Rouquier blocks of the cyclotomic Hecke algebras of complex reflection groups

Rouquier blocks of the cyclotomic Hecke algebras of complex reflection groups

Maria Chlouveraki MSRI February 22, 2008

Preliminaries

O is a Noetherian and integrally closed domain with field of fractions F.

Ais an O-algebra, free and finitely generated as an O-module. K is a finite Galois extension ofF such that the algebra KA:=K ⊗_OA is split semisimple.

Definition

The blocksofA are the primitive idempotents ofZA.

Since KA is semisimple, we have a bijection Irr(KA) ↔ Bl(KA)

χ ↔ eχ

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 2 / 21

Preliminaries

O is a Noetherian and integrally closed domain with field of fractions F.

Ais an O-algebra, free and finitely generated as an O-module. K is a finite Galois extension ofF such that the algebra KA:=K ⊗_OA is split semisimple.

Definition

The blocksofA are the primitive idempotents ofZA.

Since KA is semisimple, we have a bijection Irr(KA) ↔ Bl(KA)

χ ↔ eχ

Preliminaries

O is a Noetherian and integrally closed domain with field of fractions F.

Ais an O-algebra, free and finitely generated as an O-module.

K is a finite Galois extension ofF such that the algebra KA:=K ⊗_OA is split semisimple.

Definition

The blocksofA are the primitive idempotents ofZA.

Since KA is semisimple, we have a bijection Irr(KA) ↔ Bl(KA)

χ ↔ eχ

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 2 / 21

Preliminaries

O is a Noetherian and integrally closed domain with field of fractions F.

Ais an O-algebra, free and finitely generated as an O-module.

K is a finite Galois extension ofF such that the algebra KA:=K ⊗_OA is split semisimple.

Definition

The blocksofA are the primitive idempotents ofZA.

Since KA is semisimple, we have a bijection Irr(KA) ↔ Bl(KA)

χ ↔ eχ

Preliminaries

O is a Noetherian and integrally closed domain with field of fractions F.

Ais an O-algebra, free and finitely generated as an O-module.

K is a finite Galois extension ofF such that the algebra KA:=K ⊗_OA is split semisimple.

Definition

The blocksofA are the primitive idempotents ofZA.

Since KA is semisimple, we have a bijection Irr(KA) ↔ Bl(KA)

χ ↔ eχ

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 2 / 21

Preliminaries

O is a Noetherian and integrally closed domain with field of fractions F.

Ais an O-algebra, free and finitely generated as an O-module.

K is a finite Galois extension ofF such that the algebra KA:=K ⊗_OA is split semisimple.

Definition

The blocksofA are the primitive idempotents ofZA.

Since KA is semisimple, we have a bijection Irr(KA) ↔ Bl(KA)

χ ↔ e_χ

Proposition

There exists a unique partition Bl(A) ofIrr(KA) minimal for the property :

For all B∈Bl(A), e_B :=P

χ∈Beχ ∈A.

In particular, the set {e_B}_B∈Bl(A) is the set of all the blocks ofA.

Ifχ∈B for B ∈Bl(A), we say that “χ belongs to the blockeB”.

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 3 / 21

Proposition

There exists a unique partition Bl(A) ofIrr(KA) minimal for the property : For all B∈Bl(A), e_B :=P

χ∈Beχ∈A.

In particular, the set {e_B}_B∈Bl(A) is the set of all the blocks ofA.

Ifχ∈B for B ∈Bl(A), we say that “χ belongs to the blockeB”.

Proposition

There exists a unique partition Bl(A) ofIrr(KA) minimal for the property : For all B∈Bl(A), e_B :=P

χ∈Beχ∈A.

In particular, the set {e_B}_B∈Bl(A) is the set of all the blocks ofA.

Ifχ∈B for B ∈Bl(A), we say that “χ belongs to the blockeB”.

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 3 / 21

Proposition

There exists a unique partition Bl(A) ofIrr(KA) minimal for the property : For all B∈Bl(A), e_B :=P

χ∈Beχ∈A.

In particular, the set {e_B}_B∈Bl(A) is the set of all the blocks ofA.

Ifχ∈B for B ∈Bl(A), we say that “χ belongs to the blocke_B”.

Symmetric algebras

Definition

We say that a linear mapt :A→ Ois a symmetrizing form on Aor that A is a symmetric algebraif

t is a trace function,i.e., t(ab) =t(ba) for all a,b ∈A.

The morphism

ˆt :A→HomO(A,O), a7→(x 7→ˆt(a)(x) :=t(ax)) is an isomorphism ofA-modules-A.

Example:

IfO=Zand A=Z[G](G a finite group), we can define the following symmetrizing form (“canonical”) on A

t :Z[G]→Z, X

g∈G

agg 7→a1.

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 4 / 21

Symmetric algebras

Definition

We say that a linear mapt :A→ Ois a symmetrizing form on Aor that A is a symmetric algebraif

t is a trace function,i.e., t(ab) =t(ba) for all a,b ∈A.

