Representation Theory of Hecke algebras & connections with Cherednik algebras

Representation Theory of Hecke algebras

& connections with Cherednik algebras

Cohomology in Lie Theory, Oxford Maria Chlouveraki

Universit´e de Versailles - St Quentin

Iwahori-Hecke algebras

Let (W,S) be a finite Coxeter system, W =hS | s²= 1, ststst. . .

| {z }

m_st

=tststs. . .

| {z }

m_st

∀s,t ∈S i.

S_n=

s₁,s₂, . . . ,s_n−1

s_i²= 1,s_is_i+1s_i=s_i+1s_is_i+1,s_is_j =s_js_i if |i−j|>1 LetL:S →Zbe a function such thatL(s) =L(t) whenevers andt are conjugate inW. Letq be an indeterminate.

Hq(W)=h(T_s)_s∈S|T_sT_tT_s. . .

| {z }

m_st

=T_tT_sT_t. . .

| {z }

m_st

,(T_s−q^L(s))(T_s+q^−L(s)) = 0∀s,ti.

Hq(Sn) =

*

T1,T2, . . . ,Tn−1

TiTi+1Ti =Ti+1TiTi+1, T_iT_j=T_jT_i if|i−j|>1, (Ti−q^K)(Ti+q^−K) = 0

+

where K :=L(s1) =· · ·=L(sn−1).

The a-function

Letw ∈W. Letw =s_i1s_i2. . .s_ir be areduced expressionforw. Set

Tw:=Ts_i1Ts_i2. . .Ts_ir. The algebraHq(W) is a freeC[q,q⁻¹]-module with basis (Tw)_w∈W.

Letτ :Hq(W)→C[q,q⁻¹] be the linear map defined byτ(T₁) = 1 and

τ(T_w) = 0 ifw 6= 1. The map τ is asymmetrising trace. By extension of scalars toC(q), we have

τ= X

V∈Irr(C(q)H_q(W))

1 sV

χ_V

for some sV ∈C[q,q⁻¹] (Schur elements).

The algebraC(q)Hq(W) is (split) semisimple, hence Irr(W) ↔ Irr(C(q)Hq(W))

E 7→ VE

.

LetE ∈Irr(W). We set

a(E) :=−valuation(s_VE) and A(E) :=−degree(s_VE).

Canonical basic sets

Letθ:C[q,q⁻¹]→C, q7→ξbe a ring homomorphism such that ξ∈C^×. The algebraCHξ(W) is not necessarily semisimple. We obtain adecomposition matrix

Dξ = ([VE :M])_{E∈Irr(W),M∈Irr(CHξ(W))}. Acanonical basic set forHξ(W) is a subsetBξ ofIrr(W) such that

1 Irr(CH_ξ(W))↔ B_ξ, M 7→E^M ;

2 [V_EM :M] = 1 for allM∈Irr(CH_ξ(W)) ;

3 if [VE :M]6= 0 for someE ∈Irr(W), then eitherE^M =E ora(E^M)<a(E).

Dξ =























1 0 · · · 0

∗ 1 · · · 0 ... ... . .. ...

∗ ∗ · · · 1

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗























| {z }

Irr(CHξ(W))









 Bξ





















 Irr(W)

Theorem [Geck, Rouquier, Jacon, C.]

Canonical basic sets exist for finite Coxeter groups.

It is well-known that Irr(S_n)↔

(

λ= (λ₁≥ · · · ≥λ_r ≥1) :

r

X

i=1

λ_i =n )

.

Letξ be a primitive root of unity of ordere>1. Then we have Bξ ={λ|λise-regular} ifK >0 and

Bξ={λ|λise-restricted} ifK <0 where K :=L(s₁) =· · ·=L(s_n−1).

Cellular structure

Under Lusztig’s conjectures (P1)–(P15), Iwahori-Hecke algebras are cellular: Cell modules M(E)_E∈Irr(W).

Symmetric bilinear formh, ion cell modules.

Theorem [Graham-Lehrer]

Set D(E) :=M(E)/rad_h,iM(E). We have that

1 D(E) is either 0 or a simpleCHξ(W)-module.

2 {D(E)|D(E)6= 0}=Irr(CHξ(W)) .

We have

{D(E)|D(E)6= 0}={D(E)|E ∈ B_ξ}.

Rational Cherednik algebras

Lethbe the reflection representation ofW, and letV =h⊕h^∗. We denote byS the set of all reflections inW. Fors∈S, take

αs∈h^∗ : basis ofIm(s−idv)|h^∗. α^∨_s ∈h: basis ofIm(s−idv)|h.

Letc:S→Cbe a function such that c(s) =c(t) whenevers andtare conjugate inW.

Hc(W)= TV^∗oW

[x,x⁰] = 0,[y,y⁰] = 0,[y,x] =x(y)−P

s∈Sc(s)αs(y)x(α^∨_s)s forx,x⁰ ∈h^∗ andy,y⁰∈h.

W =S_n,S={s_ij = (i j)}, (x₁, . . . ,x_n) basis ofh^∗, (y₁, . . . ,y_n) basis ofh.

σ·x_i =x_σ(i)·σ, σ·y_i =y_σ(i)·σ, ∀σ∈S_n.

Takeαij:=xi−xj,α^∨_ij :=yi−yj, for 1≤i<j ≤n, andc∈C. Then the commutation relations inHc(W) are:

[xi,xj] = 0, [yi,yj] = 0, [yi,xi] = 1−cP

j6=isij and [yi,xj] =csij fori6=j.

