Chao-Ping Dongmath.sjtu.edu.cn/conference/Bannai/2018/data/20180911A/slides.pdf · For X unitary,...

Spin norm: combinatorics and representations

Chao-Ping Dong

Institute of MathematicsHunan University

September 11, 2018

Chao-Ping Dong (HNU) Spin norm September 11, 2018 1 / 38

Overview

This talk aims to introduce the following preprints in 2017.J. Ding, C.-P. Dong, Unitary representations with Diraccohomology: a finiteness result, arXiv:1702.01876.C.-P. Dong, Unitary representations with Dirac cohomology forcomplex E6, arXiv:1707.01380.C.-P. Dong, Unitary representations with Dirac cohomology:finiteness in the real case, arXiv:1708.00383.

For a real reductive Lie group G(R), we report a finiteness theorem for

the structure for G(R)d—all the irreducible unitary Harish-Chandra

modules (up to equivalence) for G(R) with non-zero Dirac cohomology.


Outline

1 Combinatorics

2 Representations


A game

The following problem was given at the International Olympiad ofMathematics in 1986.Five integers with positive sum are arranged on a circle. Thefollowing game is played. If there is at least one negative number,the player may pick up one of them, add it to its neighbors, andreverse its sign. The game terminates when all the numbers arenonnegative. Prove that this game must always terminate.


Elementary Solution (Demetres Chrisofides)

Take T = (a− c)2 + (b − d)2 + (c − e)2 + (d − a)2 + (e − b)2.After replacing a,b, c by a + b,−b,b + c, we get

T ′ = T + 2b(a + b + c + d + e) < T .


Some examples

The underlying structure: Coxeter group of A4.e.g. consider A2: [−1,−1] 7→ [1,−2] 7→ [−1,2] 7→ [1,1].The Cartan matrix [

2 −1−1 2

]


The A2 picture


Some examples (continued)

e.g. consider G2:

[−1,−1] 7→ [−4,1] 7→ [4,−3] 7→ [−5,3] 7→ [5,−2] 7→[−1,2] 7→ [1,1].

The Cartan matrix [2 −3−1 2

]


The G2 picture


The underlying algorithm

Given an arbitrary integral weight

λ =∑

i

λi$i = [λ1, . . . , λl ].

How to effectively conjugate it to the dominant Weyl chamber?The algorithm: select an arbitrary index i such that λi < 0, thenapply the simple reflection si ; continue this process whennecessary.si(λ) = λ− λi

∑lj=1 aji$j . It uses the i-th column of the Cartan

matrix A.Why is the algorithm effective? See Theorem 4.3.1 of A. Björner,F. Brenti, Combinatorics of Coxeter groups, GTM 231, Springer,New York (2005).


Spin norm (for complex Lie groups)

For any dominant weight µ. The spin norm of µ:

‖µ‖spin = ‖µ− ρ+ ρ‖.

Here ρ = $1 + · · ·+$l = [1, . . . ,1];and µ− ρ is the unique dominant weight to which µ− ρ isconjugate.e.g. −ρ = ρ. Thus ‖0‖spin = ‖2ρ‖. Moreover, ‖2ρ‖spin = ‖2ρ‖,and ‖ρ‖spin = ‖ρ‖Note that ‖µ‖spin ≥ ‖µ‖, and equality holds if and only if µ isregular. It becomes subtle and interesting when µ is irregular.This notion was raised in my 2011 thesis. Origin: Vµ ⊗ Vρ.


Pencils

The pencil starting with µ:

P(µ) = µ+ nβ | n ∈ Z≥0,

where β is the highest root.e.g. P(0) consists of 0, β, 2β, · · · .Reference: D. Vogan, Singular unitary representations,Noncommutative harmonic analysis and Lie groups (Marseille,1980), 506–535.Motivation: describe the K -types pattern of an infinite-dimensionalrepresentation.


The u-small convex hull (for complex Lie groups)

The u-small convex hull: the convex hull of the W -orbit of 2ρ.Reference: S. Salamanca-Riba, D. Vogan, On the classification ofunitary representations of reductive Lie groups, Ann. of Math. 148(1998), 1067–1133.Motivation: describe a unifying conjecture on the shape of theunitary dual.Pavle’s 2010 Nankai U Lecture: a work joint with Prof. Renard.


The complex G2 case, where β = $2


Distribution of spin norm along pencils

TheoremLet g be any finite-dimensional complex simple Lie algebra. The spinnorm increases strictly along any pencil once it goes beyond theu-small convex hull.

Reference: C.-P. Dong, Spin norm, pencils, and the u-small convexhull, Proc. Amer. Math. Soc. 144 (2016), 999–1013.

RemarkClassical groups: two weeks; Exceptional groups: about two years.


Outline

1 Combinatorics

2 Representations


Dirac operator in physics

In 1928, by using matrix algebra, Dirac discovered the laternamed Dirac operator in his description of the wave function ofthe spin−1/2 massive particles such as electrons and quarks.Reference: P. Dirac, The quantum theory of the electron, Proc.Roy. Soc. London Ser. A 117 (1928), 610–624.Atiyah’s remark: using Hamilton quaternionsH = ±1,±i ,±j ,±k, ij = −ji , i2 = −1, we have

−∆ = − ∂2

∂x2 −∂2

∂y2 −∂2

∂z2 = (i∂

∂x+ j

∂

∂y+ k

∂

∂z)2.


