Aiguille du Midi

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Aiguille du Midi

Departure: 12h55at the Amis de la Nature

Consequently: please be ready to pick your lunch bag at the restaurant at 12h30

VERY COLD!!!

Sun glasses...

Pay individuallyin Chamonix for the lift (around 42 euros) The bus is paid by the conference

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Calogero-Moser cells

(joint work with R. Rouquier)

C´edric Bonnaf´e

CNRS (UMR 5149) - Universit´e de Montpellier 2

Les Houches - Janvier 2011

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What does Kazhdan-Lusztig theory do for you?

(W,S) finite Coxeter system

c :S →R such thatcs =c_s⁰ is s ands⁰ areW-conjugate Hecke algebra H(W,c) overZ[q^R]

Kazhdan-Lusztig basis (1979, 1984, 1999)

=⇒ Partition ofW into c-KL-cells (left, right, two-sided) + partial order

=⇒ W →ZIrr(W) (cell module)

=⇒ IfΓ is a c-KL-two-sided cell,Irr^KL_Γ (W)⊂Irr(W) (c-KL-family)

=⇒ Irr(W) =[˙

Γ

Irr^KL_Γ (W)

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What does Kazhdan-Lusztig theory do for you?

c :S →R such thatcs =c_s⁰ is s ands⁰ areW-conjugate Hecke algebra H(W,c) overZ[q^R]

=⇒ Irr(W) =[˙

Γ

Irr^KL_Γ (W)

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What does Kazhdan-Lusztig theory do for you?

c :S →R such thatc_s =c_s⁰ is s ands⁰ are W-conjugate

Hecke algebra H(W,c) overZ[q^R] Kazhdan-Lusztig basis (1979, 1984, 1999)

=⇒ Irr(W) =[˙

Γ

Irr^KL_Γ (W)

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What does Kazhdan-Lusztig theory do for you?

c :S →R such thatc_s =c_s⁰ is s ands⁰ are W-conjugate Hecke algebra H(W,c) overZ[q^R]

=⇒ Irr(W) =[˙

Γ

Irr^KL_Γ (W)

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What does Kazhdan-Lusztig theory do for you?

=⇒ Irr(W) =[˙

Γ

Irr^KL_Γ (W)

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What does Kazhdan-Lusztig theory do for you?

=⇒ Irr(W) =[˙

Γ

Irr^KL_Γ (W)

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What does Kazhdan-Lusztig theory do for you?

=⇒ Irr(W) =[˙

Γ

Irr^KL_Γ (W)

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What does Kazhdan-Lusztig theory do for you?

=⇒ Irr(W) =[˙

Γ

Irr^KL_Γ (W)

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What does Kazhdan-Lusztig theory do for you?

=⇒ Irr(W) =[˙

Γ

Irr^KL_Γ (W)

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What does Kazhdan-Lusztig theory do for you?

=⇒ Irr(W) =[˙

Γ

Irr^KL_Γ (W)

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Interests

Links with the geometry of flag varieties (singularities of Schubert cells)

Representations of complex Lie algebras, of reductive groups in positive characteristic...

Representations of finite reductive groups Unipotent classes

Decomposition numbers of Hecke algebras at root of unity...

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Interests

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Interests

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Interests

Representations of finite reductive groups

Unipotent classes

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Interests

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Interests

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Interests

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W complex reflection group?

Difficulty. Kazhdan-Lusztig basis highly depends on the choice ofS. Strategy. Use rational Cherednik algebrasatt =0to construct partitions of W (resp. Irr(W)) into Calogero-Moser cells(resp. families), a map W →ZIrr(W) and attach to a cellΓ a subsetIrr^CM_Γ (W)of Irr(W)

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W complex reflection group?

Difficulty. Kazhdan-Lusztig basis highly depends on the choice ofS.

Strategy. Use rational Cherednik algebrasatt =0to construct partitions of W (resp. Irr(W)) into Calogero-Moser cells(resp. families), a map W →ZIrr(W) and attach to a cellΓ a subsetIrr^CM_Γ (W)of Irr(W)

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W complex reflection group?

Difficulty. Kazhdan-Lusztig basis highly depends on the choice ofS. Strategy. Use rational Cherednik algebrasatt =0 to construct partitions of W (resp. Irr(W)) into Calogero-Moser cells(resp. families), a map W →ZIrr(W) and attach to a cellΓ a subset Irr^CM_Γ (W)of Irr(W)

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W complex reflection group?

