Kazhdan-Lusztig theory and Ariki’s Theorem

Kazhdan-Lusztig theory and Ariki’s Theorem

C´edric Bonnaf´e

(joint work with Nicolas Jacon)

CNRS (UMR 6623) - Universit´e de Franche-Comt´e (Besan¸con)

Oberwolfach, March 2009

(W_n,S_n) Weyl group of type B_n S_n ={t,s₁,s₂, . . . ,s_n−1}

i i i · · · ⁱ

t s₁ s₂ s_n−1

`:W_n→N={0,1,2, . . .}length function R =Z[Q,Q⁻¹,q,q⁻¹], Q, q indeterminates

H_n=H_R(W_n,S_n,Q,q): Hecke algebra of type B_n with parametersQ and q. H_n =⊕w∈WnRT_w







TxTy =Txy if `(xy) =`(x) +`(y) (Tt−Q)(Tt+Q⁻¹) =0

(Ts_i −q)(Ts_i +q⁻¹) =0 if 16i 6n−1

(W_n,S_n) Weyl group of type B_n S_n ={t,s₁,s₂, . . . ,s_n−1}

i i i · · · ⁱ

t s₁ s₂ s_n−1

`:W_n→N={0,1,2, . . .}length function R =Z[Q,Q⁻¹,q,q⁻¹], Q, q indeterminates

H_n=H_R(W_n,S_n,Q,q): Hecke algebra of type B_n with parametersQ and q. H_n =⊕w∈WnRT_w







TxTy =Txy if `(xy) =`(x) +`(y) (Tt−Q)(Tt+Q⁻¹) =0

(Ts_i −q)(Ts_i +q⁻¹) =0 if 16i 6n−1

(W_n,S_n) Weyl group of type B_n S_n ={t,s₁,s₂, . . . ,s_n−1}

i i i · · · ⁱ

t s₁ s₂ s_n−1

`:W_n→N={0,1,2, . . .}length function

R =Z[Q,Q⁻¹,q,q⁻¹], Q, q indeterminates

H_n=H_R(W_n,S_n,Q,q): Hecke algebra of type B_n with parametersQ and q. H_n =⊕w∈WnRT_w







TxTy =Txy if `(xy) =`(x) +`(y) (Tt−Q)(Tt+Q⁻¹) =0

(Ts_i −q)(Ts_i +q⁻¹) =0 if 16i 6n−1

(W_n,S_n) Weyl group of type B_n S_n ={t,s₁,s₂, . . . ,s_n−1}

i i i · · · ⁱ

t s₁ s₂ s_n−1

`:W_n→N={0,1,2, . . .}length function R =Z[Q,Q⁻¹,q,q⁻¹], Q, q indeterminates

H_n=H_R(W_n,S_n,Q,q): Hecke algebra of type B_n with parametersQ and q.

H_n =⊕w∈WnRT_w







TxTy =Txy if `(xy) =`(x) +`(y) (Tt−Q)(Tt+Q⁻¹) =0

(Ts_i −q)(Ts_i +q⁻¹) =0 if 16i 6n−1

(W_n,S_n) Weyl group of type B_n S_n ={t,s₁,s₂, . . . ,s_n−1}

i i i · · · ⁱ

t s₁ s₂ s_n−1

`:W_n→N={0,1,2, . . .}length function R =Z[Q,Q⁻¹,q,q⁻¹], Q, q indeterminates

H_n=H_R(W_n,S_n,Q,q): Hecke algebra of type B_n with parametersQ and q. H_n =⊕w∈WnRT_w







TxTy =Txy if `(xy) =`(x) +`(y) (Tt−Q)(Tt+Q⁻¹) =0

(Ts_i −q)(Ts_i +q⁻¹) =0 if 16i 6n−1

(W_n,S_n) Weyl group of type B_n S_n ={t,s₁,s₂, . . . ,s_n−1}

i i i · · · ⁱ

t s₁ s₂ s_n−1

`:W_n→N={0,1,2, . . .}length function R =Z[Q,Q⁻¹,q,q⁻¹], Q, q indeterminates

H_n=H_R(W_n,S_n,Q,q): Hecke algebra of type B_n with parametersQ and q. H_n =⊕w∈WnRT_w







TxTy =Txy if `(xy) =`(x) +`(y) (Tt−Q)(Tt+Q⁻¹) =0

(Ts_i −q)(Ts_i +q⁻¹) =0 if 16i 6n−1

K =Frac(R),KH_n=K ⊗_R H_n split semisimple

IrrKHn={Vλ |λ∈Bip(n)}, whereBip(n) ={bipartitions of n}.

Q₀,q₀ ∈C^×, specialization −→CH_n =C⊗_R H_n

R₀(CH_n) := Grothendieck group of CH_n 'ZIrrCH_n decomposition mapdn:R0(KHn)−→R0(CHn)

Hypothesis and notation Q₀² = −q₀^2d, d ∈Z e =order of q₀², e >2.

