The 2-Representation Theory of Soergel Bimodules of ﬁnite Coxeter type

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The 2-Representation Theory of Soergel Bimodules of finite Coxeter type

Marco Mackaay

joint with Mazorchuk, Miemietz, Tubbenhauer and Zhang

CAMGSD and Universidade do Algarve

July 12, 2019

The 2-Representation Theory of Soergel Bimodules of finite Coxeter type

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Outline

• Introduction.

• Coxeter groups, Hecke algebras, Soergel bimodules.

• Lusztig’s asymptotic Hecke algebras and asymptotic Soergel categories.

• Conjectural relation between the 2-representation theories of Soergel bimodules and asymptotic Soergel bimodules for finite Coxeter type.

• Duflo involutions and cell 2-representations.

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Outline

• Introduction.

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Outline

• Introduction.

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Outline

• Introduction.

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Outline

• Introduction.

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Introduction

LetS be the monoidal category of Soergel bimodules for any finite Coxeter type.

Classification Problem

Classify all graded, simple transitive 2-representations ofS up to equivalence.

WARNING:

• Etingof-Nikshych-Ostrik: IfC is semisimple, then

# {simple transitive 2-representations ofC}/'<∞.

• This is not true in general, e.g. C:=Tn−mod, where Tn is the Taft Hopf algebra.

• S is not even abelian, let alone semisimple...

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Introduction

WARNING:

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Introduction

WARNING:

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Introduction

WARNING:

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Introduction

Recall that every simple transitive (graded) 2-representation has an apex and that the classification problem can be studied per apex.

Prior to our recent results, a complete classification was only known in the following cases:

• Arbitrary finite Coxeter type and strongly regular apex (e.g. in Coxeter type A_n, for all n ≥1) [Mazorchuk-Miemietz].

• Coxeter typeBn and arbitrary apex, forn≤4 [Zimmermann, M-Mazorchuk-Miemietz-Zhang].

• Arbitrary finite Coxeter type and subregular apex [Kildetoft-M-Mazorchuk-Zimmermann, M-Tubbenhauer].

• Coxeter type I₂(n) and arbitrary apex, for alln≥2 [Kildetoft-M-Mazorchuk-Zimmermann, M-Tubbenhauer].

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Introduction

Recall that every simple transitive (graded) 2-representation has an apex and that the classification problem can be studied per apex.

Prior to our recent results, a complete classification was only known in the following cases:

• Arbitrary finite Coxeter type and strongly regular apex (e.g.

in Coxeter type A_n, for all n ≥1) [Mazorchuk-Miemietz].

• Coxeter typeBn and arbitrary apex, forn≤4 [Zimmermann, M-Mazorchuk-Miemietz-Zhang].

• Arbitrary finite Coxeter type and subregular apex [Kildetoft-M-Mazorchuk-Zimmermann, M-Tubbenhauer].

• Coxeter type I₂(n) and arbitrary apex, for all n≥2 [Kildetoft-M-Mazorchuk-Zimmermann, M-Tubbenhauer].

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Coxeter groups, Hecke algebras, Soergel bimodules

LetM = (m_ij)ⁿ_i_,j₌₁∈Mat(n,N) be a symmetric matrix such that

mij =

(1 ifi =j;

≥2 ifi 6=j.

Definition (Coxeter system)

A Coxeter system (W,S) with Coxeter matrixM is given by a set S ={s₁, . . . ,sn}(simple reflections) and a groupW with

presentation

hs_i ∈S |i = 1, . . . ,ni/((sisj)^m^ij =e). We calln the rank of (W,S).

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Examples

• The only Coxeter groups of rank 2 are the dihedral groups (Coxeter typeI₂(n)):

D_2n=hs,t |s² =t² =e ∧ (st)ⁿ=ei.

The isomorphism with the usual presentation hρ, σ|σ² =ρⁿ=e ∧ ρσ=σρ⁻¹i is given by s 7→σ andt →σρ.