Paul Dirac

Figure 1: Paul Dirac in 1933.


Dirac operator in Lie theory

In 1972, Parthasarthy introduced the Dirac operator for G andsuccessfully used it to construct most of the discrete series.Reference: R. Parthasarathy, Dirac operators and the discreteseries, Ann. of Math. 96 (1972), 1–30.Let Zini=1 be an o.n.b. of p0 w.r.t. B. The algebraic Diracoperator is defined as:

D :=n∑

i=1

Zi ⊗ Zi ∈ U(g)⊗ C(p).

Note that we have

D2 = −(Ωg ⊗ 1 + ‖ρ‖2) + (Ωk∆+ ‖ρc‖2).


Dirac cohomology

Let X be a (g,K )-module. Then

D : X ⊗ S → X ⊗ S,

and in the 1997 MIT Lie groups seminar, Vogan introduced theDirac cohomology of X to be

HD(X ) = Ker D/(Ker D ∩ Im D).

Moreover, Vogan conjectured that when HD(X ) is nonzero, itshould reveal the infinitesimal character of X .This conjecture was verified by Huang and Pandžic in 2002.Reference: J.-S. Huang, P. Pandžic, Dirac cohomology, unitaryrepresentations and a proof of a conjecture of Vogan, J. Amer.Math. Soc. 15 (2002), 185–202.


The classification problem

Problem: classify all the equivalence classes of irreducible unitaryrepresentations with non-zero Dirac cohomology.For X unitary, we have HD(X ) = Ker D = Ker D2.

These representations are extreme ones among the unitary dualin the following sense: they are exactly the ones on whichParthasarathy’s Dirac inequality becomes equality.Cohomological induction is an important way of constructingunitary representations.When the inducing module is one-dimensional, we meetAq(λ)-modules. Under the admissible condition,J.-S. Huang, Y.-F. Kang, P. Pandžic, Dirac cohomology of someHarish-Chandra modules, Transform. Groups. 14 (2009),163–173.


Within the good range

The inducing module could be infinite-dimensional. Under thegood range condition,C.-P. Dong, J.-S. Huang, Dirac cohomology of cohomologicallyinduced modules for reductive Lie groups, Amer. J. Math. 137(2015), 37–60.P. Pandžic, Dirac cohomology and the bottom layer K-types, Glas.Mat. Ser. III 45 (65) (2010), no. 2, 453–460.What will happen beyond the good range?This point has perplexed us for quite a long time. There could beno unifying formula...


Complex Lie groups

Let G be a complex connected Lie group, K, H.A powerful reduction: J(λ,−sλ), s ∈W is an involution, 2λ isdominant integral. Here µ := λ+ sλ is the LKT.Reference: D. Barbasch, P. Pandžic, Dirac cohomology andunipotent representations of complex groups, Noncommutativegeometry and global analysis, 1–22, Contemp. Math., 546, Amer.Math. Soc., 2011.Fix λ (say, = ρ/2), and let s varies.


Complex Lie groups (continued)

Idea: fix an arbitrary involution s, and let λ varies.We call Λ(s) and the corresponding representations J(λ,−sλ) ans-family, where

Λ(s) := λ = [λ1, . . . , λl ] | 2λi ∈ P and λ+ sλ is integral .

For any involution s ∈W , put I(s) = i | s($i) = $i.i ∈ I(s) if and only if sαi does not occur in some reducedexpression of s, if and only if sαi does not occur in any reducedexpression of s. Thus s ∈ 〈sj | j /∈ I(s)〉.e.g., I(e) = 1, . . . , l, while I(w0) is empty.


Complex F4

There are 140 involutions in W (F4). Among them, 103 involutionshave the property that I(s) is empty.

F4d

consists of 10 scattered representations, and 30 strings ofrepresentations.


Table 1: The scattered part of F d4

#s λ spin LKT u-small mult25 [1/2,1/2,1/2,1] [1,3,0,1] Yes 138 ρ/2 ρ Yes 162 [1,1,1/2,1/2] [0,0,1,4] Yes 163 [1/2,1/2,1,1] [7,1,0,0] Yes 163 ρ/2 ρ Yes 176 [1,1/2,1/2,1] [4,2,0,0] Yes 192 [1,1/2,1/2,1/2] [2,2,0,1] Yes 1122 ρ/2 ρ Yes 1140 [1,1,1/2,1/2] [0,0,0,4] Yes 1140 ρ [0,0,0,0] Yes 1


Table 2: The string part of F d4 (middle part omitted)

#s λ spin LKT mult1 [a,b, c,d ] LKT 12 [1,b, c,d ] LKT 13 [a,1, c,d ] LKT 14 [a,b,1,d ] LKT 15 [a,b, c,1] LKT 1· · · · · · · · · 134 [1,1,1/2,d ] [3,0,0,2d + 3] 134 [1,1/2,1/2,d ] [1,2,0,2d + 1] 147 [1,1,1,d ] LKT 150 [a,1,1,1] LKT 150 [a,1,1/2,1/2] [2a + 2,0,2,0] 1

Here a,b, c,d run over the set 1/2,1,3/2,2, . . . .


