Low-degree cohomology for finite groups of Lie type Niles Johnson Joint with UGA VIGRE Algebra Group Department of Mathematics University of Georgia September 2011 UGA VIGRE Algebra (UGA) Low-degree cohomology September 2011 1 / 30
Introduction UGA VIGRE Algebra Group Faculty Graduate Students Brian D. Boe Brian Bonsignore Jon F. Carlson Theresa Brons Leonard Chastkofsky Adrian M. Brunyate Daniel K. Nakano Wenjing Li Lisa Townsley Phong Tanh Luu Tiago Macedo Nham Vo Ngo Postdoctoral Fellows Duc Duy Nguyen Christopher M. Drupieski Brandon L. Samples Niles Johnson Andrew J. Talian Benjamin F. Jones Benjamin J. Wyser We would like to acknowledge NSF VIGRE grant DMS-0738586 for its financial support of the project. UGA VIGRE Algebra (UGA) Low-degree cohomology September 2011 2 / 30
Introduction Overview Low-degree cohomology of finite algebraic groups SL n ( F q ), SO n ( F q ), Sp 2 n ( F q ), etc. ◮ q = p r Simple coefficient module M = L ( λ ). ◮ λ below a fundamental dominant weight Modular case: characteristic p . ◮ char ( M ) � | | G ( F q ) | ⇒ H ∗ ( G ( F q ) , M ) = 0. Small primes ◮ new techniques are necessary. Combinatorial, topological, and scheme-theoretic techniques applied to problems in cohomology of finite groups, Hopf algebras, Lie algebras. UGA VIGRE Algebra (UGA) Low-degree cohomology September 2011 3 / 30
Introduction Motivation Interest in finite group cohomology; modular case, small primes Generalize vanishing results of Cline-Parshall-Scott (1974) ◮ Wiles’s proof of Fermat’s Last Theorem Reproduce and extend degree-two results ◮ Avrunin (1978): certain minimal weights ◮ Bell (1978): type A analyzed completely Relationship between finite and algebraic groups UGA VIGRE Algebra (UGA) Low-degree cohomology September 2011 4 / 30
Introduction Background Algebraic Group Schemes k , algebraically closed field of positive characteristic p . G , (affine) algebraic group scheme over k ↔ Hopf algebra k [ G ]. ◮ A scheme is a geometric object, parametrizing (matrix) groups over k -algebras: ( SL n ( R ), SO n ( R ), Sp 2 n ( R )). M , (rational) G -module ↔ comodule over k [ G ]; Simple, simply-connected algebraic groups: classified by Lie type (Dynkin diagrams ↔ root systems, Φ) ◮ A n , B n , C n , D n ; rank n ≥ 1 ◮ G 2 , F 4 , E 6 , E 7 , E 8 UGA VIGRE Algebra (UGA) Low-degree cohomology September 2011 5 / 30
Introduction Background Example: A n = SL n SL n ( R ) = { ( a ij ) | det ( a ij ) = 1 } ∗ * ... · · · B ( R ) = 0 ∗ 1 * ... U ( R ) = 0 1 ∗ 0 ... T ( R ) = 0 ∗ Wikipedia: Root system A2.svg UGA VIGRE Algebra (UGA) Low-degree cohomology September 2011 6 / 30
Introduction Background Group Cohomology Algebraic group cohomology: H ∗ ( G , M ) = Ext ∗ G ( k , M ) = Ext ∗ k [ G ]-comod ( k , M ). Finite group cohomology: H ∗ ( G ( F q ) , M ) = Ext ∗ G ( F q ) ( k , M ) = Ext ∗ kG ( F q ) ( k , M ). ◮ M = M ( F q ) Maximal torus T ≤ G . ◮ Simultaneous diagonalization of commuting matricies ⇒ decomposition of representations into weight spaces ◮ X ( T ) = weight lattice; fundamental dominant weights ω 1 , . . . , ω n ◮ Weights are partially ordered. Highest-weight modules M = L ( λ ), λ ∈ X + ( T ) (dominant weights). ◮ Unique simple modules with highest-weight λ . UGA VIGRE Algebra (UGA) Low-degree cohomology September 2011 7 / 30