K =Frac(R),KH_n=K ⊗_R H_n split semisimple IrrKHn={Vλ |λ∈Bip(n)}, whereBip(n) ={bipartitions of n}.

Q₀,q₀ ∈C^×, specialization −→CH_n =C⊗_R H_n

R₀(CH_n) := Grothendieck group of CH_n 'ZIrrCH_n decomposition mapdn:R0(KHn)−→R0(CHn)

Hypothesis and notation Q₀² = −q₀^2d, d ∈Z e =order of q₀², e >2.

K =Frac(R),KH_n=K ⊗_R H_n split semisimple IrrKHn={Vλ |λ∈Bip(n)}, whereBip(n) ={bipartitions of n}.

Q₀,q₀ ∈C^×, specialization −→CH_n =C⊗_R H_n

R₀(CH_n) := Grothendieck group of CH_n 'ZIrrCH_n decomposition mapdn:R0(KHn)−→R0(CHn)

Hypothesis and notation Q₀² = −q₀^2d, d ∈Z e =order of q₀², e >2.

K =Frac(R),KH_n=K ⊗_R H_n split semisimple IrrKHn={Vλ |λ∈Bip(n)}, whereBip(n) ={bipartitions of n}.

Q₀,q₀ ∈C^×, specialization −→CH_n =C⊗_R H_n

R₀(CH_n) := Grothendieck group of CH_n 'ZIrrCH_n

decomposition mapdn:R0(KHn)−→R0(CHn) Hypothesis and notation

Q₀² = −q₀^2d, d ∈Z e =order of q₀², e >2.

K =Frac(R),KH_n=K ⊗_R H_n split semisimple IrrKHn={Vλ |λ∈Bip(n)}, whereBip(n) ={bipartitions of n}.

Q₀,q₀ ∈C^×, specialization −→CH_n =C⊗_R H_n

R₀(CH_n) := Grothendieck group of CH_n 'ZIrrCH_n decomposition mapdn:R0(KHn)−→R0(CHn)

Hypothesis and notation Q₀² = −q₀^2d, d ∈Z e =order of q₀², e >2.

K =Frac(R),KH_n=K ⊗_R H_n split semisimple IrrKHn={Vλ |λ∈Bip(n)}, whereBip(n) ={bipartitions of n}.

Q₀,q₀ ∈C^×, specialization −→CH_n =C⊗_R H_n

R₀(CH_n) := Grothendieck group of CH_n 'ZIrrCH_n decomposition mapdn:R0(KHn)−→R0(CHn)

Hypothesis and notation Q₀² = −q₀^2d, d ∈Z e =order of q₀², e >2.

Fock space:

Bip= a

n>0

Bip(n), r >0, v indeterminate Fock space: F_r := ⊕

λ∈BipC(v) |λ,ri

F_r is endowed with an action of U_v( ^sl_e) depending on r Uglovhas constructed an involution¯:Fr →Fr and there exists a unique G(λ,r)∈F_r such that

G(λ,r) =G(λ,r)

G(λ,r)≡|λ,ri mod vC[v] Write G(µ,r) = X

λ∈Bip

d_λµ^r (v) |λ,ri (note thatd_λλ^r (v) =1). (|λ,ri)_λ∈Bip is called thestandard basis

(G(λ,r))λ∈Bip is called theKashiwara-Lusztig canonical basis

Fock space:

Bip= a

n>0

Bip(n), r >0, v indeterminate

Fock space: F_r := ⊕

λ∈BipC(v) |λ,ri

F_r is endowed with an action of U_v( ^sl_e) depending on r Uglovhas constructed an involution¯:Fr →Fr and there exists a unique G(λ,r)∈F_r such that

G(λ,r) =G(λ,r)

G(λ,r)≡|λ,ri mod vC[v] Write G(µ,r) = X

λ∈Bip

d_λµ^r (v) |λ,ri (note thatd_λλ^r (v) =1). (|λ,ri)_λ∈Bip is called thestandard basis

(G(λ,r))λ∈Bip is called theKashiwara-Lusztig canonical basis

Fock space:

Bip= a

n>0

Bip(n), r >0, v indeterminate Fock space: F_r := ⊕

λ∈BipC(v) |λ,ri

F_r is endowed with an action of U_v( ^sl_e) depending on r Uglovhas constructed an involution¯:Fr →Fr and there exists a unique G(λ,r)∈F_r such that

G(λ,r) =G(λ,r)

G(λ,r)≡|λ,ri mod vC[v] Write G(µ,r) = X

λ∈Bip

d_λµ^r (v) |λ,ri (note thatd_λλ^r (v) =1). (|λ,ri)_λ∈Bip is called thestandard basis

(G(λ,r))λ∈Bip is called theKashiwara-Lusztig canonical basis

Fock space:

Bip= a

n>0

Bip(n), r >0, v indeterminate Fock space: F_r := ⊕

λ∈BipC(v) |λ,ri F_r is endowed with an action of U_v( ^sl_e)

depending on r Uglovhas constructed an involution¯:Fr →Fr and there exists a unique G(λ,r)∈F_r such that

G(λ,r) =G(λ,r)

G(λ,r)≡|λ,ri mod vC[v] Write G(µ,r) = X

λ∈Bip

d_λµ^r (v) |λ,ri (note thatd_λλ^r (v) =1). (|λ,ri)_λ∈Bip is called thestandard basis

(G(λ,r))λ∈Bip is called theKashiwara-Lusztig canonical basis

Fock space:

Bip= a

n>0

Bip(n), r >0, v indeterminate Fock space: F_r := ⊕

λ∈BipC(v) |λ,ri

F_r is endowed with an action of U_v( ^sl_e) depending on r

Uglovhas constructed an involution¯:Fr →Fr and there exists a unique G(λ,r)∈F_r such that

G(λ,r) =G(λ,r)

G(λ,r)≡|λ,ri mod vC[v] Write G(µ,r) = X

λ∈Bip

d_λµ^r (v) |λ,ri (note thatd_λλ^r (v) =1). (|λ,ri)_λ∈Bip is called thestandard basis

(G(λ,r))λ∈Bip is called theKashiwara-Lusztig canonical basis

Fock space:

Bip= a

n>0

Bip(n), r >0, v indeterminate Fock space: F_r := ⊕

λ∈BipC(v) |λ,ri

F_r is endowed with an action of U_v( ^sl_e) depending on r Uglovhas constructed an involution¯:Fr →Fr and there exists a unique G(λ,r)∈F_r such that

G(λ,r) =G(λ,r)

G(λ,r)≡|λ,ri mod vC[v]

Write G(µ,r) = X

λ∈Bip

d_λµ^r (v) |λ,ri (note thatd_λλ^r (v) =1). (|λ,ri)_λ∈Bip is called thestandard basis

(G(λ,r))λ∈Bip is called theKashiwara-Lusztig canonical basis

Fock space:

Bip= a

n>0

Bip(n), r >0, v indeterminate Fock space: F_r := ⊕

λ∈BipC(v) |λ,ri

F_r is endowed with an action of U_v( ^sl_e) depending on r Uglovhas constructed an involution¯:Fr →Fr and there exists a unique G(λ,r)∈F_r such that

G(λ,r) =G(λ,r)

G(λ,r)≡|λ,ri mod vC[v] Write G(µ,r) = X

λ∈Bip

d_λµ^r (v)|λ,ri (note thatd_λλ^r (v) =1).

(|λ,ri)_λ∈Bip is called thestandard basis

(G(λ,r))λ∈Bip is called theKashiwara-Lusztig canonical basis

Fock space:

Bip= a

n>0

Bip(n), r >0, v indeterminate Fock space: F_r := ⊕

λ∈BipC(v) |λ,ri

F_r is endowed with an action of U_v( ^sl_e) depending on r Uglovhas constructed an involution¯:Fr →Fr and there exists a unique G(λ,r)∈F_r such that

G(λ,r) =G(λ,r)

G(λ,r)≡|λ,ri mod vC[v] Write G(µ,r) = X

λ∈Bip

d_λµ^r (v)|λ,ri (note thatd_λλ^r (v) =1).

(|λ,ri)_λ∈Bip is called the standard basis

(G(λ,r))λ∈Bip is called the Kashiwara-Lusztig canonical basis

Ariki’s Theorem (Ariki, Uglov, Geck-Jacon). Assume that r ≡d mod e. There exists a subset Bip_e,r(n) of Bip(n) and a bijection

Bip_e,r(n) −→ IrrCHn

λ 7−→ D_λ^e,r

such that

Uv( ^sle) |∅,ri= ⊕

λ∈Bip_e,rC(v) G(λ,r), where Bip_e,r = a

n>0

Bip_e,r(n)

If λ∈Bip(n), then d_n[V_λ] = X

µ∈Bip_e,r(n)

d_λµ^r (1) [D_µ^e,r]

Remark - d_λµ^r (v) is “computable”

Ariki’s Theorem (Ariki, Uglov, Geck-Jacon). Assume that r ≡d mod e. There exists a subset Bip_e,r(n) of Bip(n) and a bijection

Bip_e,r(n) −→ IrrCHn

λ 7−→ D_λ^e,r

such that

Uv( ^sle) |∅,ri= ⊕

λ∈Bip_e,rC(v) G(λ,r), where Bip_e,r = a

n>0

Bip_e,r(n)

If λ∈Bip(n), then d_n[V_λ] = X

µ∈Bip_e,r(n)

d_λµ^r (1) [D_µ^e,r]

Remark - d_λµ^r (v) is “computable”

Ariki’s Theorem (Ariki, Uglov, Geck-Jacon). Assume that r ≡d mod e. There exists a subset Bip_e,r(n) of Bip(n) and a bijection

