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Cocenter-Representation duality for affine Hecke algebras Xuhua He HKUST/Univ. Maryland May 21, 2014 Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 1 / 27

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Page 1: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Cocenter-Representation duality for affine Hecke algebras

Xuhua He

HKUST/Univ. Maryland

May 21, 2014

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 1 / 27

Page 2: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 2 / 27

Page 3: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 3 / 27

Page 4: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Overview

Finite dimensional representations of affine Hecke algebras have beenstudied by many people:

Geometric method: Kazhdan, Lusztig, Ginzburg, Reeder, Kato, etc.

Analytic method: Opdam, Solleveld

Algebraic method: joint with Nie, joint with Ciubotaru

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 4 / 27

Page 5: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Overview

Finite dimensional representations of affine Hecke algebras have beenstudied by many people:

Geometric method: Kazhdan, Lusztig, Ginzburg, Reeder, Kato, etc.

Analytic method: Opdam, Solleveld

Algebraic method: joint with Nie, joint with Ciubotaru

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 4 / 27

Page 6: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Overview

Finite dimensional representations of affine Hecke algebras have beenstudied by many people:

Geometric method: Kazhdan, Lusztig, Ginzburg, Reeder, Kato, etc.

Analytic method: Opdam, Solleveld

Algebraic method: joint with Nie, joint with Ciubotaru

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 4 / 27

Page 7: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Affine Weyl groups

(X ,R,Y ,R∨,Π) a based root datum

W0 = W (R) a finite Weyl group

S the set of simple reflections in W0.

Wa = ZR o W0 an affine Weyl group

S the set of simple reflections in Wa

W = X o W0 an extended affine Weyl group.

Example

The group Zn o Sn is the extended affine Weyl group of GLn. It is aquasi-Coxeter group.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 5 / 27

Page 8: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Iwahori-Matsumoto presentation of H

Fix the parameter q = q(s) ∈ C× : s ∈ S such that q(s) = q(t) ifs, t ∈ S are conjugate in W .

Definition (Iwahori-Matsumoto presentation)

The affine Hecke algebra H = Hq is the C-algebra generated byTw : w ∈ W subject to the relations:

1 Tw · Tw ′ = Tww ′ , if `(ww ′) = `(w) + `(w ′);

2 (Ts + 1)(Ts − q(s)2) = 0, if s ∈ S .

The Iwahori-Matsumoto presentation is related to the quasi-Coxeterstructure of W .

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 6 / 27

Page 9: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Bernstein-Lusztig presentation of H

Bernstein-Lusztig presentation: H is the C-algebra generated byθxTw ; x ∈ X ,w ∈W0. This is related to the semi-direct productW = X o W0.

For any J ⊂ Π, let HJ be the parabolic subalgebra of H generated byθxTw ; x ∈ X ,w ∈WJ. This is the affine Hecke algebra for the parabolicsubgroup WJ = X o WJ , but for (probably) different parameters.

Roughly speaking, HJ consists of two parts: the semisimple part HssJ and

the central part.

For any representation σ of HssJ and any central character χt of HJ , we

have the parabolically induced module iJ(σ χt).

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 7 / 27

Page 10: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Bernstein-Lusztig presentation of H

Bernstein-Lusztig presentation: H is the C-algebra generated byθxTw ; x ∈ X ,w ∈W0. This is related to the semi-direct productW = X o W0.

For any J ⊂ Π, let HJ be the parabolic subalgebra of H generated byθxTw ; x ∈ X ,w ∈WJ. This is the affine Hecke algebra for the parabolicsubgroup WJ = X o WJ , but for (probably) different parameters.

Roughly speaking, HJ consists of two parts: the semisimple part HssJ and

the central part.

For any representation σ of HssJ and any central character χt of HJ , we

have the parabolically induced module iJ(σ χt).

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 7 / 27

Page 11: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Bernstein-Lusztig presentation of H

Bernstein-Lusztig presentation: H is the C-algebra generated byθxTw ; x ∈ X ,w ∈W0. This is related to the semi-direct productW = X o W0.

For any J ⊂ Π, let HJ be the parabolic subalgebra of H generated byθxTw ; x ∈ X ,w ∈WJ. This is the affine Hecke algebra for the parabolicsubgroup WJ = X o WJ , but for (probably) different parameters.

