The Permutation Module on Flag Varieties in Cross Characteristic

Abstract

Let $\mathbf{G}$ be a connected reductive group over ${\bar{\mathbb{F}}}_q$, the algebraically closure of ${\mathbb{F}}_q$ (the finite field with $q=p^e$ elements), with the standard Frobenius map $F$. Let $\mathbf{B}$ be an $F$-stable Borel subgroup. Let $\mathbb{k}$ be a field of characteristic $r\ne p$. In this paper, we completely determine the composition factors of the induced module $In{d}_B^{G}tr=\mathbb{k}G{\otimes }_{\mathbb{k}\mathbf{B}}$ tr (here $\mathbb{k}H$ is the group algebra of the group $H$, and tr is the trivial $B$-module). In particular, we find a new family of infinite dimensional irreducible abstract representations of $G$.

Accepted by Mathematische Zeitschrift