Difficulty. Kazhdan-Lusztig basis highly depends on the choice ofS. Strategy. Use rational Cherednik algebrasatt =0 to construct partitions of W (resp. Irr(W)) into Calogero-Moser cells(resp. families), a map W →ZIrr(W) and attach to a cellΓ a subset Irr^CM_Γ (W)of Irr(W)

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Weakness.

Our construction depends on an “uncontrolable” choice We don’t get the partial order

Hard to compute (one month for completingB2) Strengths.

shares many properties with KL-two-sided cells The semicontinuity is trivial

Coincide with KL two-sided cells for A2,B2,G2

Very flexible (tons of notions of cells) Geometric methods

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Weakness.

Our construction depends on an “uncontrolable” choice

We don’t get the partial order

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Weakness.

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Weakness.

Hard to compute

(one month for completing B2) Strengths.

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Weakness.

Hard to compute (one month for completingB2)

Strengths.

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Weakness.

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Weakness.

shares many properties with KL-two-sided cells

The semicontinuity is trivial

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Weakness.

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Weakness.

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Weakness.

Very flexible (tons of notions of cells)

Geometric methods

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Weakness.

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Set-up

dim_CV <∞ W ⊂GL_C(V)

|W|<∞

W =hR´ef(W)i, where

R´ef(W) ={s ∈W | codim_CKer(s−IdV) =1}. C ={c :R´ef(W)/∼−→C}

C_s :C →C,c 7→c_s

C[C] =S(C^∗) =C[(Cs)_s∈R´_ef(W_)/_∼]

ifs ∈R´ef(W), letα_s ∈V^∗ andα^∨_s ∈V be such that

Ker(s−Id_V) =Ker(α_s) and Im(s−Id_V) =Cα^∨_s.

ε:W →C^×,w 7→det(w).

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Set-up

|W|<∞

R´ef(W) ={s ∈W | codim_CKer(s−IdV) =1}.

C ={c :R´ef(W)/∼−→C} C_s :C →C,c 7→c_s

C[C] =S(C^∗) =C[(Cs)_s∈R´_ef(W_)/_∼]

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Set-up

|W|<∞

C ={c :R´ef(W)/∼−→C}

C[C] =S(C^∗) =C[(C_s)_s∈R´_ef(W_)/_∼]

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Set-up

|W|<∞

C ={c :R´ef(W)/∼−→C}

C[C] =S(C^∗) =C[(C_s)_s∈R´_ef(W_)/_∼]

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Set-up

|W|<∞

C ={c :R´ef(W)/∼−→C}

C[C] =S(C^∗) =C[(C_s)_s∈R´_ef(W_)/_∼]

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Rational Cherednik algebra

• His the C[C]-algebra such that H |{z}=

vector space

C[C]⊗C[V]⊗CW ⊗C[V^∗]

[x,y] = X

s∈R´ef(W)

(1−ε(s))C_s hx, α_si · hα^∨_s,yi hα_s, α^∨_si s.

• Specialisation atc ∈C H_c |{z}=

vector space

C[V]⊗CW ⊗C[V^∗]

[x,y] = X

s∈R´ef(W)

(1−ε(s))cs

hx, α_si · hα^∨_s,yi hα_s, α^∨_si s. Hc =Cc⊗_C_[_C_]H.

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Rational Cherednik algebra

vector space

C[T]⊗C[C]⊗C[V]⊗CW ⊗C[V^∗]

[x,y] =Thx,yi+ X

s∈R´ef(W)

vector space

[x,y] = X

s∈R´ef(W)

(1−ε(s))cs

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Rational Cherednik algebra at t = 0

vector space

[x,y] = X

s∈R´ef(W)

vector space

[x,y] = X

s∈R´ef(W)

(1−ε(s))cs

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Rational Cherednik algebra at t = 0

vector space

[x,y] = X

s∈R´ef(W)

vector space

[x,y] = X

s∈R´ef(W)

(1−ε(s))cs

hx, α_si · hα^∨_s,yi hα_s, α^∨_si s.

Hc =Cc⊗_C_[_C_]H.

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Rational Cherednik algebra at t = 0

vector space

[x,y] = X

s∈R´ef(W)

vector space

[x,y] = X

s∈R´ef(W)

(1−ε(s))cs

hx, α_si · hα^∨_s,yi hα_s, α^∨_si s.