Roughly speaking, HJ consists of two parts: the semisimple part HssJ and

the central part.

For any representation σ of HssJ and any central character χt of HJ , we

have the parabolically induced module iJ(σ χt).

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 7 / 27

Page 12: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Trace map

H = H/[H,H]—the cocenter of H.

R(H)—Grothendieck group of finite dim representations of H.

Tr : H → R(H)∗, h 7→ (V 7→ TrV (h)), the trace map.

Definition

A function f ∈ R(H)∗ is good if for any J ⊂ Π and σ ∈ R(HssJ ), the

function t 7→ f (iJ(σ χt) is a regular function on t.

Theorem (Bernstein, Deligne, Kazhdan)

Let Hq be the Hecke algebra of a p-adic group (algebra of locally constant,compact support measures). Then

1 The image of Tr is R(Hq)∗good (trace Paley-Wiener Theorem).

2 The map Tr is injective (density Theorem).

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 8 / 27

Page 13: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Trace map

H = H/[H,H]—the cocenter of H.

R(H)—Grothendieck group of finite dim representations of H.

Tr : H → R(H)∗, h 7→ (V 7→ TrV (h)), the trace map.

Definition

A function f ∈ R(H)∗ is good if for any J ⊂ Π and σ ∈ R(HssJ ), the

function t 7→ f (iJ(σ χt) is a regular function on t.

Theorem (Bernstein, Deligne, Kazhdan)

Let Hq be the Hecke algebra of a p-adic group (algebra of locally constant,compact support measures). Then

1 The image of Tr is R(Hq)∗good (trace Paley-Wiener Theorem).

2 The map Tr is injective (density Theorem).

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 8 / 27

Page 14: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Trace map

H = H/[H,H]—the cocenter of H.

R(H)—Grothendieck group of finite dim representations of H.

Tr : H → R(H)∗, h 7→ (V 7→ TrV (h)), the trace map.

Definition

A function f ∈ R(H)∗ is good if for any J ⊂ Π and σ ∈ R(HssJ ), the

function t 7→ f (iJ(σ χt) is a regular function on t.

Theorem (Bernstein, Deligne, Kazhdan)

Let Hq be the Hecke algebra of a p-adic group (algebra of locally constant,compact support measures). Then

1 The image of Tr is R(Hq)∗good (trace Paley-Wiener Theorem).

2 The map Tr is injective (density Theorem).

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 8 / 27

Page 15: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Cocenter of group algebra

Naive example: Consider H1 = Z[W ]. Then

1 For any conjugacy class O of W , and w ,w ′ ∈ O. The image of wand w ′ in H1 are the same.

2 H1 has a standard basis wO. Here O runs over all the conjugacyclasses of W and wO is a representative of O.

Question

What happens for Hecke algebra H?

Notice here that in general, for w ,w ′ ∈ O, the image of Tw and Tw ′ in Hare different.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 9 / 27

Page 16: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Cocenter of group algebra

Naive example: Consider H1 = Z[W ]. Then

1 For any conjugacy class O of W , and w ,w ′ ∈ O. The image of wand w ′ in H1 are the same.

2 H1 has a standard basis wO. Here O runs over all the conjugacyclasses of W and wO is a representative of O.

Question

What happens for Hecke algebra H?

Notice here that in general, for w ,w ′ ∈ O, the image of Tw and Tw ′ in Hare different.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 9 / 27

Page 17: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Finite Coxeter groups

Theorem (Geck-Pfeiffer, Geck-Kim-Pfeiffer, H., H.-Nie)

Let W0 be a finite Coxeter group and O be a conjugacy class. Let Omin bethe set of minimal length elements in O. Then

1 Each element w of O can be brought to a minimal length element w ′

by conjugation by simple reflections which reduce or keep constantthe length. We write w → w ′.

2 Any two elements in Omin are “strongly conjugate”, i.e., conjugate inthe associated Braid group.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 10 / 27

Page 18: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Finite Coxeter groups

Theorem (Geck-Pfeiffer, Geck-Kim-Pfeiffer, H., H.-Nie)

Let W0 be a finite Coxeter group and O be a conjugacy class. Let Omin bethe set of minimal length elements in O. Then

1 Each element w of O can be brought to a minimal length element w ′

by conjugation by simple reflections which reduce or keep constantthe length. We write w → w ′.

