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: 2 / 8
Motivation: The Alperin-McKay conjecture Notation: • G a finite group and ℓ a prime with ℓ | | G | . 2 / 8
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. 2 / 8
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. 2 / 8
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 O G with defect group D . 2 / 8
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 O G with defect group D . • b the Brauer correspondent of B in N G ( D ). 2 / 8
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 O G with defect group D . • b the Brauer correspondent of B in N G ( D ). Conjecture (Alperin-McKay conjecture) | Irr 0 ( B ) | = | Irr 0 ( b ) | , 2 / 8
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 O G with defect group D . • b the Brauer correspondent of B in N G ( D ). Conjecture (Alperin-McKay conjecture) | Irr 0 ( B ) | = | Irr 0 ( b ) | , where Irr 0 ( B ) = { χ ∈ Irr ( B ) | χ ( 1 ) ℓ = | G : D | ℓ } . 2 / 8
The reduction theorem 3 / 8
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. 3 / 8
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 : Irr 0 ( B ) → Irr 0 ( b ) , 3 / 8
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 : Irr 0 ( B ) → Irr 0 ( b ) , preserving Clifford theory with respect to G ✁ G ⋊ Aut ( G ) B , D . 3 / 8
Representation theory of finite reductive groups 4 / 8
Representation theory of finite reductive groups Let G be a connected reductive group with Frobenius F : G → G defining an F q -structure, ℓ ∤ q . 4 / 8
Representation theory of finite reductive groups Let G be a connected reductive group with Frobenius F : G → G defining an F q -structure, ℓ ∤ q . Let ( G ∗ , F ∗ ) denote the dual group of ( G , F ). Theorem (Brou´ e-Michel ’89) We have a decomposition: O G F − mod ∼ O G F e G F � − mod = s ( s ) where ( s ) runs over the set of ( G ∗ ) F ∗ -conjugacy classes of semisimple elements of ( G ∗ ) F ∗ of ℓ ′ -order. 4 / 8
Representation theory of finite reductive groups Let G be a connected reductive group with Frobenius F : G → G defining an F q -structure, ℓ ∤ q . Let ( G ∗ , F ∗ ) denote the dual group of ( G , F ). Theorem (Brou´ e-Michel ’89) We have a decomposition: O G F − mod ∼ O G F e G F � − mod = s ( s ) 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 O G F e G F − mod for s a fixed series ( s ). 4 / 8
Jordan decomposition for characters Let L Levi subgroup of G with F ( L ) = L and Levi decomposition P = L ⋉ U . 5 / 8
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 Y G U = { g U ∈ G / U | g − 1 F ( g ) ∈ U F ( U ) } with G F × ( L F ) opp -action. 5 / 8
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 Y G U = { g U ∈ G / U | g − 1 F ( g ) ∈ U F ( U ) } with G F × ( L F ) opp -action. Theorem (Bonnaf´ e–Rouquier ’03) dim ( Y G U ) U , O ) e L F ( Y G Let C G ∗ ( s ) ⊆ L ∗ . Then the bimodule H induces c s a Morita equivalence between O G F e L F − mod and O L F e G F − mod . s s 5 / 8
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 Y G U = { g U ∈ G / U | g − 1 F ( g ) ∈ U F ( U ) } with G F × ( L F ) opp -action. Theorem (Bonnaf´ e–Rouquier ’03) dim ( Y G U ) U , O ) e L F ( Y G Let C G ∗ ( s ) ⊆ L ∗ . Then the bimodule H induces c s a Morita equivalence between O G F e L F − mod and O L F e G F − mod . s s • Reduces questions about blocks to a question about quasi-isolated blocks of Levi subgroups. 5 / 8
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 Y G U = { g U ∈ G / U | g − 1 F ( g ) ∈ U F ( U ) } with G F × ( L F ) opp -action. Theorem (Bonnaf´ e–Rouquier ’03) dim ( Y G U ) U , O ) e L F ( Y G Let C G ∗ ( s ) ⊆ L ∗ . Then the bimodule H induces c s a Morita equivalence between O G F e L F − mod and O L F e G F − mod . s s • Reduces questions about blocks to a question about quasi-isolated blocks of Levi subgroups. • O L F e L F and O G F e G F splendid Rickard equivalence ⇒ s s Rickard equivalences on the level of local subgroups (Bonnaf´ e–Dat–Rouquier ’17). 5 / 8
Application to the inductive Alperin–McKay conditions We want to apply the Bonnaf´ e–Dat–Rouquier reduction to the iAM -condition. 6 / 8
Application to the inductive Alperin–McKay conditions We want to apply the Bonnaf´ e–Dat–Rouquier reduction to the iAM -condition. Bl ( L F ) ∋ C B ∈ Bl ( G F ) Bl ( N L F ( D )) ∋ c b ∈ Bl ( N G F ( D )) 6 / 8
Application to the inductive Alperin–McKay conditions We want to apply the Bonnaf´ e–Dat–Rouquier reduction to the iAM -condition. Bl ( L F ) ∋ C B ∈ Bl ( G F ) Bl ( N L F ( D )) ∋ c b ∈ Bl ( N G F ( D )) Two main tasks: 6 / 8
Application to the inductive Alperin–McKay conditions We want to apply the Bonnaf´ e–Dat–Rouquier reduction to the iAM -condition. Bl ( L F ) ∋ C B ∈ Bl ( G F ) Bl ( N L F ( D )) ∋ c b ∈ Bl ( N G F ( D )) Two main tasks: • Lift to Morita equivalence to include automorphisms coming from E = � γ, F 0 � similar as in Julian’s talk. 6 / 8
Application to the inductive Alperin–McKay conditions We want to apply the Bonnaf´ e–Dat–Rouquier reduction to the iAM -condition. Bl ( L F ) ∋ C B ∈ Bl ( G F ) Bl ( N L F ( D )) ∋ c b ∈ Bl ( N G F ( D )) Two main tasks: • Lift to Morita equivalence to include automorphisms coming from E = � γ, F 0 � similar as in Julian’s talk. • Find a local equivalence on the level of normalizers satisfying similar properties. 6 / 8
Main results Theorem (R. ’19) Suppose that the order of γ is coprime to ℓ . Then O L F E e L F − mod and O G F E e G F − mod are Morita equivalent. s s 7 / 8
Main results Theorem (R. ’19) Suppose that the order of γ is coprime to ℓ . Then O L F E e L F − mod and O G F E e G F − mod are Morita equivalent. s s dim ( Y G U ) U , O ) e L F ( Y G Proof: The module H extends to an c s O G F × ( L F ) opp ∆( E )-module M . 7 / 8
Main results Theorem (R. ’19) Suppose that the order of γ is coprime to ℓ . Then O L F E e L F − mod and O G F E e G F − mod are Morita equivalent. s s dim ( Y G U ) U , O ) e L F ( Y G Proof: The module H extends to an c s O G F × ( L F ) opp ∆( E )-module M . Then Ind G F E × ( L F E ) opp G F × ( L F ) opp ∆( E ) ( M ) induces a Morita equivalence between O L F E e L F − mod and s O G F E e G F − mod (Marcus ’96). s 7 / 8
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