Bip_e,r(n) −→ IrrCHn

λ 7−→ D_λ^e,r

such that

Uv( ^sle) |∅,ri= ⊕

λ∈Bip_e,rC(v) G(λ,r), where Bip_e,r = a

n>0

Bip_e,r(n)

If λ∈Bip(n), then d_n[V_λ] = X

µ∈Bip_e,r(n)

d_λµ^r (1) [D_µ^e,r]

Remark - d_λµ^r (v) is “computable”

Ariki’s Theorem (Ariki, Uglov, Geck-Jacon). Assume that r ≡d mod e. There exists a subset Bip_e,r(n) of Bip(n) and a bijection

Bip_e,r(n) −→ IrrCHn

λ 7−→ D_λ^e,r

such that

Uv( ^sle) |∅,ri= ⊕

λ∈Bip_e,rC(v) G(λ,r), where Bip_e,r = a

n>0

Bip_e,r(n)

If λ∈Bip(n), then d_n[V_λ] = X

µ∈Bip_e,r(n)

d_λµ^r (1) [D_µ^e,r]

Remark - d_λµ^r (v) is “computable”

Comments:

|Bip_e,r(n)|=|Bip_e,r+ke(n)| if k ∈Z (for a bijection, see Jacon, Jacon-Lecouvey)

If r >n−1,Bip_e,r(n) ={Kleshchev bipartitions} (see Ariki: it is related to the Dipper-James-Mathas or to the Graham-Lehrer cellular structure).

Bip_d

0,e(n) ={FLOTW bipartitions} (Jacon). Here, d₀ ≡d mod e and d₀ ∈{0,1,2, . . . ,e−1}.

Comments:

|Bip_e,r(n)|=|Bip_e,r+ke(n)| if k ∈Z (for a bijection, see Jacon, Jacon-Lecouvey)

If r >n−1,Bip_e,r(n) ={Kleshchev bipartitions} (see Ariki: it is related to the Dipper-James-Mathas or to the Graham-Lehrer cellular structure).

Bip_d

0,e(n) ={FLOTW bipartitions} (Jacon). Here, d₀ ≡d mod e and d₀ ∈{0,1,2, . . . ,e−1}.

Comments:

|Bip_e,r(n)|=|Bip_e,r+ke(n)| if k ∈Z (for a bijection, see Jacon, Jacon-Lecouvey)

If r >n−1,Bip_e,r(n) ={Kleshchev bipartitions} (see Ariki: it is related to the Dipper-James-Mathas or to the Graham-Lehrer cellular structure).

Bip_d

0,e(n) ={FLOTW bipartitions} (Jacon). Here, d₀ ≡d mod e and d₀ ∈{0,1,2, . . . ,e−1}.

R =Z[Z²]

= ⊕

γ∈Z²

Ze^γ. Q =e^(1,0), q =e^(0,1).

For Kazhdan-Lusztig theory (with unequal parameters) you need another ingredient: a total orderon Z² (compatible with addition).

Fix θ∈R⁺,irrational (!): let 6θ be the total order on Z² defined by

(m,n)6θ (m⁰,n⁰)⇐⇒mθ+n6m⁰θ+n⁰ (roughly speaking, “Q =q^θ”...)

Let¯:H_n →H_n, T_w 7→T_w⁻¹−1, Q 7→Q⁻¹, q7→q⁻¹ (i.e. e^γ7→e^−γ) antilinear involution.

R_<θ0 = ⊕

γ∈Z²_<θ0

Ze^γ, H^<_n^θ0 = ⊕

w∈Wn

R_<θ0Tw.

R =Z[Z²] = ⊕

γ∈Z²

Ze^γ. Q =e^(1,0), q =e^(0,1).

For Kazhdan-Lusztig theory (with unequal parameters) you need another ingredient: a total orderon Z² (compatible with addition).

Fix θ∈R⁺,irrational (!): let 6θ be the total order on Z² defined by

(m,n)6θ (m⁰,n⁰)⇐⇒mθ+n6m⁰θ+n⁰ (roughly speaking, “Q =q^θ”...)

Let¯:H_n →H_n, T_w 7→T_w⁻¹−1, Q 7→Q⁻¹, q7→q⁻¹ (i.e. e^γ7→e^−γ) antilinear involution.

R_<θ0 = ⊕

γ∈Z²_<θ0

Ze^γ, H^<_n^θ0 = ⊕

w∈Wn

R_<θ0Tw.

R =Z[Z²] = ⊕

γ∈Z²

Ze^γ. Q =e^(1,0), q =e^(0,1).

For Kazhdan-Lusztig theory (with unequal parameters) you need another ingredient: a total orderon Z² (compatible with addition).

Fix θ∈R⁺,irrational (!): let 6θ be the total order on Z² defined by

(m,n)6θ (m⁰,n⁰)⇐⇒mθ+n6m⁰θ+n⁰ (roughly speaking, “Q =q^θ”...)