2 Any two elements in Omin are “strongly conjugate”, i.e., conjugate inthe associated Braid group.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 10 / 27

Page 19: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Finite Coxeter groups

Theorem (Geck-Pfeiffer, Geck-Kim-Pfeiffer, H., H.-Nie)

Let W0 be a finite Coxeter group and O be a conjugacy class. Let Omin bethe set of minimal length elements in O. Then

1 Each element w of O can be brought to a minimal length element w ′

by conjugation by simple reflections which reduce or keep constantthe length. We write w → w ′.

2 Any two elements in Omin are “strongly conjugate”, i.e., conjugate inthe associated Braid group.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 10 / 27

Page 20: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Finite Coxeter groups

Theorem (Geck-Pfeiffer, Geck-Kim-Pfeiffer, H., H.-Nie)

Let W0 be a finite Coxeter group and O be a conjugacy class. Let Omin bethe set of minimal length elements in O. Then

1 Each element w of O can be brought to a minimal length element w ′

by conjugation by simple reflections which reduce or keep constantthe length. We write w → w ′.

2 Any two elements in Omin are “strongly conjugate”, i.e., conjugate inthe associated Braid group.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 10 / 27

Page 21: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Elliptic conjugacy classes of type E8

Figure: Geck-Pfeiffer, Characters of finite Coxeter groups and Iwahori-Heckealgebras, table B.6.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 11 / 27

Page 22: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Finite Hecke algebras

Theorem (Geck-Pfeiffer)

Let H be the Hecke algebra for the finite Weyl group W0. Then

For any conjugacy class O of W0 and w ,w ′ ∈ Omin, the image of Tw

and Tw ′ in H are the same. We denote it by TO.

TO is a basis of H.

Here minimal length representatives give the same image in the cocenterfollows from the fact that these elements are “strongly conjugate”.

TO spans H follows from the fact that each element w can be reducedto a minimal length element w ′ via w → w ′.

And TO is linearly independent follows from the fact that

] irreducible representations of W0 = ] conjugacy classes of W0.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 12 / 27

Page 23: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Finite Hecke algebras

Theorem (Geck-Pfeiffer)

Let H be the Hecke algebra for the finite Weyl group W0. Then

For any conjugacy class O of W0 and w ,w ′ ∈ Omin, the image of Tw

and Tw ′ in H are the same. We denote it by TO.

TO is a basis of H.

Here minimal length representatives give the same image in the cocenterfollows from the fact that these elements are “strongly conjugate”.

TO spans H follows from the fact that each element w can be reducedto a minimal length element w ′ via w → w ′.

And TO is linearly independent follows from the fact that

] irreducible representations of W0 = ] conjugacy classes of W0.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 12 / 27

Page 24: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Finite Hecke algebras

Theorem (Geck-Pfeiffer)

Let H be the Hecke algebra for the finite Weyl group W0. Then

For any conjugacy class O of W0 and w ,w ′ ∈ Omin, the image of Tw

and Tw ′ in H are the same. We denote it by TO.

TO is a basis of H.

Here minimal length representatives give the same image in the cocenterfollows from the fact that these elements are “strongly conjugate”.

TO spans H follows from the fact that each element w can be reducedto a minimal length element w ′ via w → w ′.

And TO is linearly independent follows from the fact that

] irreducible representations of W0 = ] conjugacy classes of W0.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 12 / 27

Page 25: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Finite Hecke algebras

Theorem (Geck-Pfeiffer)

Let H be the Hecke algebra for the finite Weyl group W0. Then

For any conjugacy class O of W0 and w ,w ′ ∈ Omin, the image of Tw

and Tw ′ in H are the same. We denote it by TO.

TO is a basis of H.

Here minimal length representatives give the same image in the cocenterfollows from the fact that these elements are “strongly conjugate”.

TO spans H follows from the fact that each element w can be reducedto a minimal length element w ′ via w → w ′.

And TO is linearly independent follows from the fact that

] irreducible representations of W0 = ] conjugacy classes of W0.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 12 / 27

Page 26: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Finite Hecke algebras

Theorem (Geck-Pfeiffer)

Let H be the Hecke algebra for the finite Weyl group W0. Then

For any conjugacy class O of W0 and w ,w ′ ∈ Omin, the image of Tw

and Tw ′ in H are the same. We denote it by TO.

