Post on 27-Mar-2020
Geometric Structure in the Representation Theory ofReductive p-adic Groups
Paul BaumPenn State
San Francisco JMM
January 16, 2010
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 1 / 35
Reference
The Hecke algebra of a reductive p-adic group: a geometricconjecture
by
Anne-Marie Aubert, Paul Baum. and Roger Plymen
in
book edited by Katia Consani and Matilde Marcolli based on meeting atMPI Bonn 2004
Title of book : Non-Commutative Geometry and Number Theory
Publisher: Vieweg Verlag (2006)
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 2 / 35
Reference
Geometric structure in the representation theory of p-adic groups
by
Anne-Marie Aubert, Paul Baum,and Roger Plymen
Comptes Rendus de l’Academie des Sciences de Paris
Ser.I 345 (2007), 573-578
arXiv:math.RT/0607381 v1
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 3 / 35
Reference
Geometric structure in the principal series of the p-adic group G2
by
Anne-Marie Aubert, Paul Baum,and Roger Plymen
To appear in Representation Theory
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 4 / 35
ABP Conjecture
ABP = Aubert-Baum-Plymen
The conjecture can be stated at four levels :
K-theory
Periodic cyclic homology
Geometric equivalence of finite type algebras
Representation theory
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 5 / 35
ABP Conjecture
ABP = Aubert-Baum-Plymen
The conjecture can be stated at four levels :
K-theory
Periodic cyclic homology
Geometric equivalence of finite type algebras
Representation theory
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 6 / 35
Let G be a reductive p-adic group.
Examples are:
GL(n, F ) SL(n, F )
where F is any finite extension of the p-adic numbers Qp
Definition
A representation of G is a group homomorphism
φ : G→ AutC(V )
where V is a vector space over the complex numbers C.
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 7 / 35
The p-adic numbers Qp in its natural topology is a locally compact andtotally disconnected topological field. Hence G is a locally compact andtotally disconnected topological group.
Definition
A representationφ : G→ AutC(V )
of G is smooth if for every v ∈ V ,
Gv = {g ∈ G | φ(g)v = v}
is an open subgroup of G.
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 8 / 35
The smooth (or admissible) dual of G, denoted G, is the set ofequivalence classes of smooth irreducible representations of G.
G = {Smooth irreducible representations of G}/ ∼
Problem: Describe G.
Since G is locally compact we may fix a (left-invariant) Haar measure dgfor G.
The Hecke algebra of G, denoted HG, is then the convolution algebra ofall locally-constant compactly-supported complex-valued functionsf : G→ C.
(f + h)(g) = f(g) + h(g)
(f ∗ h)(g0) =∫Gf(g)h(g−1g0)dg
g ∈ Gg0 ∈ Gf ∈ HGh ∈ HG
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 9 / 35
Definition
A representation of the Hecke algebra HG is a homomorphism of Calgebras
ψ : HG→ EndC(V )
where V is a vector space over the complex numbers C.
Definition
A representationψ : HG→ EndC(V )
of the Hecke algebra HG is irreducible if V 6= {0} and @ a vector subspaceW of V such that W is preserved by the action of HG and {0} 6= W 6= V .
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 10 / 35
Definition
A primitive ideal I in HG is the null space of an irreducible representationof HG.
Thus
0 // I� � // HG
ψ // EndC(V )
is exact where ψ is an irreducible representation of HG.
There is a (canonical) bijection of sets
G←→ Prim(HG)
where Prim(HG) is the set of primitive ideals in HG.
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 11 / 35
Bijection (of sets)G←→ Prim(HG)
What has been gained from this bijection?
On Prim(HG) have a topology — the Jacobson topology.
If S is a subset of Prim(HG) then the closure S (in the Jacobson toplogy)of S is
S = {J ∈ Prim(HG) | J ⊃⋂I∈S
I}
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Prim(HG) (with the Jacobson topology) is the disjoint union of itsconnected components.
πoPrim(HG) denotes the set of connected components of Prim(HG).
πoPrim(HG) is a countable set and has no further structure.
πoPrim(HG) is also known as the Bernstein spectrum of G.
πoPrim(HG) = {(L, σ)}/ ∼ where (L, σ) is a cuspidal pair i.e. L is a Levisubgroup of G and σ is an irreducible super-cuspidal representation of L.∼ is the conjugation action of G combined with twisting σ by unramifiedcharacters of L. Thus (L, σ) ∼ (M,ϕ) iff there exists g ∈ G and anunramified character θ : L→ C× with g(L, θσ) = (M,ϕ).
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 13 / 35
For each α ∈ πoPrim(HG), Xα denotes the connected component ofPrim(HG).
The problem of describing G now becomes the problem of describing eachXα.
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 14 / 35
Notation
C× denotes the (complex) affine variety C− {0}.
Definition
A complex torus is a (complex) affine variety T such that there exists anisomorphism of affine varieties
T ∼= C× × C× × · · · × C×.
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 15 / 35
Bernstein assigns to each α ∈ πoPrim(HG) a complex torus Tα and afinite group Γα acting on Tα.
He then forms the quotient variety Tα/Γα and proves that there is asurjective map πα mapping Xα onto Tα/Γα .
Xα
πα
��Tα/Γα
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 16 / 35
This map πα is referred to as the infinitesimal character or the centralcharacter.
In Bernstein’s work Xα is a set (i.e. is only a set) so πα
Xα
πα
��Tα/Γα
is a map of sets.
πα is surjective, finite-to-one and generically one-to-one.
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 17 / 35
The extended quotient
Let Γ be a finite group acting on an affine variety X.