The morphism

ˆt :A→HomO(A,O), a7→(x 7→ˆt(a)(x) :=t(ax)) is an isomorphism ofA-modules-A.

Example:

IfO=Zand A=Z[G] (G a finite group), we can define the following symmetrizing form (“canonical”) on A

t :Z[G]→Z, X

g∈G

agg 7→a1.

Theorem (Geck)

1 We have

t = X

χ∈Irr(KA)

1 s_χ

χ,

wheres_χ is theSchur element of χ with respect tot. We have s_χ ∈ O_K.

2 For all χ∈Irr(KA), the primitive idempotent ofZKA associated toχ is

eχ =ˆt⁻¹(χ) s_χ .

Example:

IfO=Z, A=Z[G](G a finite group) and t is the canonical form on A, we have

s_χ = |G| χ(1).

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 5 / 21

Theorem (Geck)

1 We have

t = X

χ∈Irr(KA)

1 s_χ

χ,

wheres_χ is theSchur element of χ with respect tot. We have s_χ ∈ O_K.

2 For all χ∈Irr(KA), the primitive idempotent ofZKA associated toχ is

eχ =ˆt⁻¹(χ) s_χ .

Example:

IfO=Z, A=Z[G](G a finite group) and t is the canonical form on A, we have

s_χ = |G| χ(1).

Theorem (Geck)

1 We have

t = X

χ∈Irr(KA)

1 s_χχ,

wheres_χ is theSchur element of χ with respect tot. We have s_χ ∈ O_K.

2 For all χ∈Irr(KA), the primitive idempotent ofZKA associated toχ is

eχ =ˆt⁻¹(χ) s_χ .

Example:

IfO=Z, A=Z[G](G a finite group) and t is the canonical form on A, we have

s_χ = |G| χ(1).

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 5 / 21

Theorem (Geck)

1 We have

t = X

χ∈Irr(KA)

1 s_χχ,

wheres_χ is theSchur element of χ with respect tot. We have s_χ ∈ O_K.

2 For all χ∈Irr(KA), the primitive idempotent of ZKA associated toχ is

eχ= ˆt⁻¹(χ) s_χ .

Example:

IfO=Z, A=Z[G](G a finite group) and t is the canonical form on A, we have

s_χ = |G| χ(1).

Theorem (Geck)

1 We have

t = X

χ∈Irr(KA)

1 s_χχ,

wheres_χ is theSchur element of χ with respect tot. We have s_χ ∈ O_K.

2 For all χ∈Irr(KA), the primitive idempotent of ZKA associated toχ is

eχ= ˆt⁻¹(χ) s_χ .

Example:

IfO=Z, A=Z[G](G a finite group) and t is the canonical form on A, we have

s_χ = |G| χ(1).

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 5 / 21

Hecke algebras of complex reflection groups

Every complex reflection group W has a nice “presentation a la Coxeter” :

I G2=<s,t|ststst=tststs,s²=t²= 1>

I G4=<s,t|sts=tst,s³=t³= 1>

and a field of realization K (with ring of integers ZK) : K_G2=QetK_G4 =Q(ζ₃).

We choose a set of indeterminates

u= (u_s,j)_{s,0≤j≤o(s)−1}

wheres runs over the set of generators ofW ando(s) denotes the order of s (if s andt are conjugate inW, then us,j =ut,j for all j).

Hecke algebras of complex reflection groups

Every complex reflection group W has a nice “presentation a la Coxeter” :

I G2=<s,t|ststst=tststs,s²=t²= 1>

I G4=<s,t|sts=tst,s³=t³= 1>

and a field of realization K (with ring of integers ZK) : K_G2=QetK_G4 =Q(ζ₃).

We choose a set of indeterminates

u= (u_s,j)_{s,0≤j≤o(s)−1}

wheres runs over the set of generators ofW ando(s) denotes the order of s (if s andt are conjugate inW, then us,j =ut,j for all j).

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 6 / 21

Hecke algebras of complex reflection groups

Every complex reflection group W has a nice “presentation a la Coxeter” :

I G2=<s,t|ststst=tststs,s²=t²= 1>

I G4=<s,t|sts=tst,s³=t³= 1>

and a field of realization K (with ring of integers ZK) : K_G2=QetK_G4 =Q(ζ₃).

We choose a set of indeterminates

u= (u_s,j)_{s,0≤j≤o(s)−1}

wheres runs over the set of generators ofW ando(s) denotes the order of s (if s andt are conjugate inW, then us,j =ut,j for all j).

Hecke algebras of complex reflection groups

Every complex reflection group W has a nice “presentation a la Coxeter” :

I G2=<s,t|ststst=tststs,s²=t²= 1>

I G4=<s,t|sts=tst,s³=t³= 1>

and a field of realization K (with ring of integers ZK) : K_G2=QetK_G4 =Q(ζ₃).

