on I; if ak−1
was constant and k− 1 was contained in T 1, then I could not be non-zero. Hence considerthe first possiblity. The entries of I must then have the form
al − ε, al−1 − ε, . . . , am1 − ε), (am1−1 + ε, . . . am2 − ε), . . . , (amq−1 + ε, . . . ak + ε
for some ε since they must correspond to a weight in HW 1(wα). Because all the entriesgrouped within parentheses must equal each other, according to Fact 5.1.1 this gives us theconditions
ak + ε = ak−1 (5.3)ami+1 + ε = ami+1 − ε (5.4)
for all i < q, which translate to
ε = ak−1 − ak
=ami+1−ami+1
2
(5.5)
for all i < q. We would like to show that these conditions are impossible to satisfy. Proposi-tion 4.5.2 and Definition 5.3.8 give us a description of each of the ai. We restrict the proofto case C, which contains all the elements of the general proof.
Let [λ1(i), λ2(i)] be the partition dual to shape T (i). Proposition 4.5.2 implies that
ak−1 = −λ1(k − 1) + 2ak = −λ2(k − 1) + 2
am2 = −λ1(m2) + 2am1+1 = −λ2(m1 + 1).
Equations 5.5 translate to
ε = −λ1(k − 1) + λ2(k − 1)
= −λ1(m2)+2+λ2(m1+1)2 .
(5.6)
However, λ1(k−1)−λ2(k−1) = λ1(l)−λ2(l) because I is a dependent interval. Furthermore,the form of the entries in I implies that λ2(l) > λ2(m1 + 1) and λ1(l) < λ1(m2). But thisimplies that it is impossible to satisfy equations 5.6. Hence we cannot find constants ci sothat w(c1, . . . cs) ∈ HW 1(wα). The only difference in proof for the other classical types arethe precise values for the ai.
Now suppose that k, l ∈ NT1 . The entries corresponding to the interval I must have
the form
al + β, al−1 − ε, . . . , am1 − ε), (am1−1 + ε, . . . am2 − ε), . . . , (amq−1 + ε, . . . ak+1 + ε, ak + β.
76
Because I is non-simple, this means that the interval k + 1, . . . , l − 1 cannot be simpleeither. This time, we need to solve the equations
ε = ak + β − ak+1
= al−1 − al − β
=ami+1−ami+1
2
(5.7)
First, we find that β = (al−1−al)+(ak+1−ak)2 . This means that we still need to solve
ε = (al−1−al)−(ak+1−ak)2
=ami+1−ami+1
2
(5.8)
By an analysis similar to the above, divided into each classical type, 5.8 again cannot besatisfied, and Fact A holds.
Proof of Fact B. If I2 is a zero interval, then the proof is identical to the proof of Fact A,as the only property we needed was the expression for the term ak1−1, which is the same inthe zero case. Now assume that I1 is to the left of I2 in the coordinate expression for wγ ofthis section; the other possibility has a symmetric proof. There are again two cases in theproof. First assume that k, n /∈ NT
1 . The two intervals must then have the form
al1 − ε, al1−1 − ε, . . . , am1 − ε), (am1−1 + ε, . . . am2 − ε), . . . , (amq−1 + ε, . . . ak2 + ε
andal2 − µ, al2−1 − µ, . . . , am′ − µ), (am′−1 + µ, . . . , ak2 + µ
with the additional restriction that al2−µ = ak1+ε. Write ρ in coordinates as (ρn, ρn−1, . . . , ρ1).The proof of Fact D and Fact C imply that either µ = 0, or µ = al2 − am′ + ρl2 − ρm′ . Thefirst possiblity was considered above. As for the second, following the outline of the proofof Fact A, we would like to solve the equations
ε = al2 − µ− ak
=ami+1−ami+1
2
(5.9)
for all i < q. In each of the classical types, Proposition 4.5.2 gives us values for the ai, andwe can similarly give an explicit description of ρ. In a manner similar to the proof of FactA, we can now show that a solution to 5.9 does not exists. A similar analysis works for thecase when k, l ∈ NT
1 and Fact B holds.
Proof of Fact C. Assume that bp = bk−1 as the proof for the other possiblity in symmetric.Utilizing the notation suggested by Fact 5.1.2, the entries of I must have the form
bl, bl−1, . . . bm), (bm−1, . . . bk.