Γ×X → X
The quotient variety X/Γ is obtained by collapsing each orbit to a point.
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 18 / 35
For x ∈ X, Γx denotes the stabilizer group of x.
Γx = {γ ∈ Γ | γx = x}
c(Γx) denotes the set of conjugacy classes of Γx.
The extended quotient is obtained by replacing the orbit of x by c(Γx).
This is done as follows:
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 19 / 35
Set X = {(γ, x) ∈ Γ×X | γx = x}
X ⊂ Γ×X
X is an affine variety and is a sub-variety of Γ×X.
Γ acts on X.
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 20 / 35
Γ× X → X
g(γ, x) = (gγg−1, gx)
The extended quotient, denoted X//Γ, is X/Γ.
i.e. The extended quotient X//Γ is the ordinary quotient for the action ofΓ on X.
The extended quotient is an affine variety.
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 21 / 35
X = {(γ, x) ∈ Γ×X | γx = x}
The projection X → X
(γ, x) 7→ x
Passes to quotient spaces to give a map
ρ : X//Γ→ X/Γ
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Conjecture
There is a certain resemblance between
Tα//Γα
ρα
��
Xα
πα
��
and
Tα/Γα Tα/Γα
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 23 / 35
Conjecture
Tα//Γα
ρα
��
Xα
πα
��
and
Tα/Γα Tα/Γα
are almost the same.
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 24 / 35
How can this conjecture be made precise?
The precise conjecture consists of two statements.
Conjecture
#1. The infinitesimal character
πα : Xα → Tα/Γα
is one-to-one if and only if the action of Γα on Tα is free.
#2. There exists a bijection
να : Tα//Γα ←→ Xα
with the following properties:
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 25 / 35
α ∈ πoPrim(HG)Within the admissible dual G have the tempered dual Gtempered.
Gtempered = {smooth tempered irreducible representations of G}/ ∼Gtempered = Support of the Plancherel measureKα = maximal compact subgroup of Tα.Kα is a compact torus. The action of Γα on Tα preserves the maximalcompact subgroup Kα , so can form the compact orbifold Kα//Γα.
Conjecture : Properties of the bijection να
The bijection να : Tα//Γα ←→ Xα mapsKα//Γα onto Xα ∩ GtemperedKα//Γα ←→ Xα ∩ Gtempered
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 26 / 35
Conjecture : Properties of the bijection ναFor many α the diagram
Tα//Γα
ρα
��
να // Xα
πα
��Tα/Γα
I// Tα/Γα
does not commute.I = the identity map of Tα/Γα.
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 27 / 35
Conjecture : Properties of the bijection ναIn the possibly non-commutative diagram
Tα//Γα
ρα
��
να // Xα
πα
��Tα/Γα
I// Tα/Γα
the bijection να : Tα//Γα −→ Xα is continuous where Tα//Γα hasthe Zariski topology and Xα has the Jacobson topologyAND the composition
πα ◦ να : Tα//Γα −→ Tα/Γα
is a morphism of algebraic varieties.
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 28 / 35
Conjecture : Properties of the bijection να
For each α ∈ πoPrim(HG) there is an algebraic family
θt : Tα//Γα −→ Tα/Γα
of morphisms of algebraic varieties, with t ∈ C×, such that
θ1 = ρα and θ√q = πα ◦ να
C× = C− {0}q = order of the residue field of the p-adic field F over which G isdefinedπα = infinitesimal character of Bernstein
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 29 / 35
Conjecture : Properties of the bijection να
Fix α ∈ πoPrim(HG) For each irreducible component c ⊂ Tα//Γαthere is a cocharacter
hc : C× −→ Tα
such that
θt(x) = λ(hc(t) · x)
for all x ∈ c.
cocharacter = homomorphism of algebraic groups C× −→ Tαλ : Tα −→ Tα/Γα is the usual quotient map from Tα to Tα/Γα.
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 30 / 35
Question
Where are these correcting co-characters coming from?
Answer
The correcting co-characters are produced by the SL(2,C) part of theLanglands parameters.
W × SL(2,C) −→ LG
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 31 / 35
Example
G = GL(2, F )F can be any finite extension of the p-adic numbers Qp.q denotes the order of the residue field of F .Xα = { Smooth irreducible representations of GL(2, F ) having a non-zeroIwahori fixed vector}
Tα = {unramified characters of the maximal torus of GL(2, F )}= C× × C×
Γα = the Weyl group of GL(2, F ) = Z/2Z
0 6= γ ∈ Z/2Z γ(ζ1, ζ2) = (ζ2, ζ1) (ζ1, ζ2) ∈ C× × C×
C× × C×//(Z/2Z) = C× × C×/(Z/2Z)⊔
C×
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 32 / 35
C× × C×/(Z/2Z)
C× × C×//(Z/2Z) = C× × C×/(Z/2Z)⊔
C×
Locus of reducibility
ζ1ζ−12 =
{q
q−1
{ζ1, ζ2} such that
ζ1 = ζ2
{ζ1, ζ2} such that
correcting cocharacter C× −→ C× × C× is t 7→ (t, t−1)
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Projection of theextended quotient onthe ordinary quotient
Infinitesimalcharacter
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Baum-Connes
ABP
Local Langlands
Theorem (V.Lafforgue)
Baum-Connes is valid for any reductive p-adic group G.
Theorem (M.Harris and R.Taylor, G.Henniart)
Local Langlands is valid for GL(n, F).
Theorem (ABP)
ABP is valid for GL(n, F).
Paul Baum (San Francisco JMM) Geometric Structure January 16, 2010 35 / 35