We choose a set of indeterminates

u= (u_s,j)_{s,0≤j≤o(s)−1}

wheres runs over the set of generators ofW ando(s) denotes the order of s (if s andt are conjugate inW, then us,j =ut,j for all j).

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 6 / 21

Hecke algebras of complex reflection groups

Every complex reflection group W has a nice “presentation a la Coxeter” :

I G2=<s,t|ststst=tststs,s²=t²= 1>

I G4=<s,t|sts=tst,s³=t³= 1>

and a field of realization K (with ring of integers ZK) :

K_G2=QetK_G4 =Q(ζ₃).

We choose a set of indeterminates

u= (u_s,j)_{s,0≤j≤o(s)−1}

wheres runs over the set of generators ofW ando(s) denotes the order of s (if s andt are conjugate inW, then us,j =ut,j for all j).

Hecke algebras of complex reflection groups

Every complex reflection group W has a nice “presentation a la Coxeter” :

I G2=<s,t|ststst=tststs,s²=t²= 1>

I G4=<s,t|sts=tst,s³=t³= 1>

and a field of realization K (with ring of integers ZK) : K_G2=QetK_G4 =Q(ζ₃).

We choose a set of indeterminates

u= (u_s,j)_{s,0≤j≤o(s)−1}

wheres runs over the set of generators ofW ando(s) denotes the order of s (if s andt are conjugate inW, then us,j =ut,j for all j).

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 6 / 21

Hecke algebras of complex reflection groups

Every complex reflection group W has a nice “presentation a la Coxeter” :

I G2=<s,t|ststst=tststs,s²=t²= 1>

I G4=<s,t|sts=tst,s³=t³= 1>

and a field of realization K (with ring of integers ZK) : K_G2=QetK_G4 =Q(ζ₃).

We choose a set of indeterminates

u= (u_s,j)_{s,0≤j≤o(s)−1}

wheres runs over the set of generators ofW ando(s) denotes the order of s (if s andt are conjugate inW, then us,j =ut,j for all j).

The associated generic Hecke algebraH(W) is an algebra over the Laurent polynomial ring Z[u,u⁻¹] and has a presentation of the form:

H(G₂) =<S,T | STSTST =TSTSTS, (S−u0)(S−u1) = 0, (T −w₀)(T −w₁) = 0> .

I u_j7→(−1)^j (j= 0,1),H(G2)7→Z[G₂].

H(G₄) =<S,T | STS =TST, (S−u₀)(S−u₁)(S−u₂) = 0, (T −u₀)(T −u₁)(T −u₂) = 0> .

I uj7→ζ₃^j (j = 0,1,2),H(G4)7→ZK[G4].

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 7 / 21

The associated generic Hecke algebraH(W) is an algebra over the Laurent polynomial ring Z[u,u⁻¹] and has a presentation of the form:

H(G₂) =<S,T | STSTST =TSTSTS, (S−u0)(S−u1) = 0, (T −w₀)(T −w₁) = 0> .

I u_j7→(−1)^j (j= 0,1),H(G2)7→Z[G₂].

H(G₄) =<S,T | STS =TST, (S−u₀)(S−u₁)(S−u₂) = 0, (T −u0)(T −u1)(T −u2) = 0> .

I uj7→ζ₃^j (j = 0,1,2),H(G4)7→ZK[G4].

The associated generic Hecke algebraH(W) is an algebra over the Laurent polynomial ring Z[u,u⁻¹] and has a presentation of the form:

H(G₂) =<S,T | STSTST =TSTSTS, (S−u0)(S−u1) = 0, (T −w₀)(T −w₁) = 0> .

I uj7→(−1)^j (j= 0,1),H(G2)7→Z[G2].

H(G₄) =<S,T | STS =TST, (S−u₀)(S−u₁)(S−u₂) = 0, (T −u0)(T −u1)(T −u2) = 0> .

I uj7→ζ₃^j (j = 0,1,2),H(G4)7→ZK[G4].

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 7 / 21

Assumptions

The algebra H(W) is a freeZ[u,u⁻¹]-module of rank|W|.

There exists a unique linear formt :H(W)→Z[u,u⁻¹] such that:

I t is a symmetrizing form onH(W).

I Via the specializationus,j 7→ζ_o(s)^j , the form t becomes the canonical form on the group algebraZK[W].

I t satisfies some other condition.

Assumptions

The algebra H(W) is a freeZ[u,u⁻¹]-module of rank|W|.

There exists a unique linear formt :H(W)→Z[u,u⁻¹] such that:

I t is a symmetrizing form onH(W).

I Via the specializationus,j 7→ζ_o(s)^j , the form t becomes the canonical form on the group algebraZK[W].

I t satisfies some other condition.

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 8 / 21

Assumptions

The algebra H(W) is a freeZ[u,u⁻¹]-module of rank|W|.

There exists a unique linear formt :H(W)→Z[u,u⁻¹] such that:

I t is a symmetrizing form onH(W).