As in the proof of Fact A, there are two possiblities. Either l, k−1 ∈ T 2\T 1 and k ∈ T 1\T 2,or l, k ∈ NT
1 . The precise statement of the first case is due to the non-zero assumptionon I. We examine the first case. The second in analogous. Write ρ in coordinates as
77
(ρn, . . . , ρ1). The entries of wγ have the form
al − ε, al−1 − ε, . . . , am − ε), (am−1 + ε, . . . , ak + ε
where entries grouped by parentheses must equal since wγ ∈ HW 1(wα). This conditionfurther forces ak−1 = ak + ε, or in other words,
ε = ak − ak−1 (5.10)
After examining the definition of the permutation σI , we need to verify that
al+i − ε + ρl+1 = am+i + ρm+i (5.11)
for all i < (l − k)/2. This then implies Fact C. We proceed for type G = A. First of all,ρl+i = n + 1− 2(l + i). Hence we would like to know whether the equality
ali − ε + n + 1− 2(l + i) = am+i + n + 1− 2(m + i)
holds. Proposition 4.5.2 implies that al+i = al and am+i = ak for all of the above i.The above equation becomes al − ak + k − l + 1 = ε. This is possible iff this equation iscompatible with 5.10. To verify this, we note that repeated application of Proposition 4.5.2implies ak−1 = −λ1(k) + (λ1−λ1(k)) = λ1− 2λ1(k) which also equals al + (k− l + 1). Thisimplies that al − ak + (k − l + 1) = ak−1 − ak, as desired. Hence Fact C holds in type A.The proof for the groups of other types are analogous, only complicated by the appearanceof horizontal dominos. However, dominos falling in cases (N2) or (N3) do not affect thedependent intervals because of Fact 4.5.1. Case (N1) is dealt with precisely as in the proofof Fact A.
Proof of Fact D. If I2 is a zero dependent interval, then the proof is identical to the proofof Fact C. We would like to show that in fact, if I1 is a non-zero simple dependent interval,then I2 must be a zero dependent interval. We can assume that I1 is to the left of I2 in thecoordinate notation we have grown accustomed to. As in Fact C, the interval I1 has theform
al1 − ε, al1−1 − ε, . . . , am − ε), (am−1 + ε, . . . , ak1 + ε
while the interval I2 has the form
al2 − µ, al2−1 − µ, . . . , am′ − µ), (am′−1 + µ, . . . , ak2 + µ
with the additional constraint that l2− 1 = k2. We would like to show that µ = 0. Becausewγ ∈ HW (wα), we know that al2 −µ = akl
+ ε. But our proof of Fact C implies that infact,al2 = akl
+ ε, forcing µ to be zero. This implies Fact D.
5.4 An Example
We find wγ for all orbital varieties that arise in D4.
78
Sphe
rica
lO
rbit
alV
arie
ties
inD
4
Part
itio
nofO
2·I
C1(O
)V T
2·w
α2·H
W(w
α)
2·H
W1(w
α)
2·H
W1(w
α)+
2ρ
[18]
(6,4
,2,0
)
1 2 3 4
((0,0
,0,0
))((
0,0
,0,0
))((
0,0
,0,0
))(6
,4,2
,0)
[22,1
4]
(4,2
,2,0
)
12
3 4
((−2
,−2,0
),0)
((−2
,−2,−
ε 1),
ε 1)
((−2
,−2,−
2),
2)
(4,2
,0,2
)
13
2 4
((−2
,−2),
(0,0
))((−2
,−2−
ε 1),
(ε1,0
))((−2
,−2),
(0,0
))(4
,2,2
,0)
14
2 3
(−4,(
0,0
,0))
(−4−
ε 1,(
ε 1,0
,0))
(−4,(
0,0
,0))
(2,4
,2,0
)
1 2
3 4
((−2
,−2,0
,0))
((−2
,−2,ε
1,ε
1))
((−2
,−2,−
2,−
2))
(4,2
,0,2
)
[24] I
(3,3
,1,1
)
12
34
(−2,(−2
,0),
0)
(−2−
ε 1,(−2
+ε 1
,−ε 2
),ε 2
)(−
2−
ε 1,(−2
+ε 1
,−2
+ε 1
),2−
ε 1)
(4−
ε 1,2−
ε 1,ε
1,2−
ε 1)
ε 1=
1
13
24
((−2
,−2),
(0,0
))((−2−
ε 1,−
2−
ε 2),
(ε2,ε
1))
((−2
−ε 1
,−2−
ε 1),
(ε1,ε
1))
(4−
ε 1,2
−ε 1
,2+
ε 1,ε
1)
ε 1=
1
1 2 3 4((−2
,−2,0
,0))
((−2
+ε 1
,−2
+ε 1
,ε2,ε
2))
((−2
+ε 1
,−2
+ε 1
,−2
+ε 1
,−2
+ε 1
,))
(4,2
,0,2
)ε 1
=0
79
Sphe
rica
lO
rbit
alV
arie
ties
inD
4(c
ont.
