Global local conjectures and the Bonnaf eDatRouquier Morita - - PowerPoint PPT Presentation

Global local conjectures and the Bonnaf´ e–Dat–Rouquier Morita equivalence

Lucas Ruhstorfer Bergische Universit¨ at Wuppertal June 12th, 2019

Motivation: The Alperin-McKay conjecture

Notation:

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Motivation: The Alperin-McKay conjecture

Notation:

• G a finite group and ℓ a prime with ℓ | |G|.

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Motivation: The Alperin-McKay conjecture

Notation:

• G a finite group and ℓ a prime with ℓ | |G|.
• (K, O, k) an ℓ-modular system large enough.

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Motivation: The Alperin-McKay conjecture

Notation:

• G a finite group and ℓ a prime with ℓ | |G|.
• (K, O, k) an ℓ-modular system large enough.
• Irr(G) the set of irreducible K-characters.

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Motivation: The Alperin-McKay conjecture

Notation:

• G a finite group and ℓ a prime with ℓ | |G|.
• (K, O, k) an ℓ-modular system large enough.
• Irr(G) the set of irreducible K-characters.
• B be an ℓ-block of OG with defect group D.

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Motivation: The Alperin-McKay conjecture

Notation:

• G a finite group and ℓ a prime with ℓ | |G|.
• (K, O, k) an ℓ-modular system large enough.
• Irr(G) the set of irreducible K-characters.
• B be an ℓ-block of OG with defect group D.
• b the Brauer correspondent of B in NG(D).

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Motivation: The Alperin-McKay conjecture

Notation:

• G a finite group and ℓ a prime with ℓ | |G|.
• (K, O, k) an ℓ-modular system large enough.
• Irr(G) the set of irreducible K-characters.
• B be an ℓ-block of OG with defect group D.
• b the Brauer correspondent of B in NG(D).

Conjecture (Alperin-McKay conjecture) |Irr0(B)| = |Irr0(b)|,

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Motivation: The Alperin-McKay conjecture

Notation:

• G a finite group and ℓ a prime with ℓ | |G|.
• (K, O, k) an ℓ-modular system large enough.
• Irr(G) the set of irreducible K-characters.
• B be an ℓ-block of OG with defect group D.
• b the Brauer correspondent of B in NG(D).

Conjecture (Alperin-McKay conjecture) |Irr0(B)| = |Irr0(b)|, where Irr0(B) = {χ ∈ Irr(B) | χ(1)ℓ = |G : D|ℓ}.

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The reduction theorem

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The reduction theorem

Theorem (Sp¨ ath ’13) The Alperin-McKay conjecture holds for all groups and primes if the so-called inductive Alperin-McKay (iAM) condition holds for all blocks of quasi-simple groups.

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The reduction theorem

Theorem (Sp¨ ath ’13) The Alperin-McKay conjecture holds for all groups and primes if the so-called inductive Alperin-McKay (iAM) condition holds for all blocks of quasi-simple groups. The iAM-condition holds for an ℓ-block B of a quasi-simple group G if there exists an Aut(G)B,D-equivariant bijection Ω : Irr0(B) → Irr0(b),

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The reduction theorem

Theorem (Sp¨ ath ’13) The Alperin-McKay conjecture holds for all groups and primes if the so-called inductive Alperin-McKay (iAM) condition holds for all blocks of quasi-simple groups. The iAM-condition holds for an ℓ-block B of a quasi-simple group G if there exists an Aut(G)B,D-equivariant bijection Ω : Irr0(B) → Irr0(b), preserving Clifford theory with respect to G ✁ G ⋊ Aut(G)B,D.

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Representation theory of finite reductive groups

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Representation theory of finite reductive groups

Let G be a connected reductive group with Frobenius F : G → G defining an Fq-structure, ℓ ∤ q.

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Representation theory of finite reductive groups

Let G be a connected reductive group with Frobenius F : G → G defining an Fq-structure, ℓ ∤ q. Let (G∗, F ∗) denote the dual group of (G, F). Theorem (Brou´ e-Michel ’89) We have a decomposition: OGF − mod ∼ =

• (s)

OGFeGF

s

− mod where (s) runs over the set of (G∗)F ∗-conjugacy classes of semisimple elements of (G∗)F ∗ of ℓ′-order.

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Representation theory of finite reductive groups

Let G be a connected reductive group with Frobenius F : G → G defining an Fq-structure, ℓ ∤ q. Let (G∗, F ∗) denote the dual group of (G, F). Theorem (Brou´ e-Michel ’89) We have a decomposition: OGF − mod ∼ =

• (s)

OGFeGF

s

− mod where (s) runs over the set of (G∗)F ∗-conjugacy classes of semisimple elements of (G∗)F ∗ of ℓ′-order. Aim: Understand the Representation theory of OGFeGF

s

− mod for a fixed series (s).

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Jordan decomposition for characters

Let L Levi subgroup of G with F(L) = L and Levi decomposition P = L ⋉ U.

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Jordan decomposition for characters

Let L Levi subgroup of G with F(L) = L and Levi decomposition P = L ⋉ U. Consider the Deligne–Lusztig variety YG

U = {gU ∈ G/U | g−1F(g) ∈ UF(U)}

with GF × (LF)opp-action.

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Jordan decomposition for characters

Let L Levi subgroup of G with F(L) = L and Levi decomposition P = L ⋉ U. Consider the Deligne–Lusztig variety YG

U = {gU ∈ G/U | g−1F(g) ∈ UF(U)}

with GF × (LF)opp-action. Theorem (Bonnaf´ e–Rouquier ’03) Let CG∗(s) ⊆ L∗. Then the bimodule H

dim(YG

U)

c

(YG

U, O)eLF s

induces a Morita equivalence between OGFeLF

s

− mod and OLFeGF

s

− mod.

