Irreducibility of Infinite Dimensional Steinberg Modules of Reductive Groups with Frobenius Maps

TL;DR

This paper proves the irreducibility of infinite dimensional Steinberg modules for certain reductive groups over algebraically closed fields of positive characteristic, and explores their quasi-finiteness properties in specific cases.

Contribution

It establishes the irreducibility of infinite dimensional Steinberg modules for reductive groups with Frobenius maps, extending previous understanding of their structure.

Findings

Infinite dimensional Steinberg modules are irreducible under specified conditions.

Certain Steinberg modules are not quasi-finite for some linear groups.

Results apply to groups over fields with characteristic different from the defining field.

Abstract

Let G be a connected reductive group over an algebraic closure of a finite field Fq. In this paper it is proved that the infinite dimensional Steinberg module of kG defined by N. Xi in 2014 is irreducible when k is a field of positive characteristic and char k is not char Fq. For certain special linear groups, we show that the Steinberg modules of the groups are not quasi-finite with respect to some natural quasi-finite sequences of the groups.