)
Part
itio
nofO
2·I
C1(O
)V T
2·w
α2·H
W(w
α)
2·H
W1(w
α)
2·H
W1(w
α)+
2ρ
[24] I
I(3
,3,1
,−1)
1
23
4((−2
,−2),
(0,0
))((−2
+ε 1
,−2−
ε 2),
(ε2,ε
1))
((−2
−ε 1
,−2−
ε 1),
(−ε 1
,ε1))
(4+
ε 1,2
+ε 1
,2−
ε 1,ε
1)
ε 1=
−1
12
3 4((−2
,−2,0
),0)
((−2
+ε 1
,−2+
ε 1,−
ε 2),
ε 2)
((−2
+ε 1
,−2+
ε 1,−
2+
ε 1),
2−
ε 1)
(4+
ε 1,2
+ε 1
,ε1,2−
ε 1)
ε 1=
−1
1 2
34
(−2,(−2
,0,0
))(−
2−
ε 1,(−2
+ε 1
,ε2,ε
2))
(−2−
ε 1,(−2
+ε 1
,−2+
ε 1,−
2+
ε 1))
(4−
ε 1,2
+ε 1
,ε1,−
2+
ε 1)
a=−1
[3,1
5]
(4,2
,1,0
)(4
,2,0
,0)
12
3 4
((−2
,−2,−
2),
0)
((−2
,−2,ε
1),
ε 2)
((−2
,−2,−
2),
ε 2)
(4,2
,0,ε
2)
ε 2=
0,
1
13
2 4
((−2
,−4),
(0,0
))((−2
,ε1),
(ε2,0
))((−2
,−2),
(0,0
))(4
,2,2
,0)∗
14
2 3
(−6,(
0,0
,0))
(−6
+ε 1
,(ε 2
,0,0
))(−
6+
ε 1,(
0,0
,0))
(ε1,4
,2,0
)ε 1
=0,1
[3,2
2,1
](3
,2,1
,0)
12
4
3((−3
,−4),
(0,0
))((−3
,ε1),
(ε2,0
))((−3
,−3),
(0,0
))(3
,1,2
,0)
13
4
2(−
3,(−2
,−2),
0)
(−3−
ε 1,(−2
+ε 1
,ε1),
ε 2)
(−3−
ε 1,(−2
+ε 1
,−2−
ε 1),
ε 2)
(3−
ε 1,2
+ε 1
,ε1,ε
2)
ε 1=
0,
ε 2=
1
80
Bibliography
[Barbasch] Dan Barbasch, The unitary dual for complex classical Liegroups. Invent. Math. 96 (1989), no. 1, 103–176.
[Barbasch-Vogan] D. Barbasch and D. Vogan, Primitive ideals and orbital integralsin complex classical groups. Math. Ann. 259 (1982), no. 2, 153–199.
[Borho-Brylinski] W. Borho and J.-L. Brylinski, Differential operators on homo-geneous spaces. III. Characteristic varieties of Harish-Chandramodules and of primitive ideals. Invent. Math. 80 (1985), no. 1,1–68.
[Carre-Leclerc] C. Carre and B. Leclerc, Splitting the Square of a Schur Func-tion into its Symmetric and Antisymmetric Parts, J. AlgebraicCombin. 4 (1995), no. 3, 201–231.
[Collingwood-McGovern] D. Collingwood and W. M. McGovern. Nilpotent orbits insemisimple Lie algebras. Van Nostrand Reinhold MathematicsSeries. Van Nostrand Reinhold Co., New York, 1993.
[Garfinkle1] D. Garfinkle, On the classification of primitive ideals for complexclassical Lie algebras. I. Compositio Math. 75 (1990), no. 2, 135–169.
[Garfinkle2] D. Garfinkle, On the classification of primitive ideals for complexclassical Lie algebra. II. Compositio Math. 81 (1992), no. 3, 307–336.
[Garfinkle3] D. Garfinkle, On the classification of primitive ideals for complexLie algebras III, Compositio Math., 88 (1993), 187–234
[Gerstenhaber] M. Gerstenhaber, Dominance over the classical groups. Ann. ofMath. (2) 74 1961 532–569.
