Second cohomology groups for algebraic groups and their Frobenius kernels

Abstract

Let G be a simple simply connected algebraic group scheme defined over an algebraically closed field of characteristic p > 0. Let T be a maximal split torus in G, B ⊃ T be a Borel subgroup of G and U its unipotent radical. Let F : G → G be the Frobenius morphism. For r 1 define the Frobenius kernel, G r , to be the kernel of F iterated with itself r times. Define U r (respectively B r ) to be the kernel of the Frobenius map restricted to U (respectively B). Let X(T ) be the integral weight lattice and X(T ) + be the dominant integral weights.The computations of particular importance are H 2 (U 1 , k), H 2 (B r , λ) for λ ∈ X(T ), H 2 (G r , H 0 (λ)) for λ ∈ X(T ) + , and H 2 (B, λ) for λ ∈ X(T ). The above cohomology groups for the case when the field has characteristic 2 are computed in this paper. These computations complete the picture started by Bendel, Nakano, and Pillen (2007) [5] for p 3. Furthermore, the computations show H 2 (G r , H 0 (λ)) has a good filtration.

On the smoothness of normalisers, the subalgebra structure of modular Lie algebras, and the cohomology of small representations

“…Then: Remark 8.3. All the values of H 2 (G r , H 0 (λ)) [−1] are known for all λ by [BNP07, Theorem 6.2] (p ≥ 3) and [Wri11] (p = 2). For example, H 2 (G 1 , k) [−1] ∼ = g * also when G is of type A 1 and p = 2.…”

Extensions for finite Chevalley groups III: Rational and generic cohomology

“…One such case is when p > h where H 2•+1 (G 1 , k) (−1) = 0 and H 2• (G 1 , k) (−1) ∼ = k [N ], where N is the nilpotent cone of g. When p is arbitrary, H n (G s , H 0 (λ)) has a good filtration for λ ∈ X(T ) + and n = 0, 1, 2, (cf. [BNP4,BNP7,Jan1,Wr]). It is suspected that this should hold for arbitrary n. In fact, if p > h, H n (G 1 , H 0 (λ)) has a good filtration for all n [KLT].…”

On the smoothness of normalisers and the subalgebra structure of modular Lie algebras

“…Remark 8.3. All the values of H 2 (G r , H 0 (λ)) [−1] are known for all λ by [BNP07, Theorem 6.2] (p ≥ 3) and [Wri11] (p = 2). For example, H 2 (G 1 , k) [−1] ∼ = g * also when G is of type A 1 and p = 2.…”