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Jordan decomposition for characters

Let L Levi subgroup of G with F(L) = L and Levi decomposition P = L ⋉ U. Consider the Deligne–Lusztig variety YG

U = {gU ∈ G/U | g−1F(g) ∈ UF(U)}

with GF × (LF)opp-action. Theorem (Bonnaf´ e–Rouquier ’03) Let CG∗(s) ⊆ L∗. Then the bimodule H

dim(YG

U)

c

(YG

U, O)eLF s

induces a Morita equivalence between OGFeLF

s

− mod and OLFeGF

s

− mod.

• Reduces questions about blocks to a question about

quasi-isolated blocks of Levi subgroups.

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Jordan decomposition for characters

Let L Levi subgroup of G with F(L) = L and Levi decomposition P = L ⋉ U. Consider the Deligne–Lusztig variety YG

U = {gU ∈ G/U | g−1F(g) ∈ UF(U)}

with GF × (LF)opp-action. Theorem (Bonnaf´ e–Rouquier ’03) Let CG∗(s) ⊆ L∗. Then the bimodule H

dim(YG

U)

c

(YG

U, O)eLF s

induces a Morita equivalence between OGFeLF

s

− mod and OLFeGF

s

− mod.

• Reduces questions about blocks to a question about

quasi-isolated blocks of Levi subgroups.

• OLFeLF

s

and OGFeGF

s

splendid Rickard equivalence ⇒ Rickard equivalences on the level of local subgroups (Bonnaf´ e–Dat–Rouquier ’17).

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Application to the inductive Alperin–McKay conditions

We want to apply the Bonnaf´ e–Dat–Rouquier reduction to the iAM-condition.

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Application to the inductive Alperin–McKay conditions

We want to apply the Bonnaf´ e–Dat–Rouquier reduction to the iAM-condition. Bl(LF) ∋ C B ∈ Bl(GF) Bl(NLF (D)) ∋ c b ∈ Bl(NGF (D))

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Application to the inductive Alperin–McKay conditions

We want to apply the Bonnaf´ e–Dat–Rouquier reduction to the iAM-condition. Bl(LF) ∋ C B ∈ Bl(GF) Bl(NLF (D)) ∋ c b ∈ Bl(NGF (D)) Two main tasks:

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Application to the inductive Alperin–McKay conditions

We want to apply the Bonnaf´ e–Dat–Rouquier reduction to the iAM-condition. Bl(LF) ∋ C B ∈ Bl(GF) Bl(NLF (D)) ∋ c b ∈ Bl(NGF (D)) Two main tasks:

• Lift to Morita equivalence to include automorphisms coming

from E = γ, F0 similar as in Julian’s talk.

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Application to the inductive Alperin–McKay conditions

We want to apply the Bonnaf´ e–Dat–Rouquier reduction to the iAM-condition. Bl(LF) ∋ C B ∈ Bl(GF) Bl(NLF (D)) ∋ c b ∈ Bl(NGF (D)) Two main tasks:

• Lift to Morita equivalence to include automorphisms coming

from E = γ, F0 similar as in Julian’s talk.

• Find a local equivalence on the level of normalizers satisfying

similar properties.

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Main results

Theorem (R. ’19) Suppose that the order of γ is coprime to ℓ. Then OLFE eLF

s

− mod and OGFE eGF

s

− mod are Morita equivalent.

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Main results

Theorem (R. ’19) Suppose that the order of γ is coprime to ℓ. Then OLFE eLF

s

− mod and OGFE eGF

s

− mod are Morita equivalent. Proof: The module H

dim(YG

U)

c

(YG

U, O)eLF s

extends to an OGF × (LF)opp∆(E)-module M.

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Main results

Theorem (R. ’19) Suppose that the order of γ is coprime to ℓ. Then OLFE eLF

s

− mod and OGFE eGF

s

− mod are Morita equivalent. Proof: The module H

dim(YG

U)

c

(YG

U, O)eLF s

extends to an OGF × (LF)opp∆(E)-module M. Then IndGF E×(LF E)opp

GF ×(LF )opp∆(E)(M)

induces a Morita equivalence between OLFE eLF

s

− mod and OGFE eGF

s

− mod (Marcus ’96).

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Main results

Theorem (R. ’19) Suppose that the order of γ is coprime to ℓ. Then OLFE eLF

s

− mod and OGFE eGF

s

− mod are Morita equivalent. Proof: The module H

dim(YG

U)

c

(YG

U, O)eLF s

extends to an OGF × (LF)opp∆(E)-module M. Then IndGF E×(LF E)opp

GF ×(LF )opp∆(E)(M)

induces a Morita equivalence between OLFE eLF

s

− mod and OGFE eGF

s

− mod (Marcus ’96). Consequences We have similar equivalence on the level of normalizers

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Main results

Theorem (R. ’19) Suppose that the order of γ is coprime to ℓ. Then OLFE eLF

s

− mod and OGFE eGF

s

− mod are Morita equivalent. Proof: The module H

dim(YG

U)

c

(YG

U, O)eLF s

extends to an OGF × (LF)opp∆(E)-module M. Then IndGF E×(LF E)opp

GF ×(LF )opp∆(E)(M)

induces a Morita equivalence between OLFE eLF

s

− mod and OGFE eGF

s

− mod (Marcus ’96). Consequences We have similar equivalence on the level of normalizers ⇒ Reduce the verification of the iAM-conditions to quasi-isolated blocks of Levi subgroups (Work in progress).

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Thank you!

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