Let¯:H_n →H_n, T_w 7→T_w⁻¹−1, Q 7→Q⁻¹, q7→q⁻¹ (i.e. e^γ7→e^−γ) antilinear involution.

R_<θ0 = ⊕

γ∈Z²_<θ0

Ze^γ, H^<_n^θ0 = ⊕

w∈Wn

R_<θ0Tw.

R =Z[Z²] = ⊕

γ∈Z²

Ze^γ. Q =e^(1,0), q =e^(0,1).

For Kazhdan-Lusztig theory (with unequal parameters) you need another ingredient: a total orderon Z² (compatible with addition).

Fix θ∈R⁺,irrational (!): let 6θ be the total order on Z² defined by

(m,n)6θ (m⁰,n⁰)⇐⇒mθ+n 6m⁰θ+n⁰

(roughly speaking, “Q =q^θ”...)

Let¯:H_n →H_n, T_w 7→T_w⁻¹−1, Q 7→Q⁻¹, q7→q⁻¹ (i.e. e^γ7→e^−γ) antilinear involution.

R_<θ0 = ⊕

γ∈Z²_<θ0

Ze^γ, H^<_n^θ0 = ⊕

w∈Wn

R_<θ0Tw.

R =Z[Z²] = ⊕

γ∈Z²

Ze^γ. Q =e^(1,0), q =e^(0,1).

For Kazhdan-Lusztig theory (with unequal parameters) you need another ingredient: a total orderon Z² (compatible with addition).

Fix θ∈R⁺,irrational (!): let 6θ be the total order on Z² defined by

(m,n)6θ (m⁰,n⁰)⇐⇒mθ+n 6m⁰θ+n⁰ (roughly speaking, “Q =q^θ”...)

Let¯:H_n →H_n, T_w 7→T_w⁻¹−1, Q 7→Q⁻¹, q7→q⁻¹ (i.e. e^γ7→e^−γ) antilinear involution.

R_<θ0 = ⊕

γ∈Z²_<θ0

Ze^γ, H^<_n^θ0 = ⊕

w∈Wn

R_<θ0Tw.

R =Z[Z²] = ⊕

γ∈Z²

Ze^γ. Q =e^(1,0), q =e^(0,1).

For Kazhdan-Lusztig theory (with unequal parameters) you need another ingredient: a total orderon Z² (compatible with addition).

Fix θ∈R⁺,irrational (!): let 6θ be the total order on Z² defined by

(m,n)6θ (m⁰,n⁰)⇐⇒mθ+n 6m⁰θ+n⁰ (roughly speaking, “Q =q^θ”...)

Let¯:H_n→H_n, T_w 7→T_w⁻¹−1, Q 7→Q⁻¹, q7→q⁻¹ (i.e.

e^γ7→e^−γ) antilinear involution.

R_<θ0 = ⊕

γ∈Z²_<θ0

Ze^γ, H^<_n^θ0 = ⊕

w∈Wn

R_<θ0Tw.

R =Z[Z²] = ⊕

γ∈Z²

Ze^γ. Q =e^(1,0), q =e^(0,1).

For Kazhdan-Lusztig theory (with unequal parameters) you need another ingredient: a total orderon Z² (compatible with addition).

Fix θ∈R⁺,irrational (!): let 6θ be the total order on Z² defined by

(m,n)6θ (m⁰,n⁰)⇐⇒mθ+n 6m⁰θ+n⁰ (roughly speaking, “Q =q^θ”...)

Let¯:H_n→H_n, T_w 7→T_w⁻¹−1, Q 7→Q⁻¹, q7→q⁻¹ (i.e.

e^γ7→e^−γ) antilinear involution.

R_<θ0 = ⊕

γ∈Z²_<θ0

Ze^γ,

H^<_n^θ0 = ⊕

w∈Wn

R_<θ0Tw.

R =Z[Z²] = ⊕

γ∈Z²

Ze^γ. Q =e^(1,0), q =e^(0,1).

For Kazhdan-Lusztig theory (with unequal parameters) you need another ingredient: a total orderon Z² (compatible with addition).

Fix θ∈R⁺,irrational (!): let 6θ be the total order on Z² defined by

(m,n)6θ (m⁰,n⁰)⇐⇒mθ+n 6m⁰θ+n⁰ (roughly speaking, “Q =q^θ”...)

Let¯:H_n→H_n, T_w 7→T_w⁻¹−1, Q 7→Q⁻¹, q7→q⁻¹ (i.e.

e^γ7→e^−γ) antilinear involution.

R_<θ0 = ⊕

γ∈Z²_<θ0

Ze^γ, H^<_n^θ0 = ⊕

w∈Wn

R_<θ0T_w.