I Via the specializationus,j 7→ζ_o(s)^j , the form t becomes the canonical form on the group algebraZK[W].

I t satisfies some other condition.

Assumptions

The algebra H(W) is a freeZ[u,u⁻¹]-module of rank|W|.

There exists a unique linear formt :H(W)→Z[u,u⁻¹] such that:

I t is a symmetrizing form onH(W).

I Via the specializationus,j 7→ζ_o(s)^j , the form t becomes the canonical form on the group algebraZK[W].

I t satisfies some other condition.

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 8 / 21

Assumptions

The algebra H(W) is a freeZ[u,u⁻¹]-module of rank|W|.

There exists a unique linear formt :H(W)→Z[u,u⁻¹] such that:

I t is a symmetrizing form onH(W).

I Via the specializationus,j 7→ζ_o(s)^j , the form t becomes the canonical form on the group algebraZK[W].

I t satisfies some other condition.

Assumptions

The algebra H(W) is a freeZ[u,u⁻¹]-module of rank|W|.

There exists a unique linear formt :H(W)→Z[u,u⁻¹] such that:

I t is a symmetrizing form onH(W).

I Via the specializationus,j 7→ζ_o(s)^j , the form t becomes the canonical form on the group algebraZK[W].

I t satisfies some other condition.

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 8 / 21

Theorem (Malle)

Let v= (vs,j)s,j be a set of indeterminates such that, for alls,j, we have v_s,j^|µ(K)|:=ζ_o(s)^−j u_s,j.

Then the K(v)-algebra K(v)H is split semisimple

By “Tits’ deformation theorem”, we know that the specialization v_s,j 7→1 induces a bijection

Irr(K(v)H) ↔ Irr(W)

χ_v 7→ χ

s_χ(v) 7→ |W|/χ(1)

Theorem (Malle)

Let v= (vs,j)s,j be a set of indeterminates such that, for alls,j, we have v_s,j^|µ(K)|:=ζ_o(s)^−j u_s,j.

Then the K(v)-algebra K(v)H is split semisimple

By “Tits’ deformation theorem”, we know that the specialization v_s,j 7→1 induces a bijection

Irr(K(v)H) ↔ Irr(W)

χ_v 7→ χ

s_χ(v) 7→ |W|/χ(1)

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 9 / 21

Theorem (Malle)

Let v= (vs,j)s,j be a set of indeterminates such that, for alls,j, we have v_s,j^|µ(K)|:=ζ_o(s)^−j u_s,j.

Then the K(v)-algebra K(v)H is split semisimple

By “Tits’ deformation theorem”, we know that the specialization v_s,j 7→1 induces a bijection

Irr(K(v)H) ↔ Irr(W)

χ_v 7→ χ

s_χ(v) 7→ |W|/χ(1)

Generic Schur elements

Theorem (C.)

The Schur element s_χ(v) associated with the irreducible character χ_v of K(v)His an element ofZK[v,v⁻¹] whose irreducible factors (inK[v,v⁻¹]) are of the form

Ψ(M) where

Ψ is aK-cyclotomic polynomial in one variable, M is a primitive monomial of degree 0, i.e., ifM =Q

s,jv_s,j^as,j, then gcd(a_s,j) = 1 andP

s,ja_s,j = 0.

The primitive monomials appearing in the factorization of s_χ(v) are unique up to inversion.

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 10 / 21

Generic Schur elements

Theorem (C.)

The Schur element s_χ(v) associated with the irreducible character χ_v of K(v)His an element ofZK[v,v⁻¹] whose irreducible factors (inK[v,v⁻¹]) are of the form

Ψ(M) where

Ψ is aK-cyclotomic polynomial in one variable, M is a primitive monomial of degree 0, i.e., ifM =Q

s,jv_s,j^as,j, then gcd(a_s,j) = 1 andP

s,ja_s,j = 0.

The primitive monomials appearing in the factorization of s_χ(v) are unique up to inversion.

Schur elements of G₂ :

X₀² :=u₀, X₁² :=−u₁, Y₀² :=w₀,Y₁² :=−w₁.

s₁ = Φ₄(X₀X₁⁻¹)·Φ₄(Y₀Y₁⁻¹)·Φ₃(X₀Y₀X₁⁻¹Y₁⁻¹)· Φ6(X0Y0X₁⁻¹Y₁⁻¹)

s₂ = 2·X₁²X₀⁻²·Φ₃(X₀Y₀X₁⁻¹Y₁⁻¹)·Φ₆(X₀Y₁X₁⁻¹Y₀⁻¹)

Φ4(x) =x²+ 1, Φ3(x) =x²+x+ 1, Φ6(x) =x²−x+ 1.

