The Permutation Module on Flag Varieties in Cross Characteristic

TL;DR

This paper determines the composition factors of a specific induced module for a reductive group over an algebraic closure of a finite field, revealing new infinite-dimensional irreducible representations in cross characteristic.

Contribution

It completely classifies the composition factors of the induced module and introduces a new family of infinite-dimensional irreducible representations.

Findings

Explicit composition factors of the induced module are identified.

A new family of infinite-dimensional irreducible representations is constructed.

The results extend understanding of modular representations in cross characteristic.

Abstract

Let ${\bf G}$ 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 ${\bf B}$ be an $F$-stable Borel subgroup. Let $\Bbbk$ be a field of characteristic $r\neq p$. In this paper, we completely determine the composition factors of the induced module $Ind_{B}^{G}{tr}=\Bbbk{G}\otimes_{\Bbbk{\bf B}}$ tr (here $\Bbbk{H}$ 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 $G$.