� � � � � Introduction Background Example: A n = SL n k [ SL n ] = k [ X ij ] / det − 1 Frobenius F : ( a ij ) �→ ( a p ij ) ( SL n ) r = ker F r If R = F p , ( SL n ( R )) 1 = ker F = 1; more interesting when R has nilpotents, roots of unity. Strategy: (top row) G G r B r U r Fact: simple module L ( λ ) restricts to simple ( r =1) univ p ( u ⊕ r ) modules for G ( F q ) and G r . gr G ( F q ) U r ( F q ) UGA VIGRE Algebra (UGA) Low-degree cohomology September 2011 8 / 30
Introduction Background Fundamental exact sequence Consider the long exact sequence in cohomology induced by 0 → k → G r → G r / k → 0 . res 0 0 − → Hom G ( k , L ( λ )) − − → Hom G ( F q ) ( k , L ( λ )) − → Hom G ( k , L ( λ ) ⊗ G r / k ) res 1 Ext 1 Ext 1 Ext 1 − → − − → − → G ( k , L ( λ ) ⊗ G r / k ) G ( k , L ( λ )) G ( F q ) ( k , L ( λ )) res 2 Ext 2 Ext 2 Ext 2 − → G ( k , L ( λ )) − − → G ( F q ) ( k , L ( λ )) − → G ( k , L ( λ ) ⊗ G r / k ) − → · · · G r = ind G G ( F q ) ( k ) As a G -module, G r / k admits a filtration with layers of the form H 0 ( µ ) ⊗ H 0 ( µ ∗ ) ( r ) . UGA VIGRE Algebra (UGA) Low-degree cohomology September 2011 9 / 30
Introduction Results Results: Comparison with algebraic group Assume that p > 2 for Φ = A n , D n . p > 3 for Φ = B n , C n , E 6 , E 7 , F 4 . p > 5 for Φ = E 8 , G 2 . q ≥ 4. Theorem Suppose λ ≤ ω j for some j. Then the restriction map res t : H t ( G , L ( λ )) → H t ( G ( F q ) , L ( λ )) is an isomorphism for t = 1 and an injection for t = 2 . UGA VIGRE Algebra (UGA) Low-degree cohomology September 2011 10 / 30
Introduction Results Results: Comparison with algebraic group Assume the following Prime-Power Restrictions hold for p and q = p r : p > 3 for Φ = A n , B n , C n , D n , E 6 , E 7 , F 4 . p > 5 for Φ = E 8 , G 2 . q ≥ 7 for Φ = E 7 , F 4 . Theorem Suppose λ ≤ ω j for some j, and suppose the Weight Condition holds for λ . Then the restriction map res 2 : H 2 ( G , L ( λ )) → H 2 ( G ( F q ) , L ( λ )) is an isomorphism. We say that λ ∈ X ( T ) + satisfies the Weight Condition if � � − ( ν, γ ∨ ) : γ ∈ ∆ , ν a weight of Ext 1 max U r ( k , L ( λ )) < q . UGA VIGRE Algebra (UGA) Low-degree cohomology September 2011 11 / 30
Introduction Results Problem weights: Weight Condition fails to hold Type Weights A 2 , q = 5 ω 1 , ω 2 α = ω 2 if n ≥ 3) B n α 0 = ω 1 (and � C n α 0 = ω 2 α = ω 2 � D n α = ω 2 E 6 � E 7 α = ω 1 � E 8 α = ω 8 � α 0 = ω 4 , � α = ω 1 F 4 α 0 = ω 1 , � α = ω 2 G 2 Table: Highest short roots are denoted by α 0 , and highest long roots by � α . UGA VIGRE Algebra (UGA) Low-degree cohomology September 2011 12 / 30