Theorem (Kazhdan-Lusztig, 1979). For each w ∈W_n, there exists a uniqueC_w^θ ∈ Hn such that

C^θ_w =C_w^θ

C_w^θ ≡T_w mod H_n^<θ0

Case n=2 (write s =s₁)

C₁^θ = 1, C_t^θ =T_t+Q⁻¹, C_s^θ=T_s+q⁻¹ C_st^θ = Tst +Q⁻¹Ts+q⁻¹Tt+Q⁻¹q⁻¹ C_ts^θ = Tts +Q⁻¹Ts+q⁻¹Tt+Q⁻¹q⁻¹ C_sts^θ = T_sts +q⁻¹(T_st +T_ts) +

q⁻²Tt+Q⁻¹q⁻¹(1+q²)(Ts+q⁻¹) if θ >1

Q⁻¹q⁻¹T_s+Q⁻¹q⁻¹(1−Q²)(T_t+Q⁻¹) if 0< θ <1 C_tst^θ = Ttst +Q⁻¹(Tst +Tts) +

Q⁻¹q⁻¹Ts+Q⁻¹q⁻¹(1−q²)(Tt+Q⁻¹) ifθ >1 Q⁻¹q⁻¹T_t+Q⁻¹q⁻¹(1+q²)(T_s +q⁻¹) if 0< θ <1 C_w^θ₀ = Tw0+Q⁻¹Tsts+q⁻¹Ttst +Q⁻¹q⁻¹(Tst +Tts)

+Q⁻²q⁻¹T_s+Q⁻¹q⁻²T_t+Q⁻²q⁻²

If x, y ∈W, we write x ←^L,θ−y if there existsh ∈ H_n such that C_x^θ occurs inhC_y^θ

Let6^θL be the transitive closure of ←^L,θ−: it is a preorder (reflexive and transitive)

Let∼^θL be the equivalence relation associated to 6^θL (i.e. x ∼^θL y if and only ifx 6L y and y 6^θL x)

Definition

Aθ-left cell is an equivalence class for the relation ∼^θL.

If C is a θ-left cell, we set









 I₆^θ

LC = ⊕

x6^θ_LC

RC_x^θ I_<^θ

LC = ⊕

x<^θ_LC

RC_x^θ V_C^θ =I₆^θ

LC/I_<^θ

LC

By construction, I₆^θ

LC and I_<^θ

LC are left ideals of H_n and V_C^θ is a leftHn-module: V_C^θ is called the left cell representation associated toC.

If x, y ∈W, we write x ←^L,θ−y if there existsh ∈ H_n such that C_x^θ occurs inhC_y^θ

Let6^θL be the transitive closure of ←^L,θ−:

it is a preorder (reflexive and transitive)

Let∼^θL be the equivalence relation associated to 6^θL (i.e. x ∼^θL y if and only ifx 6L y and y 6^θL x)

Definition

Aθ-left cell is an equivalence class for the relation ∼^θL.

If C is a θ-left cell, we set









 I₆^θ

LC = ⊕

x6^θ_LC

RC_x^θ I_<^θ

LC = ⊕

x<^θ_LC

RC_x^θ V_C^θ =I₆^θ

LC/I_<^θ

LC

By construction, I₆^θ

LC and I_<^θ

LC are left ideals of H_n and V_C^θ is a leftHn-module: V_C^θ is called the left cell representation associated toC.

If x, y ∈W, we write x ←^L,θ−y if there existsh ∈ H_n such that C_x^θ occurs inhC_y^θ

Let6^θL be the transitive closure of ←^L,θ−: it is a preorder (reflexive and transitive)

Let∼^θL be the equivalence relation associated to 6^θL (i.e. x ∼^θL y if and only ifx 6L y and y 6^θL x)

Definition

Aθ-left cell is an equivalence class for the relation ∼^θL.

If C is a θ-left cell, we set









 I₆^θ

LC = ⊕

x6^θ_LC

RC_x^θ I_<^θ

LC = ⊕

x<^θ_LC

RC_x^θ V_C^θ =I₆^θ

LC/I_<^θ

LC

By construction, I₆^θ

LC and I_<^θ

LC are left ideals of H_n and V_C^θ is a leftHn-module: V_C^θ is called the left cell representation associated toC.

If x, y ∈W, we write x ←^L,θ−y if there existsh ∈ H_n such that C_x^θ occurs inhC_y^θ

Let6^θL be the transitive closure of ←^L,θ−: it is a preorder (reflexive and transitive)

Let∼^θL be the equivalence relation associated to 6^θL (i.e. x ∼^θL y if and only ifx 6L y and y 6^θL x)

Definition

Aθ-left cell is an equivalence class for the relation ∼^θL.

If C is a θ-left cell, we set









 I₆^θ

LC = ⊕

x6^θ_LC

RC_x^θ I_<^θ

LC = ⊕

x<^θ_LC

RC_x^θ V_C^θ =I₆^θ

LC/I_<^θ

LC

By construction, I₆^θ

LC and I_<^θ

LC are left ideals of H_n and V_C^θ is a leftHn-module: V_C^θ is called the left cell representation associated toC.