TO is a basis of H.

Here minimal length representatives give the same image in the cocenterfollows from the fact that these elements are “strongly conjugate”.

TO spans H follows from the fact that each element w can be reducedto a minimal length element w ′ via w → w ′.

And TO is linearly independent follows from the fact that

] irreducible representations of W0 = ] conjugacy classes of W0.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 12 / 27

Page 27: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Finite Hecke algebras

Theorem (Geck-Pfeiffer)

Let H be the Hecke algebra for the finite Weyl group W0. Then

For any conjugacy class O of W0 and w ,w ′ ∈ Omin, the image of Tw

and Tw ′ in H are the same. We denote it by TO.

TO is a basis of H.

Here minimal length representatives give the same image in the cocenterfollows from the fact that these elements are “strongly conjugate”.

TO spans H follows from the fact that each element w can be reducedto a minimal length element w ′ via w → w ′.

And TO is linearly independent follows from the fact that

] irreducible representations of W0 = ] conjugacy classes of W0.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 12 / 27

Page 28: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

From finite Weyl groups to affine Weyl groups

We’d like to understand the behaviour of elements in a given conjugacyclass O of an (extended) affine Weyl group and to see if

Any element w ∈ O can be reduced to an element w ′ ∈ Omin.

Any two elements in Omin are “strongly conjugate”.

The main difficulty is that in general, O contains infinite elements.

This make it impossible to check by computer via brute force even for asimple affine Weyl group such as A2.

How can we see the infinity?

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 13 / 27

Page 29: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

From finite Weyl groups to affine Weyl groups

We’d like to understand the behaviour of elements in a given conjugacyclass O of an (extended) affine Weyl group and to see if

Any element w ∈ O can be reduced to an element w ′ ∈ Omin.

Any two elements in Omin are “strongly conjugate”.

The main difficulty is that in general, O contains infinite elements.

This make it impossible to check by computer via brute force even for asimple affine Weyl group such as A2.

How can we see the infinity?

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 13 / 27

Page 30: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

From finite Weyl groups to affine Weyl groups

We’d like to understand the behaviour of elements in a given conjugacyclass O of an (extended) affine Weyl group and to see if

Any element w ∈ O can be reduced to an element w ′ ∈ Omin.

Any two elements in Omin are “strongly conjugate”.

The main difficulty is that in general, O contains infinite elements.

This make it impossible to check by computer via brute force even for asimple affine Weyl group such as A2.

How can we see the infinity?

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 13 / 27

Page 31: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

MIThenge

Figure: MIT Infinite Corridor, January 2001. Photograph by Matt Yourst.

Predictions: 11/11/2014-11/14/2014, Time: 4:18:52 to 4:22:38.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 14 / 27

Page 32: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

MIThenge

Figure: MIT Infinite Corridor, January 2001. Photograph by Matt Yourst.

Predictions: 11/11/2014-11/14/2014, Time: 4:18:52 to 4:22:38.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 14 / 27

Page 33: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Arithmetic invariants of conjugacy classes in W

Definition

For w ∈ W , w |W0| = tλ for some λ ∈ X . Set νw = λ/|W0| ∈ XQ, theNewton point of w . Let νw be the unique dominant element in W0 · νw .

The map f : W → W /Wa × XQ, w 7→ (wWa, νw ) is constant on eachconjugacy class of W .

Theorem (Kottwitz, H.)

Let G be a split p-adic group with Iwahori-Weyl group W and σ be thestandard Frobenius map. Then

1 The map f factors as W /conjΦ // G/σ − conj

1−1 // Im(f ) .

2 The map Φ is surjective and has a natural cross-section given by theset of straight conjugacy classes of W .

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 15 / 27

Page 34: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Arithmetic invariants of conjugacy classes in W

Definition

For w ∈ W , w |W0| = tλ for some λ ∈ X . Set νw = λ/|W0| ∈ XQ, theNewton point of w . Let νw be the unique dominant element in W0 · νw .

The map f : W → W /Wa × XQ, w 7→ (wWa, νw ) is constant on eachconjugacy class of W .