H_c =Cc⊗_C_[_C_]H.

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Some features (Etingof-Ginzburg 2002)

Freeness

P =C[C]⊗C[V]^W ⊗C[V^∗]^W ⊂ Z(H) =:Q H is a freeP-module of rank |W|³

Q is a freeP-module of rank |W| (in particular,Q is Cohen-Macaulay)

Satake isomorphism: the natural map Q→EndH(He) = (eHe)^op is an isomorphism (with e = _|_W¹_|P

w∈W w) H'End_Q(He)

Graduation(s)

N×N-graduation : deg^N^×^N(V) = (1,0), deg^N^×^N(V^∗) = (0,1), deg^N^×^N(W) = (0,0), deg^N^×^N(C) = (1,1).

Specialisation

Hc is not N×N-graded

Q andQ_c are normal integral domains

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Some features (Etingof-Ginzburg 2002)

Freeness

P =C[C]⊗C[V]^W ⊗C[V^∗]^W ⊂ Z(H) =:Q

H is a freeP-module of rank |W|³

Graduation(s)

Specialisation

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Some features (Etingof-Ginzburg 2002)

Freeness

Graduation(s)

Specialisation

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Some features (Etingof-Ginzburg 2002)

Freeness

Q is a freeP-module of rank |W|

(in particular,Q is Cohen-Macaulay)

Graduation(s)

Specialisation

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Some features (Etingof-Ginzburg 2002)

Freeness

Graduation(s)

Specialisation

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Some features (Etingof-Ginzburg 2002)

Freeness

w∈W w)

H'End_Q(He) Graduation(s)

Specialisation

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Some features (Etingof-Ginzburg 2002)

Freeness

Graduation(s)

Specialisation

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Some features (Etingof-Ginzburg 2002)

Freeness

Graduation(s)

Specialisation

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Some features (Etingof-Ginzburg 2002)

Freeness

Graduation(s)

Specialisation

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Example: specialisation at c=0

H0 =C[V ×V^∗]oW and Q0=C[V ×V^∗]^W H₀e 'C[V ×V^∗]

(becauseW does not contain any reflection... for its action on V ×V^∗)

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Example: specialisation at c=0

H0 =C[V ×V^∗]oW and Q0=C[V ×V^∗]^W

H₀e 'C[V ×V^∗]

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Example: specialisation at c=0

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Example: specialisation at c=0

C(V ×V^∗)oW =End_C_(V_×V^∗₎W(C(V ×V^∗))

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Example: specialisation at c=0

C[V ×V^∗]oW =End_C_[V_×V^∗_]W(C[V ×V^∗])

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Example: specialisation at c=0

C[V ×V^∗]oW =End_C_[V_×V^∗_]W(C[V ×V^∗])(becauseW does not contain any reflection...

for its action on V ×V^∗)

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Example: specialisation at c=0

C[V ×V^∗]oW =End_C_[V_×V^∗_]W(C[V ×V^∗])(becauseW does not contain any reflection... for its action on V ×V^∗)

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The P = C [ C × V /W × V

^∗

/W ]-algebra H

Relatively classical: take a point

p = (c,v,v^∗)∈P =C ×V/W ×V^∗/W and viewC=Cc,v,v^∗ as a P-algebra via evaluation atp =⇒ H_c,v,v^∗ =Cc,v,v^∗⊗_P H.

Restricted Cherednik algebra Take v =0 and v^∗ =0. You get

H_c |{z}=

vector space

C[V]^co(W⁾⊗CW ⊗C[V^∗]^co(W⁾.

This will lead toCalogero-Moser families (Gordon).

New (!): takeK=Frac(P)

This will lead toCalogero-Moser cells(B.-Rouquier)

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The P = C [ C × V /W × V

^∗

/W ]-algebra H

p = (c,v,v^∗)∈P =C ×V/W ×V^∗/W and viewC=Cc,v,v^∗ as a P-algebra via evaluation atp

=⇒ H_c,v,v^∗ =Cc,v,v^∗⊗_P H.

H_c |{z}=

vector space

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The P = C [ C × V /W × V

^∗

/W ]-algebra H

p = (c,v,v^∗)∈P =C ×V/W ×V^∗/W and viewC=Cc,v,v^∗ as a P-algebra via evaluation atp =⇒ H_c,v,v^∗ =Cc,v,v^∗⊗_PH.

