Varieties attached to an $\textrm {SL}_ 2(2^ k)$-module

Abstract:

Let ${G_k} = \operatorname {SL}_2({2^k})$ and $V$ be an $F{G_k}$-module with $F$ a field containing $\operatorname {GF}(2^k)$. We show that $V$ is irreducible if and only if there is a subgroup ${U_0}$ contained in a $2$-Sylow of ${G_k}$ such that $V$ affords the regular representation of ${U_0}$. We further show how to construct a variety, defined over an algebraic closure of $\operatorname {GF}(2)$, whose $\operatorname {GF}(2^k)$-rational points parameterize those conjugacy classes of subgroups of ${G_k}$, isomorphic to ${U_0}$, that are not represented regularly on $V$.