Theorem (Kottwitz, H.)

Let G be a split p-adic group with Iwahori-Weyl group W and σ be thestandard Frobenius map. Then

1 The map f factors as W /conjΦ // G/σ − conj

1−1 // Im(f ) .

2 The map Φ is surjective and has a natural cross-section given by theset of straight conjugacy classes of W .

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 15 / 27

Page 35: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Affine Weyl group

In the study of minimal length elements in a given conjugacy class of W ,the Newton point provide a crucial invariant. It allows us to reduce fromaffine Weyl groups to finite Weyl groups (associated to the Newton point).

Theorem (H., H.-Nie)

Let W be an extended affine Weyl group and O be a conjugacy class.Then

1 For any w ∈ O, there exists w ′ ∈ Omin such that w → w ′.

2 Any two elements in Omin are “strongly conjugate”.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 16 / 27

Page 36: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Affine Weyl group

In the study of minimal length elements in a given conjugacy class of W ,the Newton point provide a crucial invariant. It allows us to reduce fromaffine Weyl groups to finite Weyl groups (associated to the Newton point).

Theorem (H., H.-Nie)

Let W be an extended affine Weyl group and O be a conjugacy class.Then

1 For any w ∈ O, there exists w ′ ∈ Omin such that w → w ′.

2 Any two elements in Omin are “strongly conjugate”.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 16 / 27

Page 37: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Affine Weyl group

In the study of minimal length elements in a given conjugacy class of W ,the Newton point provide a crucial invariant. It allows us to reduce fromaffine Weyl groups to finite Weyl groups (associated to the Newton point).

Theorem (H., H.-Nie)

Let W be an extended affine Weyl group and O be a conjugacy class.Then

1 For any w ∈ O, there exists w ′ ∈ Omin such that w → w ′.

2 Any two elements in Omin are “strongly conjugate”.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 16 / 27

Page 38: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Affine Weyl group

In the study of minimal length elements in a given conjugacy class of W ,the Newton point provide a crucial invariant. It allows us to reduce fromaffine Weyl groups to finite Weyl groups (associated to the Newton point).

Theorem (H., H.-Nie)

Let W be an extended affine Weyl group and O be a conjugacy class.Then

1 For any w ∈ O, there exists w ′ ∈ Omin such that w → w ′.

2 Any two elements in Omin are “strongly conjugate”.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 16 / 27

Page 39: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Affine Weyl group

In the study of minimal length elements in a given conjugacy class of W ,the Newton point provide a crucial invariant. It allows us to reduce fromaffine Weyl groups to finite Weyl groups (associated to the Newton point).

Theorem (H., H.-Nie)

Let W be an extended affine Weyl group and O be a conjugacy class.Then

1 For any w ∈ O, there exists w ′ ∈ Omin such that w → w ′.

2 Any two elements in Omin are “strongly conjugate”.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 16 / 27

Page 40: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Iwahori-Matsumoto presentation of H

Theorem (H.-Nie)

Let H be an extended affine Hecke algebra. Then

For any conjugacy class O of W and w ,w ′ ∈ Omin, the image of Tw

and Tw ′ in H are the same. We denote it by TO.

TO spans of H.

We’ll see later that TO is always a basis of H. This is theIwahori-Matsumoto presentation of the cocenter.

However, the relation between Iwahori-Matsumoto presentation andrepresentation theory (the induction functor and restriction functors, etc.)is hard to understand.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 17 / 27

Page 41: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Iwahori-Matsumoto presentation of H

Theorem (H.-Nie)

Let H be an extended affine Hecke algebra. Then

For any conjugacy class O of W and w ,w ′ ∈ Omin, the image of Tw

and Tw ′ in H are the same. We denote it by TO.

TO spans of H.

We’ll see later that TO is always a basis of H. This is theIwahori-Matsumoto presentation of the cocenter.

However, the relation between Iwahori-Matsumoto presentation andrepresentation theory (the induction functor and restriction functors, etc.)is hard to understand.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 17 / 27

Page 42: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Iwahori-Matsumoto presentation of H

Theorem (H.-Nie)

Let H be an extended affine Hecke algebra. Then

For any conjugacy class O of W and w ,w ′ ∈ Omin, the image of Tw

and Tw ′ in H are the same. We denote it by TO.