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 11 / 21

Schur elements of G₄ : X_i⁶ :=ζ₃⁻ⁱu_i.

s₁ = Φ⁰⁰₉(X₀X₁⁻¹)·Φ⁰₁₈(X₀X₁⁻¹)·Φ₄(X₀X₁⁻¹)·Φ⁰₁₂(X₀X₁⁻¹)· Φ⁰⁰₁₂(X0X₁⁻¹)·Φ⁰₃₆(X0X₁⁻¹)·Φ⁰₉(X0X₂⁻¹)·Φ⁰⁰₁₈(X0X₂⁻¹)· Φ₄(X₀X₂⁻¹)·Φ⁰₁₂(X₀X₂⁻¹)·Φ⁰⁰₁₂(X₀X₂⁻¹)·Φ⁰⁰₃₆(X₀X₂⁻¹)· Φ₄(X₀²X₁⁻¹X₂⁻¹)·Φ⁰₁₂(X₀²X₁⁻¹X₂⁻¹)·Φ⁰⁰₁₂(X₀²X₁⁻¹X₂⁻¹) s₂ = −ζ₃²X₂⁶X₁⁻⁶Φ⁰₉(X₁X₀⁻¹)·Φ⁰⁰₁₈(X₁X₀⁻¹)·Φ⁰⁰₉(X₂X₀⁻¹)·

Φ⁰₁₈(X2X₀⁻¹)·Φ4(X1X₂⁻¹)·Φ⁰₁₂(X1X₂⁻¹)·Φ⁰⁰₁₂(X1X₂⁻¹)· Φ⁰₃₆(X1X₂⁻¹)·Φ₄(X₀⁻²X1X2)·Φ⁰₁₂(X₀⁻²X1X2)·Φ⁰⁰₁₂(X₀⁻²X1X2) s3 = Φ4(X₀²X₁⁻¹X₂⁻¹)·Φ⁰₁₂(X₀²X₁⁻¹X₂⁻¹)·Φ⁰⁰₁₂(X₀²X₁⁻¹X₂⁻¹)·

Φ₄(X₁²X₂⁻¹X₀⁻¹)·Φ⁰₁₂(X₁²X₂⁻¹X₀⁻¹)·Φ⁰⁰₁₂(X₁²X₂⁻¹X₀⁻¹)· Φ₄(X₂²X₀⁻¹X₁⁻¹)·Φ⁰₁₂(X₂²X₀⁻¹X₁⁻¹)·Φ⁰⁰₁₂(X₂²X₀⁻¹X₁⁻¹)

Φ₄(x) =x²+ 1, Φ⁰₉(x) =x³−ζ₃, Φ⁰⁰₉(x) =x³−ζ₃², Φ⁰⁰₁₂(x) =x²+ζ₃, Φ⁰₁₂(x) =x²+ζ₃², Φ⁰⁰₁₈(x) =x³+ζ₃, Φ⁰₁₈(x) =x³+ζ₃², Φ⁰⁰₃₆(x) =x⁶+ζ₃, Φ⁰₃₆(x) =x⁶+ζ₃².

Cyclotomic Hecke algebras

Definition

Let y be an indeterminate. Acyclotomic specialization ofH is a ZK-algebra morphism φ:ZK[v,v⁻¹]→ZK[y,y⁻¹] such that

φ:vs,j 7→y^ns,j, with ns,j ∈Z for all s andj.

The correspondingcyclotomic Hecke algebra H_φis the ZK[y,y⁻¹]-algebra obtained via the specialization of Hvia the morphism φ. It also has a symmetrizing form t_φdefined as the specialization of the form t. Examples:

The group algebra ZK[W]is the cyclotomic Hecke algebra obtained via vs,j 7→1for all s,j.

The spetsial algebraH^s(W)is the cyclotomic Hecke algebra obtained via vs,07→y, vs,j7→1for1≤j ≤o(s)−1, for all s.

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 13 / 21

Cyclotomic Hecke algebras

Definition

Let y be an indeterminate. Acyclotomic specialization ofH is a ZK-algebra morphism φ:ZK[v,v⁻¹]→ZK[y,y⁻¹] such that

φ:vs,j 7→y^ns,j, with ns,j ∈Z for all s andj.

The correspondingcyclotomic Hecke algebra H_φis the ZK[y,y⁻¹]-algebra obtained via the specialization of Hvia the morphism φ. It also has a symmetrizing form t_φdefined as the specialization of the form t.

Examples:

The group algebra ZK[W]is the cyclotomic Hecke algebra obtained via vs,j 7→1for all s,j.

The spetsial algebraH^s(W)is the cyclotomic Hecke algebra obtained via vs,07→y, vs,j7→1for1≤j ≤o(s)−1, for all s.

Cyclotomic Hecke algebras

Definition

Let y be an indeterminate. Acyclotomic specialization ofH is a ZK-algebra morphism φ:ZK[v,v⁻¹]→ZK[y,y⁻¹] such that

φ:vs,j 7→y^ns,j, with ns,j ∈Z for all s andj.

The correspondingcyclotomic Hecke algebra H_φis the ZK[y,y⁻¹]-algebra obtained via the specialization of Hvia the morphism φ. It also has a symmetrizing form t_φdefined as the specialization of the form t.