H_c |{z}=

vector space

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The P = C [ C × V /W × V

^∗

/W ]-algebra H

H_c |{z}=

vector space

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The P = C [ C × V /W × V

^∗

/W ]-algebra H

H_c |{z}=

vector space

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The P = C [ C × V /W × V

^∗

/W ]-algebra H

H_c |{z}=

vector space

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Restricted Cherednik algebra (Gordon, 2003)

Take c ∈C and let

H_c⁻ =CW ⊗C[V^∗]^co(W⁾=W n C[V^∗]^co(W⁾⊂H_c.

Si χ∈Irr(W), letMc(χ) =Hc⊗_H⁻

c

Vfχ.Then (Gordon):

Mc(χ) is indecomposable with a unique simple quotientLc(χ). Irr(W)−→Irr(H_c),χ7→Lc(χ) is a bijection.

K0(Hc)'ZIrr(W).

Calogero-Moser families

The c-Calogero-Moser families are the subsets ofIrr(W)corresponding to the blocks of H_c.

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Restricted Cherednik algebra (Gordon, 2003)

Take c ∈C and let

c

Vfχ.Then (Gordon):

K0(Hc)'ZIrr(W).

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Restricted Cherednik algebra (Gordon, 2003)

Take c ∈C and let

c

Vfχ.

Then (Gordon):

K0(Hc)'ZIrr(W).

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Restricted Cherednik algebra (Gordon, 2003)

Take c ∈C and let

c

Vfχ.Then (Gordon):

Mc(χ) is indecomposable with a unique simple quotientLc(χ).

Irr(W)−→Irr(H_c),χ7→Lc(χ) is a bijection. K0(Hc)'ZIrr(W).

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Restricted Cherednik algebra (Gordon, 2003)

Take c ∈C and let

c

Vfχ.Then (Gordon):

Irr(W)−→Irr(H_c),χ7→Lc(χ) is a bijection.

K0(Hc)'ZIrr(W).

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Restricted Cherednik algebra (Gordon, 2003)

Take c ∈C and let

c

Vfχ.Then (Gordon):

K0(Hc)'ZIrr(W).

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Restricted Cherednik algebra (Gordon, 2003)

Take c ∈C and let

c

Vfχ.Then (Gordon):

K0(Hc)'ZIrr(W).

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Restricted Cherednik algebra (continued)

Let Q¯c =Cc,0,0⊗_P Q. Then

Q¯_c ⊆Z(H_c)

but, in general, the inclusion is strict. However (M¨uller) Idempr(Q¯c) =Idempr(Z(Hc)).

Theorem (B.-Rouquier 2010)

IfF is a Calogero-Moser family corresponding to a primitive idempotent b ∈Idempr(Q¯c), then

dim_CQ¯cb = X

χ∈Irr(W)

χ(1)².

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Restricted Cherednik algebra (continued)

Q¯_c ⊆Z(H_c)

dim_CQ¯cb = X

χ∈Irr(W)

χ(1)².

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Restricted Cherednik algebra (continued)

Q¯_c ⊆Z(H_c) but, in general, the inclusion is strict.

However (M¨uller) Idempr(Q¯c) =Idempr(Z(Hc)).

dim_CQ¯cb = X

χ∈Irr(W)

χ(1)².

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Restricted Cherednik algebra (continued)

Q¯_c ⊆Z(H_c)

dim_CQ¯cb = X

χ∈Irr(W)

χ(1)².

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KH?

Recall that

H'End_Q(He). So

KH'End_KQ(KHe). But Q is an integral domain (and normal) so

KQ =Frac(Q) =:L. Moreover,

[L:K] =|W| and dim_K(KHe) =|W|². So dim_L(KHe) =|W| and

KH'Mat_|_W_|(L).

NOT SPLIT

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KH?

Recall that

H'End_Q(He).

So

KH'End_KQ(KHe). But Q is an integral domain (and normal) so

KQ =Frac(Q) =:L. Moreover,

KH'Mat_|_W_|(L).

NOT SPLIT

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KH?

Recall that

H'End_Q(He).

So

KH'End_KQ(KHe).

But Q is an integral domain (and normal) so KQ =Frac(Q) =:L. Moreover,

KH'Mat_|_W_|(L).

NOT SPLIT

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KH?

Recall that

H'End_Q(He).

So

KH'End_KQ(KHe).

But Q is an integral domain (and normal) so KQ =Frac(Q)

=:L. Moreover,

KH'Mat_|_W_|(L).