Introduction Results Finite group H 1 , λ = ω j Assume that p > 2 for Φ = A n , D n p > 3 for Φ = B n , C n , E 6 , E 7 , F 4 , G 2 p > 5 for Φ = E 8 and assume q ≥ 4. Theorem Then H 1 ( G ( F q ) , L ( ω j )) = 0 except for the following cases, in which we have H 1 ( G ( F q ) , L ( ω j )) ∼ = k: Φ has type C n , n ≥ 3 , ( n + 1) = � t i =0 b i p i with 0 ≤ b i < p and b t � = 0 , and j = 2 b i p i for some 0 ≤ i < t with b i � = 0 ; Φ is of type E 7 , p = 7 and j = 6 . UGA VIGRE Algebra (UGA) Low-degree cohomology September 2011 13 / 30
Introduction Results Finite group H 1 , λ < ω j (exceptional types) Let Φ be of exceptional type. Assume that p > 3 for Φ = E 6 , F 4 , G 2 . p > 7 for Φ = E 7 , E 8 . Theorem Suppose λ ≤ ω j for some j. Then H 1 ( G ( F q ) , L ( λ )) = 0 except for the following cases, in which we have H 1 ( G ( F q ) , L ( λ )) ∼ = k: Φ = F 4 , p = 13 , and λ = 2 ω 4 . Φ = E 7 , p = 19 , and λ = 2 ω 1 . Φ = E 8 , p = 31 , and λ = 2 ω 8 . UGA VIGRE Algebra (UGA) Low-degree cohomology September 2011 14 / 30
Introduction Results Finite group H 2 , λ ≤ ω j Assume that The Prime-Power Restrictions hold for p and q . p > n for Φ = C n if λ = ω j with j even. ( H 2 ∼ For Φ = E 8 and p = 31, λ � = ω 7 + ω 8 . = k in this case.) The Weight Condition holds for λ . Theorem Under the assumptions above, H 2 ( G ( F q ) , L ( λ )) = 0 except possibly the following cases: Φ = E 7 , p = 5 , λ = 2 ω 7 Φ = E 7 , p = 7 , λ = ω 2 + ω 7 Φ = E 8 , p = 7 , λ ∈ { 2 ω 7 , ω 1 + ω 7 , ω 2 + ω 8 } Φ = E 8 , p = 31 , λ = ω 6 + ω 8 Note: E 7 has 12 non-zero weights λ ≤ ω j for some j ; E 8 has 23. UGA VIGRE Algebra (UGA) Low-degree cohomology September 2011 15 / 30
Introduction Results Finite Group H 2 for problem weights Show, instead, that the restriction map vanishes. The finite group cohomology is isomorphic to the term in column 3. Theorem Suppose that the Prime-Power Restrictions hold for p and q, and suppose that λ does not satisfy the Weight Condition. Assume moreover: For Φ = B n and λ = � α , � α is not linked to α 0 . For Φ = C n , p � | n. For p = q and Φ � = A 2 , p > 5 . Then � 0 if λ = α 0 and Φ has two root lengths H 2 ( G ( F q ) , L ( λ ) = k otherwise UGA VIGRE Algebra (UGA) Low-degree cohomology September 2011 16 / 30
Summary of Methods General Type Summary of Methods: Fundamental exact sequence res 0 0 − → Hom G ( k , L ( λ )) − − → Hom G ( F q ) ( k , L ( λ )) − → Hom G ( k , L ( λ ) ⊗ G r / k ) res 1 Ext 1 Ext 1 Ext 1 − → G ( k , L ( λ )) − − → G ( F q ) ( k , L ( λ )) − → G ( k , L ( λ ) ⊗ G r / k ) res 2 Ext 2 Ext 2 Ext 2 − → G ( k , L ( λ )) − − → G ( F q ) ( k , L ( λ )) − → G ( k , L ( λ ) ⊗ G r / k ) − → · · · G r = ind G G ( F q ) ( k ) As a G -module, G r / k admits a filtration with layers of the form H 0 ( µ ) ⊗ H 0 ( µ ∗ ) ( r ) . Ext i ( k , L ( λ ) ⊗ H 0 ( µ ) ⊗ H 0 ( µ ∗ ) ( r ) ) ∼ = Ext i ( V ( µ ) ( r ) , L ( λ ) ⊗ H 0 ( µ )) . UGA VIGRE Algebra (UGA) Low-degree cohomology September 2011 17 / 30
Recommend
More recommend