Examples:

The group algebra ZK[W]is the cyclotomic Hecke algebra obtained via vs,j 7→1for all s,j.

The spetsial algebraH^s(W)is the cyclotomic Hecke algebra obtained via vs,07→y, vs,j7→1for1≤j ≤o(s)−1, for all s.

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 13 / 21

Cyclotomic Hecke algebras

Definition

Let y be an indeterminate. Acyclotomic specialization ofH is a ZK-algebra morphism φ:ZK[v,v⁻¹]→ZK[y,y⁻¹] such that

φ:vs,j 7→y^ns,j, with ns,j ∈Z for all s andj.

The correspondingcyclotomic Hecke algebra H_φis the ZK[y,y⁻¹]-algebra obtained via the specialization of Hvia the morphism φ. It also has a symmetrizing form t_φdefined as the specialization of the form t.

Examples:

The group algebra ZK[W]is the cyclotomic Hecke algebra obtained via vs,j 7→1for all s,j.

The spetsial algebraH^s(W)is the cyclotomic Hecke algebra obtained via vs,07→y, vs,j7→1for1≤j ≤o(s)−1, for all s.

Proposition (C.)

The algebra K(y)H_φ is split semisimple.

By “Tits’ deformation theorem”, we obtain

Irr(K(v)H) ↔ Irr(K(y)H_φ) ↔ Irr(W)

χv 7→ χ_φ 7→ χ

s_χ(v) 7→ s_χφ(y) 7→ |W|/χ(1)

Proposition

The Schur element sχ_φ(y) associated to the irreducible characterχ_φ of K(y)H_φis a Laurent polynomial iny of the form

s_χφ(y) =ψ_χ,φy^aχ,φ Y

Φ∈C_K

Φ(y)^nχ,φ,

where ψ_χ,φ ∈ZK,a_χ,φ∈Z,n_χ,φ∈NandC_K is a set ofK-cyclotomic polynomials.

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 14 / 21

Proposition (C.)

The algebra K(y)H_φ is split semisimple.

By “Tits’ deformation theorem”, we obtain

Irr(K(v)H) ↔ Irr(K(y)H_φ) ↔ Irr(W)

χv 7→ χ_φ 7→ χ

s_χ(v) 7→ s_χφ(y) 7→ |W|/χ(1) Proposition

The Schur element sχ_φ(y) associated to the irreducible characterχ_φ of K(y)H_φis a Laurent polynomial iny of the form

s_χφ(y) =ψ_χ,φy^aχ,φ Y

Φ∈C_K

Φ(y)^nχ,φ,

where ψ_χ,φ ∈ZK,a_χ,φ∈Z,n_χ,φ∈NandC_K is a set ofK-cyclotomic polynomials.

Proposition (C.)

The algebra K(y)H_φ is split semisimple.

By “Tits’ deformation theorem”, we obtain

Irr(K(v)H) ↔ Irr(K(y)H_φ) ↔ Irr(W)

χv 7→ χ_φ 7→ χ

s_χ(v) 7→ s_χφ(y) 7→ |W|/χ(1)

Proposition

The Schur element sχ_φ(y) associated to the irreducible characterχ_φ of K(y)H_φis a Laurent polynomial iny of the form

s_χφ(y) =ψ_χ,φy^aχ,φ Y

Φ∈C_K

Φ(y)^nχ,φ,

where ψ_χ,φ ∈ZK,a_χ,φ∈Z,n_χ,φ∈NandC_K is a set ofK-cyclotomic polynomials.

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 14 / 21

Proposition (C.)

The algebra K(y)H_φ is split semisimple.

By “Tits’ deformation theorem”, we obtain

Irr(K(v)H) ↔ Irr(K(y)H_φ) ↔ Irr(W)

χv 7→ χ_φ 7→ χ

s_χ(v) 7→ s_χφ(y) 7→ |W|/χ(1) Proposition

The Schur element sχ_φ(y) associated to the irreducible characterχ_φ of K(y)H_φis a Laurent polynomial iny of the form

s_χφ(y) =ψ_χ,φy^aχ,φ Y

Φ∈C_K

Φ(y)^nχ,φ,

where ψ_χ,φ ∈ZK,a_χ,φ∈Z,n_χ,φ∈NandC_K is a set ofK-cyclotomic polynomials.

Rouquier blocks

Definition

We call Rouquier ringof K theZK-subalgebra ofK(y) R:=ZK[y,y⁻¹,(yⁿ−1)⁻¹_n≥1]

Let φ:vs,j 7→y^ns,j be a cyclotomic specialization andH_φ the

corresponding cyclotomic Hecke algebra. The Rouquier blocks ofH_φ are the blocks of the algebra RH_φ.

W Weyl group : Rouquier blocks ≡“families of characters” W c.r.g. (non-Weyl) : Rouquier blocks ≡?