• The Coxeter group of typeA_n is isomorphic toS_n+1, generated by the simple transpositions s₁, . . . ,s_n, subject to

m_ii = 1 : (s_is_i)¹=e ⇔s_i² =e;

mij = 2 : (sisj)² =e ⇔sisj =sjsi ifj 6=i±1; mi(i±1)= 3 : (sisi±1)³=e ⇔sisi±1si =si±1sisi±1.

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Examples

• The only Coxeter groups of rank 2 are the dihedral groups (Coxeter typeI₂(n)):

D_2n=hs,t |s² =t² =e ∧ (st)ⁿ=ei.

The isomorphism with the usual presentation hρ, σ|σ² =ρⁿ=e ∧ ρσ=σρ⁻¹i is given by s 7→σ andt →σρ.

• The Coxeter group of typeA_n is isomorphic toS_n+1, generated by the simple transpositions s₁, . . . ,s_n, subject to

m_ii = 1 : (s_is_i)¹=e ⇔s_i² =e;

m_ij = 2 : (s_is_j)² =e ⇔s_is_j =s_js_i ifj 6=i±1;

mi(i±1)= 3 : (sisi±1)³=e⇔sisi±1si =si±1sisi±1.

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Coxeter diagrams of finite type

Weyl type non-Weyl type A_n

Bn=Cn 4

Dn

E₆ E₇ E₈

F4 4

G2 6

H₃ ⁵

H₄ ⁵

I2(n) ⁿ

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Hecke algebras

Recall thatH =H(W,S) is a deformation of Z[W] over Z[v,v⁻¹]:

s_i² =e s_i² = (v⁻²−1)s_i+v⁻².

Let{b_w |w ∈W} be the Kazhdan-Lusztig basis ofH and write b_ub_v = X

w∈W

h_u,v,wb_w,

forhu,v,w ∈Z[v,v⁻¹].

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The coinvariant algebra R

Definition

Leth^∗:=C{α_i |i = 1, . . . ,n}. The dual geometric representation ofW onh^∗ is defined by

si(αj) :=αj −2 cos π

mij

αi.

Definition

Let ˜R :=Sym(h^∗)∼=C[αi |i = 1, . . . ,n]. We define aZ-grading on R˜ by deg(h^∗) = 2 and theW-action on h^∗ extends to aW-action on ˜R by degree-preserving algebra-automorphisms. The coinvariant algebra isR := ˜R/( ˜R₊^W).

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The coinvariant algebra R

Definition

Leth^∗:=C{α_i |i = 1, . . . ,n}. The dual geometric representation ofW onh^∗ is defined by

si(αj) :=αj −2 cos π

mij

αi.

Definition

Let ˜R :=Sym(h^∗)∼=C[αi |i = 1, . . . ,n]. We define aZ-grading on R˜ by deg(h^∗) = 2 and theW-action on h^∗ extends to aW-action on ˜R by degree-preserving algebra-automorphisms. The coinvariant algebra isR := ˜R/( ˜R₊^W).

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Soergel bimodules

For everyi = 1, . . . ,n, define the R−R bimodule B_s_i :=R⊗_Rsi Rh1i.

Definition (Soergel)

LetS be the additive closure inR−bimod^fg_gr−R of the full, additive, graded, monoidal subcategory generated byBsihti, for i = 1, . . . ,n andt∈Z.

Remark: S is not abelian, e.g. the kernel of B_s_i =R⊗_Rsi R −−−−−−→^a⊗b^7→^ab R

is isomorphic toR as a right R-module but the leftR-action is twisted bys_i.

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Soergel bimodules

is isomorphic toR as a right R-module but the leftR-action is twisted bys_i.

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Soergel bimodules

is isomorphic toR as a right R-module but the leftR-action is

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Letw ∈W andw =s_i₁· · ·s_i_r a reduced expression (rex). The Bott-Samelson bimoduleis defined as

BS(w) :=B_s_i

1 ⊗_R · · · ⊗_R B_s_ir.