If x, y ∈W, we write x ←^L,θ−y if there existsh ∈ H_n such that C_x^θ occurs inhC_y^θ

Let6^θL be the transitive closure of ←^L,θ−: it is a preorder (reflexive and transitive)

Let∼^θL be the equivalence relation associated to 6^θL (i.e. x ∼^θL y if and only ifx 6L y and y 6^θL x)

Definition

Aθ-left cell is an equivalence class for the relation ∼^θL.

If C is a θ-left cell, we set









 I₆^θ

LC = ⊕

x6^θ_LC

RC_x^θ I_<^θ

LC = ⊕

x<^θ_LC

RC_x^θ V_C^θ =I₆^θ

LC/I_<^θ

LC

By construction, I₆^θ

LC and I_<^θ

LC are left ideals of H_n and V_C^θ is a leftHn-module: V_C^θ is called the left cell representation associated toC.

If x, y ∈W, we write x ←^L,θ−y if there existsh ∈ H_n such that C_x^θ occurs inhC_y^θ

Let6^θL be the transitive closure of ←^L,θ−: it is a preorder (reflexive and transitive)

Let∼^θL be the equivalence relation associated to 6^θL (i.e. x ∼^θL y if and only ifx 6L y and y 6^θL x)

Definition

Aθ-left cell is an equivalence class for the relation ∼^θL.

If C is a θ-left cell, we set









 I₆^θ

LC = ⊕

x6^θ_LC

RC_x^θ

I_<^θ

LC = ⊕

x<^θ_LC

RC_x^θ V_C^θ =I₆^θ

LC/I_<^θ

LC

By construction, I₆^θ

LC and I_<^θ

LC are left ideals of H_n and V_C^θ is a leftHn-module: V_C^θ is called the left cell representation associated toC.

If x, y ∈W, we write x ←^L,θ−y if there existsh ∈ H_n such that C_x^θ occurs inhC_y^θ

Let6^θL be the transitive closure of ←^L,θ−: it is a preorder (reflexive and transitive)

Let∼^θL be the equivalence relation associated to 6^θL (i.e. x ∼^θL y if and only ifx 6L y and y 6^θL x)

Definition

Aθ-left cell is an equivalence class for the relation ∼^θL.

If C is a θ-left cell, we set









 I₆^θ

LC = ⊕

x6^θ_LC

RC_x^θ I_<^θ

LC = ⊕

x<^θ_LC

RC_x^θ

V_C^θ =I₆^θ

LC/I_<^θ

LC

By construction, I₆^θ

LC and I_<^θ

LC are left ideals of H_n and V_C^θ is a leftHn-module: V_C^θ is called the left cell representation associated toC.

If x, y ∈W, we write x ←^L,θ−y if there existsh ∈ H_n such that C_x^θ occurs inhC_y^θ

Let6^θL be the transitive closure of ←^L,θ−: it is a preorder (reflexive and transitive)

Let∼^θL be the equivalence relation associated to 6^θL (i.e. x ∼^θL y if and only ifx 6L y and y 6^θL x)

Definition

Aθ-left cell is an equivalence class for the relation ∼^θL.

If C is a θ-left cell, we set









 I₆^θ

LC = ⊕

x6^θ_LC

RC_x^θ I_<^θ

LC = ⊕

x<^θ_LC

RC_x^θ V_C^θ=I₆^θ

LC/I_<^θ

LC

By construction, I₆^θ

LC and I_<^θ

LC are left ideals of H_n and V_C^θ is a leftHn-module: V_C^θ is called the left cell representation associated toC.

If x, y ∈W, we write x ←^L,θ−y if there existsh ∈ H_n such that C_x^θ occurs inhC_y^θ

Let6^θL be the transitive closure of ←^L,θ−: it is a preorder (reflexive and transitive)

Let∼^θL be the equivalence relation associated to 6^θL (i.e. x ∼^θL y if and only ifx 6L y and y 6^θL x)

Definition

Aθ-left cell is an equivalence class for the relation ∼^θL.

If C is a θ-left cell, we set









 I₆^θ

LC = ⊕

x6^θ_LC

RC_x^θ I_<^θ

LC = ⊕

x<^θ_LC

RC_x^θ V_C^θ=I₆^θ

LC/I_<^θ

LC

By construction, I₆^θ

LC and I_<^θ

LC are left ideals of H_n and V_C^θ is a leftHn-module

: V_C^θ is called the left cell representation associated toC.

If x, y ∈W, we write x ←^L,θ−y if there existsh ∈ H_n such that C_x^θ occurs inhC_y^θ

Let6^θL be the transitive closure of ←^L,θ−: it is a preorder (reflexive and transitive)

Let∼^θL be the equivalence relation associated to 6^θL (i.e. x ∼^θL y if and only ifx 6L y and y 6^θL x)

Definition

Aθ-left cell is an equivalence class for the relation ∼^θL.