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 15 / 21

Rouquier blocks

Definition

We call Rouquier ringof K theZK-subalgebra ofK(y) R:=ZK[y,y⁻¹,(yⁿ−1)⁻¹_n≥1]

Let φ:vs,j 7→y^ns,j be a cyclotomic specialization andH_φ the

corresponding cyclotomic Hecke algebra. The Rouquier blocks ofH_φ are the blocks of the algebra RH_φ.

W Weyl group : Rouquier blocks ≡“families of characters” W c.r.g. (non-Weyl) : Rouquier blocks ≡?

Rouquier blocks

Definition

We call Rouquier ringof K theZK-subalgebra ofK(y) R:=ZK[y,y⁻¹,(yⁿ−1)⁻¹_n≥1]

Let φ:vs,j 7→y^ns,j be a cyclotomic specialization andH_φ the

corresponding cyclotomic Hecke algebra. The Rouquier blocks ofH_φ are the blocks of the algebra RH_φ.

W Weyl group : Rouquier blocks ≡“families of characters”

W c.r.g. (non-Weyl) : Rouquier blocks ≡?

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 15 / 21

Rouquier blocks

Definition

We call Rouquier ringof K theZK-subalgebra ofK(y) R:=ZK[y,y⁻¹,(yⁿ−1)⁻¹_n≥1]

Let φ:vs,j 7→y^ns,j be a cyclotomic specialization andH_φ the

corresponding cyclotomic Hecke algebra. The Rouquier blocks ofH_φ are the blocks of the algebra RH_φ.

W Weyl group : Rouquier blocks ≡“families of characters”

W c.r.g. (non-Weyl) : Rouquier blocks ≡?

Proposition

The characters χ_φ andψ_φ are in the same Rouquier block ofH_φif and only if there exist a finite sequence χ0, χ1, . . . , χn∈Irr(W) and a finite sequence p₁, . . . ,p_n of prime ideals of Rsuch that

(χ0)φ=χφet (χn)φ=ψφ,

∀j (1≤j ≤n), (χj−1)φet (χj)φ are in the same blockR_pjH_φ.

If Ω :=ZK[y,y⁻¹], thenRpR 'Ω_pΩ for all prime ideals pof ZK.

AIM: Determine the blocks Ω_pΩH_φ.

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 16 / 21

Proposition

The characters χ_φ andψ_φ are in the same Rouquier block ofH_φif and only if there exist a finite sequence χ0, χ1, . . . , χn∈Irr(W) and a finite sequence p₁, . . . ,p_n of prime ideals of ZK such that

(χ0)φ=χφet (χn)φ=ψφ,

∀j (1≤j ≤n), (χj−1)φet (χj)φ are in the same block of R_pjRH_φ.

If Ω :=ZK[y,y⁻¹], thenRpR 'Ω_pΩ for all prime ideals pof ZK.

AIM: Determine the blocks Ω_pΩH_φ.

Proposition

The characters χ_φ andψ_φ are in the same Rouquier block ofH_φif and only if there exist a finite sequence χ0, χ1, . . . , χn∈Irr(W) and a finite sequence p₁, . . . ,p_n of prime ideals of ZK such that

(χ0)φ=χφet (χn)φ=ψφ,

∀j (1≤j ≤n), (χj−1)φet (χj)φ are in the same block of R_pjRH_φ.

If Ω :=ZK[y,y⁻¹], thenRpR 'ΩpΩ for all prime idealsp of ZK.

AIM: Determine the blocks Ω_pΩH_φ.

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 16 / 21

Proposition

The characters χ_φ andψ_φ are in the same Rouquier block ofH_φif and only if there exist a finite sequence χ0, χ1, . . . , χn∈Irr(W) and a finite sequence p₁, . . . ,p_n of prime ideals of ZK such that

(χ0)φ=χφet (χn)φ=ψφ,

∀j (1≤j ≤n), (χj−1)φet (χj)φ are in the same block of R_pjRH_φ.

If Ω :=ZK[y,y⁻¹], thenRpR 'ΩpΩ for all prime idealsp of ZK.

AIM: Determine the blocks Ω_pΩH_φ.

p-essential monomials

Let p be a prime ideal ofZK. Set A:=ZK[v,v⁻¹].

Definition

A primitive monomial M in A is calledp-essential for W if there exists an irreducible character χ ofW and aK-cyclotomic polynomial Ψ such that

1 Ψ(M) dividessχ(v)

2 Ψ(1)∈p.

Let φbe a cyclotomic specialization. A monomial M in Ais singular forφ if φ(M) = 1.

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 17 / 21

p-essential monomials

Let p be a prime ideal ofZK. Set A:=ZK[v,v⁻¹].

Definition

A primitive monomial M in Ais called p-essential for W if there exists an irreducible character χ ofW and aK-cyclotomic polynomial Ψ such that

1 Ψ(M) dividessχ(v)

2 Ψ(1)∈p.

Let φbe a cyclotomic specialization. A monomial M in Ais singular forφ if φ(M) = 1.

p-essential monomials

Let p be a prime ideal ofZK. Set A:=ZK[v,v⁻¹].