Theorem (Soergel)

S is idempotent complete and Krull-Schmidt. For every w ∈W , there is an indecomposable bimoduleB_w ∈ S, unique up to degree-preserving isomorphism, such that

(1) B_w is isomorphic to a direct summand, with multiplicity one, of BS(w) for any rex w of w ;

(2) For all t ∈Z,B_whtiis not isomorphic to a direct summand of BS(u) for any u<w and any rex u of u.

(3) Every indecomposable Soergel bimodule is isomorphic to Bwhti for some w ∈W and t∈Z.

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The categorification theorem

Theorem (Soergel, Elias-Williamson) TheZ[v,v⁻¹]-linear map given by

b_w 7→[B_w]

defines an algebra isomorphism between H and[S]_⊕ (split Grothendieck group).

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The categorification theorem

Letp =Ps

i=−raivⁱ ∈N[v,v⁻¹]. Define B^⊕p:=

s

M

i=−r

B^⊕aⁱh−ii.

Then the above theorem means: Positive Integrality

For allu,v ∈W, we have

B_u⊗_RB_v ∼= M

w∈W

B^⊕h_w ^u,v^,w,

whence

h_u,v_,w ∈N[v,v⁻¹].

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The categorification theorem

Letp =Ps

i=−raivⁱ ∈N[v,v⁻¹]. Define B^⊕p:=

s

M

i=−r

B^⊕aⁱh−ii.

Then the above theorem means:

Positive Integrality For allu,v ∈W, we have

B_u⊗_R B_v ∼= M

w∈W

B^⊕h_w ^u,v^,w,

whence

h_u,v,w ∈N[v,v⁻¹].

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Reduction to H-cells

•LetH:=L ∩ L^∗ inside some two-sided cell J. There exists a subquotient monoidal categoryS_HofS, whose indecomposable objects are all of the form Bxhtifor somex ∈ H andt ∈Z.

• Recall:

{Graded simple transitive 2-reps of S with apex J}/'

←→1:1

{Graded simple transitive 2-reps ofS_H with apex H}/'

←→1:1

{absolutely cosimple coalgebra objects in add(H)}/'_MT .

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Reduction to H-cells

•LetH:=L ∩ L^∗ inside some two-sided cell J. There exists a subquotient monoidal categoryS_HofS, whose indecomposable objects are all of the form Bxhtifor somex ∈ H andt ∈Z.

• Recall:

{Graded simple transitive 2-reps of S with apex J}/'

←→1:1

{Graded simple transitive 2-reps of S_H with apex H}/'

←→1:1

{absolutely cosimple coalgebra objects in add(H)}/'_MT.

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H-cells: Dihedral groups

The table below contains all Kazhdan-Lusztig cells ofD_2n (the H-cells are in blue).

e

s ,sts, . . . st,stst, . . . ts,tsts, . . . t ,tst, . . .

w₀

Remark: d is the so called Duflo involutionof the H-cell.

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Lusztig’s a-function

Fact: hx,y,z is symmetric invand v⁻¹. Proposition (Lusztig)

LetH:=L ∩ L^∗. There exists a∈N such that for allx,y,z ∈ H:

h_x,y,z =γ_x,y,z⁻¹v^a+· · ·+γ_x,y,z⁻¹v^−a.

Moreover, there exists a uniqued ∈ H (Duflo involution) such that d²=e in W and

γ_d,x,y⁻¹ =γ_x,d,y⁻¹ =γ_x,y⁻¹_,d =δx,y

for allx,y ∈ H.

Asymptotic limit:

γ_x,y,z⁻¹ = lim

v→+∞v^−ah_x,y,z∈N.