TO spans of H.

We’ll see later that TO is always a basis of H. This is theIwahori-Matsumoto presentation of the cocenter.

However, the relation between Iwahori-Matsumoto presentation andrepresentation theory (the induction functor and restriction functors, etc.)is hard to understand.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 17 / 27

Page 43: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Iwahori-Matsumoto presentation of H

Theorem (H.-Nie)

Let H be an extended affine Hecke algebra. Then

For any conjugacy class O of W and w ,w ′ ∈ Omin, the image of Tw

and Tw ′ in H are the same. We denote it by TO.

TO spans of H.

We’ll see later that TO is always a basis of H. This is theIwahori-Matsumoto presentation of the cocenter.

However, the relation between Iwahori-Matsumoto presentation andrepresentation theory (the induction functor and restriction functors, etc.)is hard to understand.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 17 / 27

Page 44: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Iwahori-Matsumoto presentation of H

Theorem (H.-Nie)

Let H be an extended affine Hecke algebra. Then

For any conjugacy class O of W and w ,w ′ ∈ Omin, the image of Tw

and Tw ′ in H are the same. We denote it by TO.

TO spans of H.

We’ll see later that TO is always a basis of H. This is theIwahori-Matsumoto presentation of the cocenter.

However, the relation between Iwahori-Matsumoto presentation andrepresentation theory (the induction functor and restriction functors, etc.)is hard to understand.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 17 / 27

Page 45: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Bernstein-Lusztig presentation of H

In order to give the Bernstein-Lusztig presentation of the cocenter, oneneeds to have a nice paramitrization of conjugacy classes of W .

Proposition (H.-Nie)

Let A be the set of all pairs (J,C ), where C is an elliptic conjugacy classof WJ and νw is dominant for all w ∈ C . Then

π : A/ ∼ 1−1−−→ conjugacy classes of W .

Theorem (H.-Nie)

Let iJ : HJ → H be the inclusion and iJ : HJ → H the induced map. Forany (J,C ) ∈ A,

TO = iJ(T JC ).

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 18 / 27

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Bernstein-Lusztig presentation of H

In order to give the Bernstein-Lusztig presentation of the cocenter, oneneeds to have a nice paramitrization of conjugacy classes of W .

Proposition (H.-Nie)

Let A be the set of all pairs (J,C ), where C is an elliptic conjugacy classof WJ and νw is dominant for all w ∈ C . Then

π : A/ ∼ 1−1−−→ conjugacy classes of W .

Theorem (H.-Nie)

Let iJ : HJ → H be the inclusion and iJ : HJ → H the induced map. Forany (J,C ) ∈ A,

TO = iJ(T JC ).

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 18 / 27

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Cocenter-Representation duality, I

Theorem (Ciubotaru-H.)

Cocenter-Representation duality holds for affine Hecke algebras for genericparameters.

Theorem (Ciubotaru-H.)

TO is a basis of H (basis Theorem).

For any parameters, the image of Tr is R(Hq)∗good (tracePaley-Wiener Theorem).

For generic parameter, the map Tr is injective (density Theorem).

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 19 / 27

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Cocenter-Representation duality, I

Theorem (Ciubotaru-H.)

Cocenter-Representation duality holds for affine Hecke algebras for genericparameters.

Theorem (Ciubotaru-H.)

TO is a basis of H (basis Theorem).

For any parameters, the image of Tr is R(Hq)∗good (tracePaley-Wiener Theorem).

For generic parameter, the map Tr is injective (density Theorem).

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 19 / 27

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Cocenter-Representation duality, I

Theorem (Ciubotaru-H.)

Cocenter-Representation duality holds for affine Hecke algebras for genericparameters.

Theorem (Ciubotaru-H.)

TO is a basis of H (basis Theorem).

For any parameters, the image of Tr is R(Hq)∗good (tracePaley-Wiener Theorem).

For generic parameter, the map Tr is injective (density Theorem).

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 19 / 27

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Cocenter-Representation duality, I

Theorem (Ciubotaru-H.)

Cocenter-Representation duality holds for affine Hecke algebras for genericparameters.

Theorem (Ciubotaru-H.)

TO is a basis of H (basis Theorem).

For any parameters, the image of Tr is R(Hq)∗good (tracePaley-Wiener Theorem).

