Second cohomology groups for algebraic groups and their Frobenius kernels

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, Gr, to be the kernel of F iterated with itself r times. Define Ur (respectively Br) 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 H2(U1,k), H2(Br,λ) for λ∈X(T), H2(Gr,H0(λ)) for λ∈X(T)+, and H2(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 H2(Gr,H0(λ)) has a good filtration.