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Lusztig’s a-function

Fact: hx,y,z is symmetric invand v⁻¹. Proposition (Lusztig)

LetH:=L ∩ L^∗. There exists a∈N such that for allx,y,z ∈ H:

h_x,y,z =γ_x,y,z⁻¹v^a+· · ·+γ_x,y,z⁻¹v^−a.

Moreover, there exists a uniqued ∈ H (Duflo involution) such that d²=e in W and

γ_d,x,y⁻¹ =γ_x,d,y⁻¹ =γ_x,y⁻¹_,d =δx,y

for allx,y ∈ H.

Asymptotic limit:

γ_x,y,z⁻¹ = lim

v→+∞v^−ah_x,y,z ∈N.

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Definition (Lusztig’s asymptotic Hecke algebra)

The algebraAH is spanned (over Z[v,v⁻¹]) by a_w,w ∈ H, with multiplication

a_ua_v = X

w∈H

γ_u,v,w⁻¹a_w.

The unit is given by a_d.

Lusztig defined an injective homomorphism ofZ[v,v⁻¹]-algebras φ:HH→AH⊗_ZZ[v,v⁻¹] by

b_u7→ X

v∈H

h_u,d,va_v.

He also proved thatφis invertible overQ(v).

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Definition (Lusztig’s asymptotic Hecke algebra)

The algebraAH is spanned (over Z[v,v⁻¹]) by a_w,w ∈ H, with multiplication

a_ua_v = X

w∈H

γ_u,v,w⁻¹a_w.

The unit is given by a_d.

Lusztig defined an injective homomorphism ofZ[v,v⁻¹]-algebras φ:HH→AH⊗_ZZ[v,v⁻¹] by

b_u7→ X

v∈H

h_u,d,va_v.

He also proved thatφis invertible overQ(v).

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Example: A

_H_s

for Coxeter type I

₂

(n).

First considern= 4. Recall H_s ={s,sts}. We have

b²_s = [2]_vb_s, b_sb_sts =b_stsb_s = [2]_vb_sts, b²_sts = [2]_vb_s, where [2]_v=v+v⁻¹. We see thata= 1 and

t_s²=ts, tststs =tststs =tsts, t_sts² =ts. This shows thatAH_s ∼= [U_q(so₃)-mod_ss] for q=e^πi⁴.

Proposition

For anyn∈N≥2, we have

AHs ∼= [U_q(so₃)-mod_ss] forq=e^πiⁿ.

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Example: A

_H_s

for Coxeter type I

₂

(n).

First considern= 4. Recall H_s ={s,sts}. We have

b²_s = [2]_vb_s, b_sb_sts =b_stsb_s = [2]_vb_sts, b²_sts = [2]_vb_s, where [2]_v=v+v⁻¹. We see thata= 1 and

t_s²=ts, tststs =tststs =tsts, t_sts² =ts. This shows thatAH_s ∼= [U_q(so₃)-mod_ss] for q=e^πi⁴. Proposition

For anyn ∈N≥2, we have

AH_s ∼= [U_q(so₃)-mod_ss] forq=e^πiⁿ.

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Lusztig’s asymptotic Soergel categories

LetH:=L ∩ L^∗.

Theorem (Lusztig, Elias-Williamson)

There exists a (weak) fusion category(A_H, ?,∨) s.t.

(1) For every x ∈ H, there exists a simple objectAx. (2) The A_x, for x ∈ H, form a complete set of pairwise non-isomorphic simple objects.

(3) For any x,y ∈ H, we have Ax?Ay ∼=M

z∈H

A^⊕γz ^x,y^,z⁻¹.

(4) The identity object is A_d, where d is the Duflo involution.

(5) For every x ∈ H, we haveA^∨_x ∼=A_x⁻¹.

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Soergel’s hom-formula

Theorem (Soergel, Elias-Williamson) dim hom(Bx,Byhti)

=

(δx,y, if t = 0;

0 if t <0.