Understanding the string part

C.-P. Dong, On the Dirac cohomology of complex Lie grouprepresentations, Transformation Groups 18 (1) (2013), 61–79.Erratum: Transformation Groups 18 (2) (2013), 595–597.Vogan’s encouragement: “...But we are still human, andsometimes we do make mistakes. You feel bad because you are agood mathematician, and that means not accepting errors. Yourpaper has good mathematics in it..."


Understanding the string part (continued)

Fix an involution s ∈W such that I(s) is non-empty. Ps—theθ-stable parabolic subgroup of G corresponding to the simpleroots αi | i /∈ I(s); Ls—the Levi factor.We have that

J(λ,−sλ) ∼= LS(Zλ),

where Zλ is the irreducible unitary representation of Ls withZhelobenko parameters (λ− ρ(us)/2,−s(λ− ρ(us)/2)).The good range condition is met since

〈(λ,−λ), α〉 > 0, ∀α ∈ ∆(us).

Reference: D. Vogan, Unitarizability of certain series ofrepresentations, Ann. of Math. 120 (1) (1984), 141–187.


A finiteness result

Theorem (with J. Ding, 2017, arXiv:1702.01876)

The set Gd for a connected complex simple Lie group consists of twoparts:

a) finitely many scattered modules (the scattered part); andb) finitely many strings of modules (the string part).

Moreover, modules in the string part of G are all cohomologicallyinduced from the scattered part of Ld

ss tensored with unitary charactersof Z (L), and they are all in the good range. Here L runs over theproper θ-stable Levi subgroups of G, Z (L) is the center of L, and Lssdenotes the semisimple factor of L. In particular, there are at mostfinitely many modules of Gd beyond the good range.


Some remarks

To classify Gd for G complex, it suffices to consider finitely manycandidate representations.Later, we classified Gd for complex E6 (arXiv:1707.01380).The distribution of spin norm along a pencil is very efficient inactual computation. For instance, it reduces the candidaterepresentation in an s-familiy of E6 from 124048 to 3, wheres = s4s5s6s5s1s3s2s4s1.Another important tool: atlas, version 1.0, January 2017, seewww.liegroups.org for more.Reference: J. Adams, M. van Leeuwen, P. Trapa and D. Vogan,Unitary representations of real reductive groups, preprint, 2012(arXiv:1212.2192).


Barbasch-Pandžic Conjecture

The following is Conjecture 1.1 of [Barbasch-Pandžic, 2010].Let G be a complex Lie group viewed as a real group, and π be anirreducible unitary representation such that twice the infinitesimalcharacter of π is regular and integral. Then π has nonzero Diraccohomology if and only if π is cohomologically induced from anessentially unipotent representation with nonzero Diraccohomology. Here by an essentially unipotent representation wemean a unipotent representation tensored with a unitary character.


Finiteness in the real case

Theorem (2017, arXiv:1708.00383)Let G(R) be a real reductive Lie group. For all but finitely many

exceptions, any member π in G(R)d

is cohomologically induced from amember πL(R) in Ld which is in the good range. Here L(R) is a properθ-stable Levi subgroup of G(R).

• We call the finitely many exceptions the scattered part of G(R)d.

The scattered part is the "kernel" of G(R)d.

• By [DH-AJM-2015] and cohomological induction in stages, to

classify G(R)d

for G real reductive, it suffices to consider finitelymany candidate representations.


A few remarks

• We have benefited a lot from the 2017 Atlas workshop held at U ofUtah, July 10–21.• The powerful reduction due to Barbasch–Pandžic is unavailable

for real reductive Lie groups yet. We adopted another approach.• atlas parameter (x , λ, ν), infinitesimal character

12

(1 + θ)λ+ ν ∈ h∗.


A few conjectures

Conjecture 1. Let G(R) be a real reductive Lie group. Then any

spin-lowest K -type of any π in the scattered part of G(R)d

must beu-small.Conjecture 2. Let G be a connected complex Lie group. The setGd consists exactly of the unitary representations J(λ,−sλ),where s is an involution, and λ is a weight such that• 2λ is dominant integral and regular;• λ+ sλ is an integral weight;• λ− sλ is a non-negative integer combination of simple roots.Once the Barbasch-Pandžic reduction has been worked out forreal Lie groups, an analogue of Conj. 2 should be immediate.


Possible applications

Automorphic forms: Chapter 8 [Huang–Pandžic-2006] sharpenedthe results of [Langlands-1963-AJM] and[Hotta-Parthasarathy-1974-InventMath].Dirac index polynomial: S. Mehdi, P. Pandžic, D. Vogan,Translation principle for Dirac index, Amer. J. Math. 139 (6)(2017), 1465–1491.Other settings.


Thank you for listening!


Chao-Ping Dongmath.sjtu.edu.cn/conference/Bannai/2018/data/20180911A/slides.pdf · For X unitary,...

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