For generic parameter, the map Tr is injective (density Theorem).

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 19 / 27

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Cocenter-Representation duality, I

Theorem (Ciubotaru-H.)

Cocenter-Representation duality holds for affine Hecke algebras for genericparameters.

Theorem (Ciubotaru-H.)

TO is a basis of H (basis Theorem).

For any parameters, the image of Tr is R(Hq)∗good (tracePaley-Wiener Theorem).

For generic parameter, the map Tr is injective (density Theorem).

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 19 / 27

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Cocenter-Representation duality, I

Theorem (Ciubotaru-H.)

Cocenter-Representation duality holds for affine Hecke algebras for genericparameters.

Theorem (Ciubotaru-H.)

TO is a basis of H (basis Theorem).

For any parameters, the image of Tr is R(Hq)∗good (tracePaley-Wiener Theorem).

For generic parameter, the map Tr is injective (density Theorem).

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 19 / 27

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Cocenter-Representation duality, II

To explain the basic idea of cocenter-representation duality, we start withthe group algebra for a finite group and its representation over C.

Then ] irreducible representations=] conjugacy classes. In other words,

rank of Grothendieck group=rank of cocenter.

And the trace map leads to an invertible square matrix.

Difficulty: for affine Hecke algebras, both the rank of R(H) and the rankof H are infinite.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 20 / 27

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Cocenter-Representation duality, II

To explain the basic idea of cocenter-representation duality, we start withthe group algebra for a finite group and its representation over C.

Then ] irreducible representations=] conjugacy classes. In other words,

rank of Grothendieck group=rank of cocenter.

And the trace map leads to an invertible square matrix.

Difficulty: for affine Hecke algebras, both the rank of R(H) and the rankof H are infinite.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 20 / 27

Page 55: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Cocenter-Representation duality, II

To explain the basic idea of cocenter-representation duality, we start withthe group algebra for a finite group and its representation over C.

Then ] irreducible representations=] conjugacy classes. In other words,

rank of Grothendieck group=rank of cocenter.

And the trace map leads to an invertible square matrix.

Difficulty: for affine Hecke algebras, both the rank of R(H) and the rankof H are infinite.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 20 / 27

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Cocenter-Representation duality, III

The key idea is to construct filtrations on H and on R(H) so that thecorresponding subquotient “matches”.

Question

What are the filtrations?

Elliptic Theory (Arthur): There is a standard filtration on R(H) obtainedfrom induced modules: R(H) ⊃ R(H)1 ⊃ · · · , where R(H)1 is spanned byall the induced modules. The elliptic quotient R(H)ell = R(H)/R(H)1.

However, the dual filtration on H is very complicated to understand.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 21 / 27

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Cocenter-Representation duality, III

The key idea is to construct filtrations on H and on R(H) so that thecorresponding subquotient “matches”.

Question

What are the filtrations?

Elliptic Theory (Arthur): There is a standard filtration on R(H) obtainedfrom induced modules: R(H) ⊃ R(H)1 ⊃ · · · , where R(H)1 is spanned byall the induced modules. The elliptic quotient R(H)ell = R(H)/R(H)1.

However, the dual filtration on H is very complicated to understand.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 21 / 27

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Cocenter-Representation duality, III

The key idea is to construct filtrations on H and on R(H) so that thecorresponding subquotient “matches”.

Question

What are the filtrations?

Elliptic Theory (Arthur): There is a standard filtration on R(H) obtainedfrom induced modules: R(H) ⊃ R(H)1 ⊃ · · · , where R(H)1 is spanned byall the induced modules. The elliptic quotient R(H)ell = R(H)/R(H)1.

However, the dual filtration on H is very complicated to understand.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 21 / 27

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Rigid quotient, I

Here we consider different filtrations. For simplicity, we only considersemisimple root datum here.

On the H side, we consider:

0 ⊂ H1 ⊂ · · · ⊂ H,

where H1 is spanned by TO with νO = 0.

On the R(H) side, we consider the dual filtration:

R(H) ⊃ R(H)1rigid ⊃ · · · ,

where R(H)1rigid is spanned by iJ(σ)− iJ(σ χt) for all J, σ, t. The

quotient R(H)rigid = R(H)/R(H)1rigid is called the rigid quotient.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 22 / 27

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Rigid quotient, I

Here we consider different filtrations. For simplicity, we only considersemisimple root datum here.

