Second cohomology groups for algebraic groups and their Frobenius kernels

Details Second cohomology groups for algebraic groups and their Frobenius kernels Second cohomology groups for algebraic groups and their Frobenius kernels

2008-09-17

arxiv.org/abs/0809.2833v2

Mathematics

Representation Theory

49 pages, 4 appendices, 6 tables

Scientific paper

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 $G$, $B \supset T$ be a Borel subgroup of $G$ and $U$ its unipotent radical. Let $F: G \rightarrow G$ be the Frobenius morphism. For $r \geq 1$ 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 $\h^2(U_1,k)$, $\h^2(B_r,\la)$ for $\la \in X(T)$, $\h^2(G_r,H^0(\la))$ for $\la \in X(T)_+$, and $\h^2(B,\la)$ for $\la \in X(T)$. The above cohomology groups for the case when the field has characteristic 2 one computed in this paper. These computations complete the picture started by Bendel, Nakano, and Pillen for $p \geq 3$ \cite{BNP2}.

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