[Ginsburg] V. Ginsburg, G-modules, Springer’s representations and bivari-ant Chern classes, Adv. in Math, 61 (1986), 1-48.
[Graham-Vogan] W. Graham and D. Vogan, Geometric quantization for nilpo-tent coadjoint orbits, Geometry and representation theory of realand p-adic groups, J. Tirao, D. Vogan, and J. Wolf Birkhauser,Boston-Basel-Berlin, 1998 .
81
[Guillemin-Sternberg] V. Guillemin and S. Sternberg, A Generalization of the Notionof Polarization, Ann. Glob. Analysis and Geometry, 4 (1986),327-347.
[Hotta] R. Hotta, Weyl Group Representations, Tohoku Math. Journal,36 (1984), 49-74.
[Kirillov] A. Kirillov, Unitary representations of nilpotent Lie groups, Us-pehi Mat. Nauk., 17 (1962), 57–110 .
[Kostant] B. Kostant, Symplectic Spinors, 139-152 in Geometria Simplet-tica e Fisica Matematica. Symposia MAthematica XIV, IstitutoNazionale di Alta Matematica. Academic Press, London andNew York, 1974.
[Joseph] A. Joseph, On the characteristic polynomials of orbital varieties.Ann. Sci. Ecole Norm. Sup. (4) 22 (1989), no. 4, 569–603.
[Joseph2] A. Joseph, Orbital Varieties, Goldie Rank Polynomials, and Uni-tary Highest Weight Modules, Algebraic and Analytic Methodsin Representation Theory, B. Ørsted and H. Schlichtkrull eds.,San Diego, 1997.
[van Leeuwen] M.A. van Leeuwen, The Robinson-Schensted andSchutzenberger algorithms and interpretations, Computa-tional aspects of Lie group representations and related topics,Amsterdam, (1990), 65-88 .
[Lusztig] G. Lusztig, A class of irreducible representations of a Weylgroup. Nederl. Akad. Wetensch. Indag. Math. 41 (1979), no. 3,323–335.
[McGovern] W. McGovern, Completely prime maximal ideals and quanti-zation, Memoirs of the AMS American Mathematical Society,Providence, Rhode Island , 1994
[McGovern2] W. McGovern, On the Spaltenstein-Steinberg Map for ClassicalLie Algebras, Comm. Algebra, 27 (1999), 2979–2993.
[McGovern3] W. M. McGovern, Rings of regular functions on nilpotent orbits.II. Model algebras and orbits. Comm. Algebra 22 (1994), no. 3,765–772
[McGovern4] W. M. McGovern, Rings of regular functions on nilpotent orbitsand their covers. Invent. Math. 97 (1989), no. 1, 209–217.
[Melnikov] A. Melnikov, Orbital Varieties in sln and the Smith conjecture,Journal of Algebra, 200 (1998), 1-31.
[Ozeki-Wakimoto] H. Ozeki and M. Wakimoto, On Polarizations of certain Homo-geneous Spaces, Hiroshima Math. J., 2 (1972), 445-482.
82
[Panyushev] D. Panyushev, Complexity and nilpotent orbits. ManuscriptaMath. 83 (1994), no. 3-4, 223–237.
[Panyushev2] D. Panyushev, On spherical nilpotent orbits and beyond. Ann.Inst. Fourier (Grenoble) 49 (1999), no. 5, 1453–1476.
[Spaltenstein] N. Spaltenstein, Classes Unipotentes et Sous-Groupes de Borel,Lecture Notes in Mathematics, 946 Springer-Verlag, New York,1982
[Springer-Steinberg] T. A. Springer and R. Steinberg, Conjugacy classes. 1970 Semi-nar on Algebraic Groups and Related Finite Groups (The Insti-tute for Advanced Study, Princeton, N.J., 1968/69) pp. 167–266Lecture Notes in Mathematics, Vol. 131 Springer, Berlin
[Springer] T.A. Springer, Trigonometric sums, Green functions of finitegroups and representations of Weyl groups. Invent. Math. 36(1976), 173–207
[Stanton-White] D. Stanton and D. White, A Schensted Algorithm for Rim HookTableaux, Journal of Combinatorial Theory, 40 (1985), 211-247.
[Steinberg] R. Steinberg, An occurrence of the Robinson-Schensted corre-spondence, J. Algebra, 113 (1988), no. 2, 523–528.
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