NOT SPLIT

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KH?

Recall that

H'End_Q(He).

So

KH'End_KQ(KHe).

But Q is an integral domain (and normal) so KQ =Frac(Q) =:L.

Moreover,

[L:K] =|W| and dim_K(KHe) =|W|².

So dim_L(KHe) =|W| and

KH'Mat_|_W_|(L).

NOT SPLIT

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KH?

Recall that

H'End_Q(He).

So

KH'End_KQ(KHe).

Moreover,

KH'Mat_|_W_|(L).

NOT SPLIT

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KH?

Recall that

H'End_Q(He).

So

KH'End_KQ(KHe).

Moreover,

KH'Mat_|_W_|(L).

NOT SPLIT

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How to split KH ' Mat

_|_W_|

(L)?

Let Mbe the Galois closure of the extensionL/Kand let G =Gal(M/K) and H =Gal(M/L). Then, as [L:K] =|W|, one gets

|G/H|=|W|. Then

M⊗_KL −→ ⊕_gH_∈G_/HM m⊗_Kl 7−→ ⊕_gH∈G_/Hmg(l). So

MH' ⊕_gH∈G_/HMat_|_W_|(M) and

Irr(MH)←→^∼ G/H.

|Irr(MH)|=|G/H|=|W|

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How to split KH ' Mat

_|_W_|

(L)?

Let Mbe the Galois closure of the extensionL/Kand let G =Gal(M/K) and H =Gal(M/L).

Then, as [L:K] =|W|, one gets

|G/H|=|W|. Then

|Irr(MH)|=|G/H|=|W|

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How to split KH ' Mat

_|_W_|

(L)?

|G/H|=|W|.

Then

|Irr(MH)|=|G/H|=|W|

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How to split KH ' Mat

_|_W_|

(L)?

|G/H|=|W|. Then

M⊗_KL −→ ⊕_gH_∈G_/HM m⊗_Kl 7−→ ⊕_gH∈G_/Hmg(l).

So

|Irr(MH)|=|G/H|=|W|

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How to split KH ' Mat

_|_W_|

(L)?

|G/H|=|W|. Then

So

MH' ⊕_gH_∈G_/HMat_|_W_|(M) and

|Irr(MH)|=|G/H|=|W|

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How to split KH ' Mat

_|_W_|

(L)?

|G/H|=|W|. Then

So

MH' ⊕_gH_∈G_/HMat_|_W_|(M) and

|Irr(MH)|=|G/H|=|W|

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G /H and W ?

Let p₀ =Ker(P →C[V/W ×V^∗/W])!evaluation atc =0. Recall that Q0 =Q/p0Q'C[(V ×V^∗)/W]. So

q₀ = p₀Q ∈Spec(Q) and we fix a prime ideal r₀ of R lying above q₀.

D0 =G_r^dec₀ ={g ∈G | g(r₀) = r₀} I0 =G_rⁱⁿ₀ ={g ∈G | ∀r ∈R, g(r)≡r modr₀} Then

Gal(kR(r₀)/kP(p₀))'D0/I0. But, since p₀Q∈Spec(Q), we also have:

G =D0·H =H·D0

I₀=1

k_R(r₀)/k_P(p₀) is the Galois closure ofk_Q(q₀)/k_P(p₀). But

k_P(p₀)'C(V ×V^∗)^W^×W ⊂k_Q(q₀) =C(V ×V^∗)^∆W ⊂C(V ×V^∗).

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G /H and W ?

Let p₀ =Ker(P →C[V/W ×V^∗/W])!evaluation atc =0.

Recall that Q0 =Q/p0Q'C[(V ×V^∗)/W]. So q₀ = p₀Q ∈Spec(Q) and we fix a prime ideal r₀ of R lying above q₀.

D0 =G_r^dec₀ ={g ∈G | g(r₀) = r₀} I0 =G_rⁱⁿ₀ ={g ∈G | ∀r ∈R, g(r)≡r modr₀} Then

Gal(kR(r₀)/kP(p₀))'D0/I0. But, since p₀Q∈Spec(Q), we also have:

G =D0·H =H·D0

I₀=1

k_R(r₀)/k_P(p₀) is the Galois closure ofk_Q(q₀)/k_P(p₀). But

k_P(p₀)'C(V ×V^∗)^W^×W ⊂k_Q(q₀) =C(V ×V^∗)^∆W ⊂C(V ×V^∗).