The category O

O= the category of finitely generatedH_c(W)-modules locally nilpotent for the action ofh.

Standard modules ∆(E) =Ind^Hc(W)

C[h]oW(E), E∈Irr(W).

Simple modulesL(E) =Head(∆(E)), E ∈Irr(W).

Decomposition matrix D^O = ([∆(F) :L(E)])_E,F∈Irr(W). [∆(E) :L(E)] = 1, for allE ∈Irr(W).

There exists an ordering <on the standard modules (and consequently on Irr(W)) such that if [∆(F) :L(E)]6= 0, then eitherE =F orE <F.

A famous ordering on the categoryO is the following:

E<F if and only ifc(F)−c(E)∈Z>0

where c(E) is the scalar with which theEuler element∈Z(CW) acts onE.

The KZ functor

There exists an exact functor

KZ:O −→ H_ξ(W)−mod where ξ=exp(2πic(s)/L(s)). We have:

1 KZ(L(E)) is either 0 or a simpleHξ(W)-module.

2 {KZ(L(E))|KZ(L(E))6= 0}=Irr(CHξ(W)).

3 IfKZ(L(E))6= 0, then [∆(F) :L(E)] = [KZ(∆(F)) :KZ(L(E))]. Thus,

I [KZ(∆(E)) :KZ(L(E))] = 1 ;

I If [KZ(∆(F)) :KZ(L(E))]6= 0, then eitherE =F orE <F.

Proposition [C-Gordon-Griffeth]

For allE ∈Irr(W), we have

c(E) =a(E) +A(E).

Main result

Theorem [C-Gordon-Griffeth]

Thea-function is an ordering on the categoryO. This in turn implies that

1 there exists a canonical basic setB_ξ forH_ξ(W) ;

2 KZ(L(E))6= 0 if and only ifE ∈ Bξ.

Moreover, we haveKZ(∆(E))∼=M(E) (cell module) for allE ∈Irr(W).

Remark: The above theorem holds for the complex reflection group G(`,1,n)∼= (Z/`Z)ⁿoSn (without the use of Ariki’s theorem).

The case of G (`, 1, n)

Lete∈Z>0and (s0,s1, . . . ,s`−1)∈Z<sup>`</sup>. We consider the specialised Ariki-Koike algebra defined by

generators : T0,T1, . . . ,T_n−1

relations : Tt₀ ⁴ Tt₁ Tt₂ · · · Tt_n−1

. (T0−ζ_e^s0)(T0−ζ_e^s1)· · ·(T0−ζe^s`−1) = 0

(Ti−ζe)(Ti+ 1) = 0, for alli= 1, . . . ,n−1.

We setm_j :=`s<sub>j</sub>−je andm:= (m<sub>j</sub>)<sub>0≤j≤`−1</sub>. We consider the cyclotomic Ariki-Koike algebraHq,mwith relations:

(T₀−q^m0)(T₀−ζ_{`</sub>q<sup>m1</sup>)· · ·(T<sub>0</sub>−ζ<sub>`}^{`−1</sup>q<sup>m`−1}) = 0 (T_i−q^`)(T_i+ 1) = 0, for alli= 1, . . . ,n−1







θ:q7→ζe`.

Π^{`</sup><sub>n</sub>={`-partitions of n} ↔ Irr(G(`,1,n)) ↔ Irr(Hq,m) λ= (λ<sup>(0)</sup>, . . . , λ<sup>(`−1)}) 7→ E^λ

Theorem [Geck-Jacon]

Canonical basic set ↔ {Uglov`-partitions}.

[λ] :={γ= (a,b,c)|0≤c≤`−1,a≥1,1≤b≤λ^(c)a }.

cont(γ) :=b(γ)−a(γ) and ϑ(γ) :=cont(γ) +sc.

Theorem [Dunkl-Griffeth]

Letλ, λ⁰∈Π^`_n. If [∆(E^λ) :L(E^λ0)]6= 0, then there exist orderings on the nodes γ₁, γ₂, . . . , γ_n andγ₁⁰, γ₂⁰, . . . , γ_n⁰ ofλandλ⁰ respectively, and integers

µ₁, µ₂, . . . , µ_n∈Z≥0such that, for all 1≤i ≤n,

µ_i ≡c(γ_i)−c(γ⁰_i)mod` and µ<sub>i</sub> =c(γ<sub>i</sub>)−c(γ<sub>i</sub><sup>0</sup>) + `

e(ϑ(γ⁰_i)−ϑ(γ_i)).

Theorem [C-Gordon-Griffeth]

Letλ, λ⁰∈Π^`_n. If [∆(E^λ) :L(E^λ0)]6= 0, then eitherλ⁰=λora(E^λ0)<a(E^λ).

Proof: Letγ andγ⁰ be nodes of `-partitions. We write γ≺γ⁰ if we have ϑ(γ)< ϑ(γ⁰) or ϑ(γ) =ϑ(γ⁰) and c(γ)>c(γ⁰).

Using the result by Dunkl and Griffeth, we can order the nodesγ1, γ2, . . . , γnet γ₁⁰, γ₂⁰, . . . , γ_n⁰ ofλandλ⁰ respectively such that, for all 1≤i≤n,

γi ≺γ⁰_i or γi =γ_i⁰. Then we can prove that either λ⁰ =λora(E^λ0)<a(E^λ).