This implies thatS_H is a filtered category. By the properties of h_x,y_,z, the part

X_≤−a :=add {B_whki |w ∈ H,k ≤ −a}

is lax monoidal: It is strictly associative with lax identity object B_dh−ai. Define

A_H :=X_≤−a/ X_<−a

.

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Soergel’s hom-formula

=

(δx,y, if t = 0;

0 if t <0.

X_≤−a:=add {B_whki |w ∈ H,k ≤ −a}

is lax monoidal: It is strictly associative with lax identity object B_dh−ai.

Define

A_H :=X_≤−a/ X_<−a

.

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Soergel’s hom-formula

=

(δx,y, if t = 0;

0 if t <0.

X_≤−a:=add {B_whki |w ∈ H,k ≤ −a}

is lax monoidal: It is strictly associative with lax identity object B_dh−ai. Define

AH:=X≤−a/ X<−a

.

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Classification of A

_H

Theorem (Bezrukavnikov-Finkelberg-Ostrik, Ostrik, Elias) In all but a handful of cases,A_H is biequivalent to one of the following fusion categories:

(a)Vect_G or Rep(G), with G = (Z/2Z)^k,S3,S4,S5; (b) U_q(so₃)-mod_ss for q=e^πiⁿ for some n∈N≥2.

• Recall that we have a complete classification of all cosimple coalgebra objects in these fusion categories, up to

MT-equivalence.

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Main result 1

Theorem (M-Mazorchuk-Miemietz-Tubbenhauer-Zhang)

For any finite Coxeter group W and any diagonalH-cellH of W , there exists an oplax monoidal functor

Θ :A_H −→ S_H

withΘ(Ax)∼=Bxh−aiand (non-invertible) natural transformations η_x,y: Θ(A_x?A_y)→Θ(A_x)Θ(A_y)

for all x,y ∈ H.

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Main result 2 and main conjecture

General fact: Oplax monoidal functors send coalgebra objects to coalgebra objects and comodule categories to comodule categories.

Theorem (M-Mazorchuk-Miemietz-Tubbenhauer-Zhang) Θpreserves cosimplicity and MT-equivalence and induces an injection

Θ :b {Simple transitive2-reps ofA_H}/' ,−→

{Graded simple transitive2-reps ofS_H with apex H}/'. Conjecture (M-Mazorchuk-Miemietz-Tubbenhauer-Zhang) Θ is a bijection.b

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Main result 2 and main conjecture

Θ :b {Simple transitive2-reps of A_H}/' ,−→

{Graded simple transitive2-reps ofS_H with apex H}/'.

Conjecture (M-Mazorchuk-Miemietz-Tubbenhauer-Zhang) Θ is a bijection.b

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Main result 2 and main conjecture

Θ :b {Simple transitive2-reps of A_H}/' ,−→

{Graded simple transitive2-reps ofS_H with apex H}/'. Conjecture (M-Mazorchuk-Miemietz-Tubbenhauer-Zhang) Θ is a bijection.b

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Some remarks

• We have proved the conjecture for allH which contain the longest element of a parabolic subgroup ofW.

• If true, the conjecture implies that there are finitely many equivalence classes of simple transitive 2-representations of S.

• For almost allW andH, we would get a complete

classification of the graded, simple transitive 2-representations of S_H with apex H(and therefore of those of S).

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Some remarks

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Some remarks

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The cell 2-representation

We know quite a bit about the graded, simple transtive 2-representations ofS_H in the image of Θ, e.g. the cellb 2-representationC_H with apex H.

Theorem (MMMTZ)

Let d ∈ Hbe the Duflo involution anda the a-value ofH.

• B_d is a graded Frobenius object in S_H. More precisely, B_dhai is a graded algebra object, B_dh−aia graded coalgebra object and the product and coproduct morphisms satisfy the compatibility condition.

• inj_S_H(Bdh−ai)' C_H as 2-representations of S_H.

Remark: Klein and, separately, Elias-Hogancamp conjectured that B_d is a Frobenius algebra object in S, which is a stronger statement, but we do not know how to prove that.