If C is a θ-left cell, we set









 I₆^θ

LC = ⊕

x6^θ_LC

RC_x^θ I_<^θ

LC = ⊕

x<^θ_LC

RC_x^θ V_C^θ=I₆^θ

LC/I_<^θ

LC

By construction, I₆^θ

LC and I_<^θ

LC are left ideals of H_n and V_C^θ is a leftHn-module: V_C^θ is called the left cell representation associated toC.

Theorem (B., Geck, Iancu, Jacon, Lam, Lusztig, Pietraho, Taskin)

Assume that Lusztig’s conjectures P1, P2,..., P15 hold. Then :

If C is a θ-left cell of W_n, then KV_C^θ ∈IrrKH_n. We denove by λ(C) the bipartition such that KV_C^θ 'Vλ(C).

If KV_C^θ 'KV_C^θ0, then V_C^θ 'V_C^θ0. We write S_λ^θ =V_C^θ isλ(C) =λ. S_λ^θ is endowed with an “H_n-invariant” bilinear form φ^θ_λ. We set

D_λ^θ =CS_λ^θ/KerCφ^θ_λ. LetBip_θ(n) ={λ∈Bip(n) |D_λ^θ 6=0}. Then

IrrCH_n ={D_λ^θ | λ∈Bip_θ(n)}.

Theorem (B., Geck, Iancu, Jacon, Lam, Lusztig, Pietraho, Taskin)

Assume that Lusztig’s conjectures P1, P2,..., P15 hold. Then : If C is a θ-left cell of W_n, then KV_C^θ ∈IrrKH_n.

We denove by λ(C) the bipartition such that KV_C^θ 'Vλ(C).

If KV_C^θ 'KV_C^θ0, then V_C^θ 'V_C^θ0. We write S_λ^θ =V_C^θ isλ(C) =λ. S_λ^θ is endowed with an “H_n-invariant” bilinear form φ^θ_λ. We set

D_λ^θ =CS_λ^θ/KerCφ^θ_λ. LetBip_θ(n) ={λ∈Bip(n) |D_λ^θ 6=0}. Then

IrrCH_n ={D_λ^θ | λ∈Bip_θ(n)}.

Theorem (B., Geck, Iancu, Jacon, Lam, Lusztig, Pietraho, Taskin)

Assume that Lusztig’s conjectures P1, P2,..., P15 hold. Then : If C is a θ-left cell of W_n, then KV_C^θ ∈IrrKH_n. We denove by λ(C) the bipartition such that KV_C^θ 'Vλ(C).

If KV_C^θ 'KV_C^θ0, then V_C^θ 'V_C^θ0. We write S_λ^θ =V_C^θ isλ(C) =λ. S_λ^θ is endowed with an “H_n-invariant” bilinear form φ^θ_λ. We set

D_λ^θ =CS_λ^θ/KerCφ^θ_λ. LetBip_θ(n) ={λ∈Bip(n) |D_λ^θ 6=0}. Then

IrrCH_n ={D_λ^θ | λ∈Bip_θ(n)}.

Theorem (B., Geck, Iancu, Jacon, Lam, Lusztig, Pietraho, Taskin)

Assume that Lusztig’s conjectures P1, P2,..., P15 hold. Then : If C is a θ-left cell of W_n, then KV_C^θ ∈IrrKH_n. We denove by λ(C) the bipartition such that KV_C^θ 'Vλ(C).

If KV_C^θ 'KV_C^θ0, then V_C^θ 'V_C^θ0.

We write S_λ^θ =V_C^θ isλ(C) =λ. S_λ^θ is endowed with an “H_n-invariant” bilinear form φ^θ_λ. We set

D_λ^θ =CS_λ^θ/KerCφ^θ_λ. LetBip_θ(n) ={λ∈Bip(n) |D_λ^θ 6=0}. Then

IrrCH_n ={D_λ^θ | λ∈Bip_θ(n)}.

Theorem (B., Geck, Iancu, Jacon, Lam, Lusztig, Pietraho, Taskin)

Assume that Lusztig’s conjectures P1, P2,..., P15 hold. Then : If C is a θ-left cell of W_n, then KV_C^θ ∈IrrKH_n. We denove by λ(C) the bipartition such that KV_C^θ 'Vλ(C).

If KV_C^θ 'KV_C^θ0, then V_C^θ 'V_C^θ0. We write S_λ^θ =V_C^θ isλ(C) =λ.

S_λ^θ is endowed with an “H_n-invariant” bilinear form φ^θ_λ. We set D_λ^θ =CS_λ^θ/KerCφ^θ_λ.

LetBip_θ(n) ={λ∈Bip(n) |D_λ^θ 6=0}. Then IrrCH_n ={D_λ^θ | λ∈Bip_θ(n)}.