On the H side, we consider:

0 ⊂ H1 ⊂ · · · ⊂ H,

where H1 is spanned by TO with νO = 0.

On the R(H) side, we consider the dual filtration:

R(H) ⊃ R(H)1rigid ⊃ · · · ,

where R(H)1rigid is spanned by iJ(σ)− iJ(σ χt) for all J, σ, t. The

quotient R(H)rigid = R(H)/R(H)1rigid is called the rigid quotient.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 22 / 27

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Rigid quotient, I

Here we consider different filtrations. For simplicity, we only considersemisimple root datum here.

On the H side, we consider:

0 ⊂ H1 ⊂ · · · ⊂ H,

where H1 is spanned by TO with νO = 0.

On the R(H) side, we consider the dual filtration:

R(H) ⊃ R(H)1rigid ⊃ · · · ,

where R(H)1rigid is spanned by iJ(σ)− iJ(σ χt) for all J, σ, t. The

quotient R(H)rigid = R(H)/R(H)1rigid is called the rigid quotient.

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 22 / 27

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Rigid quotient, II

Theorem (Ciubotaru-H.)

For generic parameters, the trace map Tr : H × R(H)→ C induces aperfect pairing

Tr : H1 × R(H)rigid → C.

Corollary (Ciubotaru-H.)

1 For generic parameters, the rank of R(Hq)rigid equals the number ofconjugacy classes of W of finite order.

2 For generic parameters, the rank of R(Hq)ell equals the number ofelliptic conjugacy classes of W .

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 23 / 27

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Rigid quotient, II

Theorem (Ciubotaru-H.)

For generic parameters, the trace map Tr : H × R(H)→ C induces aperfect pairing

Tr : H1 × R(H)rigid → C.

Corollary (Ciubotaru-H.)

1 For generic parameters, the rank of R(Hq)rigid equals the number ofconjugacy classes of W of finite order.

2 For generic parameters, the rank of R(Hq)ell equals the number ofelliptic conjugacy classes of W .

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 23 / 27

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Rigid quotient, II

Theorem (Ciubotaru-H.)

For generic parameters, the trace map Tr : H × R(H)→ C induces aperfect pairing

Tr : H1 × R(H)rigid → C.

Corollary (Ciubotaru-H.)

1 For generic parameters, the rank of R(Hq)rigid equals the number ofconjugacy classes of W of finite order.

2 For generic parameters, the rank of R(Hq)ell equals the number ofelliptic conjugacy classes of W .

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 23 / 27

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My wild guess for modular case

Let k be an algebraically closed field and q(s) ∈ k×. Let R(Hq,k) be theGrothendieck group of finite dimensional representation of Hq over k.

For any K ⊂ S with WK is finite, let ΩK be the subset of Ω that stabilizesK and W ]

K = WK o ΩK . Let I be the set of all K with maximal W ]K .

Question

Is the rank of R(Hq,k)ell equal to the∑

K∈I/conj rank R(HK ,q,k o ΩK )ell?

Here the left side is for affine Hecke algebras and the right side is for finiteHecke algebras (with twist).

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 24 / 27

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Example

We list the expected rank of R(Hq,k)ell for SL3 and PGL3. HereΦ3(q) = q2 + q + 1.

SL3 char(k) 6= 3 char(k) = 3

Φ3(q) 6= 0 3 3

Φ3(q) = 0 0 0

PGL3 char(k) 6= 3 char(k) = 3

Φ3(q) 6= 0 3 1

Φ3(q) = 0 2 0

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 25 / 27

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One more thing: on the real group

Xuhua He (HKUST/Univ. Maryland) Cocenter-Representation duality 26 / 27

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Thank you, David:for being a great teacher, guide and friend of my wife and I.

Happy birthday!

Page 69: Cocenter-Representation duality for a ne Hecke algebras · 2014-05-22 · Finite Coxeter groups Theorem (Geck-Pfei er, Geck-Kim-Pfei er, H., H.-Nie) Let W 0 be a nite Coxeter group

Thank you, David:for being a great teacher, guide and friend of my wife and I.

Happy birthday!