Definition

A primitive monomial M in Ais called p-essential for W if there exists an irreducible character χ ofW and aK-cyclotomic polynomial Ψ such that

1 Ψ(M) dividessχ(v)

2 Ψ(1)∈p.

Let φbe a cyclotomic specialization. A monomial M in Ais singular forφ if φ(M) = 1.

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 17 / 21

p-essential monomials

Let p be a prime ideal ofZK. Set A:=ZK[v,v⁻¹].

Definition

A primitive monomial M in Ais called p-essential for W if there exists an irreducible character χ ofW and aK-cyclotomic polynomial Ψ such that

1 Ψ(M) dividessχ(v)

2 Ψ(1)∈p.

Let φbe a cyclotomic specialization. A monomial M in Ais singular forφ if φ(M) = 1.

p-essential monomials

Let p be a prime ideal ofZK. Set A:=ZK[v,v⁻¹].

Definition

A primitive monomial M in Ais called p-essential for W if there exists an irreducible character χ ofW and aK-cyclotomic polynomial Ψ such that

1 Ψ(M) dividessχ(v)

2 Ψ(1)∈p.

Let φbe a cyclotomic specialization. A monomial M in Ais singular forφ if φ(M) = 1.

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 17 / 21

p-blocks and p-essential monomials

Let B_p(H) be the partition ofIrr(W) into p-blocks ofH.

Let B_p(H_φ) be the partition ofIrr(W) intop-blocks ofH_φ. Theorem (C.)

For every p-essential monomialM for W, there exists a unique partition B_p^M(H) ofIrr(W) with the following properties:

1 The parts ofB^M_p (H) are unions of the parts ofB_p(H).

2 The partition B_p(H_φ) is the partition generated by the partitions B_p(H) et B_p^M(H), where M runs over the set of allp-essential monomials which are singular forφ.

Moreover, the partition B^M_p (H) coincides with the blocks of the algebra A_qMH, whereq_M := (M−1)A+pA.

p-blocks and p-essential monomials

Let B_p(H) be the partition ofIrr(W) into p-blocks ofH.

Let B_p(H_φ) be the partition ofIrr(W) intop-blocks ofH_φ.

Theorem (C.)

For every p-essential monomialM for W, there exists a unique partition B_p^M(H) ofIrr(W) with the following properties:

1 The parts ofB^M_p (H) are unions of the parts ofB_p(H).

2 The partition B_p(H_φ) is the partition generated by the partitions B_p(H) et B_p^M(H), where M runs over the set of allp-essential monomials which are singular forφ.

Moreover, the partition B^M_p (H) coincides with the blocks of the algebra A_qMH, whereq_M := (M−1)A+pA.

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 18 / 21

p-blocks and p-essential monomials

Let B_p(H) be the partition ofIrr(W) into p-blocks ofH.

Let B_p(H_φ) be the partition ofIrr(W) intop-blocks ofH_φ. Theorem (C.)

For every p-essential monomialM for W, there exists a unique partition B_p^M(H) ofIrr(W) with the following properties:

1 The parts ofB^M_p (H) are unions of the parts ofB_p(H).

2 The partition B_p(H_φ) is the partition generated by the partitions B_p(H) et B_p^M(H), where M runs over the set of allp-essential monomials which are singular forφ.

Moreover, the partition B^M_p (H) coincides with the blocks of the algebra A_qMH, whereq_M := (M−1)A+pA.

p-blocks and p-essential monomials

Let B_p(H) be the partition ofIrr(W) into p-blocks ofH.

Let B_p(H_φ) be the partition ofIrr(W) intop-blocks ofH_φ. Theorem (C.)

For every p-essential monomialM for W, there exists a unique partition B_p^M(H) ofIrr(W) with the following properties:

1 The parts ofB^M_p (H) are unions of the parts ofB_p(H).

2 The partition B_p(H_φ) is the partition generated by the partitions B_p(H) et B_p^M(H), where M runs over the set of allp-essential monomials which are singular forφ.

Moreover, the partition B^M_p (H) coincides with the blocks of the algebra A_qMH, whereq_M := (M−1)A+pA.

Maria Chlouveraki (MSRI) Rouquier blocks of Hecke algebras September 14, 2008 18 / 21

p-blocks and p-essential monomials

Let B_p(H) be the partition ofIrr(W) into p-blocks ofH.

Let B_p(H_φ) be the partition ofIrr(W) intop-blocks ofH_φ. Theorem (C.)

For every p-essential monomialM for W, there exists a unique partition B_p^M(H) ofIrr(W) with the following properties:

1 The parts ofB^M_p (H) are unions of the parts ofB_p(H).

2 The partition B_p(H_φ) is the partition generated by the partitions B_p(H) et B_p^M(H), where M runs over the set of allp-essential monomials which are singular forφ.

Moreover, the partition B^M_p (H) coincides with the blocks of the algebra A_qMH, whereq_M := (M−1)A+pA.