Extensions for finite Chevalley groups II

Abstract:

Let $G$ be a semisimple simply connected algebraic group defined and split over the field ${\mathbb {F}}_p$ with $p$ elements, let $G(\mathbb {F}_{q})$ be the finite Chevalley group consisting of the ${\mathbb {F}}_{q}$-rational points of $G$ where $q = p^r$, and let $G_{r}$ be the $r$th Frobenius kernel. The purpose of this paper is to relate extensions between modules in $\text {Mod}(G(\mathbb {F}_{q}))$ and $\text {Mod}(G_{r})$ with extensions between modules in $\text {Mod}(G)$. Among the results obtained are the following: for $r >2$ and $p\geq 3(h-1)$, the $G(\mathbb {F}_{q})$-extensions between two simple $G(\mathbb {F}_{q})$-modules are isomorphic to the $G$-extensions between two simple $p^r$-restricted $G$-modules with suitably “twisted" highest weights. For $p \geq 3(h-1)$, we provide a complete characterization of $\text {H}^{1}(G(\mathbb {F}_{q}),H^{0}(\lambda ))$ where $H^{0}(\lambda )=\text {ind}_{B}^{G}\ \lambda$ and $\lambda$ is $p^r$-restricted. Furthermore, for $p \geq 3(h-1)$, necessary and sufficient bounds on the size of the highest weight of a $G$-module $V$ are given to insure that the restriction map $\operatorname {H}^{1}(G,V)\rightarrow \operatorname {H}^{1}(G(\mathbb {F}_{q}),V)$ is an isomorphism. Finally, it is shown that the extensions between two simple $p^r$-restricted $G$-modules coincide in all three categories provided the highest weights are “close" together.