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The cell 2-representation

Theorem (MMMTZ)

Remark: Klein and, separately, Elias-Hogancamp conjectured that B_d is a Frobenius algebra object in S, which is a stronger statement, but we do not know how to prove that.

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The cell 2-representation

Theorem (MMMTZ)

Remark: Klein and, separately, Elias-Hogancamp conjectured that B_d is a Frobenius algebra object in S, which is a stronger statement,

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The cell 2-representation

Proposition

• The underlying basic algebraA ofC_H is a positively graded, weakly symmetric Frobenius algebra of graded length 2a.

• Let 1 =P

w∈He_w. The action of B_wh−ai onA-mod_gr is given by tensoringA with

M

u,v∈H

Aev⊗euA^⊕γ^w,u,v⁻¹.

In particular,

B_dh−ai 7→M

u∈H

Ae_u⊗e_uA

andµ_d, δ_d, ι_d, _d are mapped to the A-A bimodule maps from my first talk (possibly up to some scalars).

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The cell 2-representation

Proposition

• Let 1 =P

M

u,v∈H

Ae_v⊗e_uA^⊕γ^w,u,v⁻¹.

In particular,

B_dh−ai 7→M

u∈H

Ae_u⊗e_uA

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The cell 2-representation

Proposition

• Let 1 =P

M

u,v∈H

Ae_v⊗e_uA^⊕γ^w,u,v⁻¹.

In particular,

B_dh−ai 7→M

u∈H

Ae_u⊗e_uA

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Graded dimension of A

Proposition

For anyu,w ∈ H, we have

grdim(euAew) =v^ah_u⁻¹_,w,d. In particular,

grdim(e_uA) =v^a X

w∈H

h_u⁻¹_,w,d.

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When is A symmetric?

For anyu ∈ H, define

λu:= X

w∈H

h_u⁻¹_,w,d(1)∈N.

Note thatλ_u= dim(e_uA).

Proposition (MMMTZ) IfAis symmetric, then

λ_u =λ_v ∀u,v ∈ H.

Fact: LetW be a Coxeter group of type E6,E7,E8,F4,H3 or H4. There are H-cells ofW which containu,v such that λ_u6=λ_v, so for thoseH-cellsA is weakly symmetric but not symmetric.

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When is A symmetric?

λu:= X

w∈H

h_u⁻¹_,w,d(1)∈N.

λ_u=λ_v ∀u,v ∈ H.

Fact: LetW be a Coxeter group of type E6,E7,E8,F4,H3 or H4. There are H-cells ofW which containu,v such that λ_u6=λ_v, so for thoseH-cellsA is weakly symmetric but not symmetric.

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When is A symmetric?

λu:= X

w∈H

h_u⁻¹_,w,d(1)∈N.

λ_u=λ_v ∀u,v ∈ H.

Fact: LetW be a Coxeter group of type E6,E7,E8,F4,H3 or H4. There are H-cells ofW which containu,v such thatλ_u6=λ_v, so for thoseH-cellsA is weakly symmetric but not symmetric.

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The asymptotic cell 2-representation

Note thatA0=⊕_w∈HCew. The asymptotic cell 2-representation of A_H is equivalent to

A₀-mod

and the action ofA_w onA₀-modis given by tensoring with M

u,v∈H

Cev ⊗euC^⊕γ^w,u,v⁻¹.

In particular, the action ofAd is given by tensoring with M

u∈H

Ce_u⊗e_uC.

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The asymptotic cell 2-representation

Note thatA0=⊕_w∈HCew. The asymptotic cell 2-representation of A_H is equivalent to

A₀-mod

and the action ofA_w onA₀-modis given by tensoring with M

u,v∈H

Cev ⊗euC^⊕γ^w,u,v⁻¹.

In particular, the action ofA_d is given by tensoring with M

u∈H

Ce_u⊗e_uC.

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THANKS!!!