The permutation module on flag varieties in cross characteristic

Abstract

Let $\mathbf{G}$ be a connected reductive group defined over $\mathbb {F}_q$ (the finite field with q elements, where q is a power of the prime number p), with the standard Frobenius map F. Let $\mathbf{B}$ be an F-stable Borel subgroup. Let $\Bbbk$ be a field (may not be algebraically closed) of characteristic $0\le r\ne p$. In this paper, we completely determine the composition factors of the permutation module module $\mathbb {M}(\text{ tr })=\Bbbk \mathbf{G}\otimes _{\Bbbk \mathbf{B}}\text{ tr }$ (here $\Bbbk \mathbf{H}$ is the group algebra of the group $\mathbf{H}$, and $\text{ tr }$ is the trivial $\mathbf{B}$-module). In particular, we find a large family of infinite dimensional absolutely irreducible abstract representations of $\mathbf{G}$ over $\Bbbk$.