Steinberg representations for groups of parahoric types: the special case

TL;DR

This paper introduces and analyzes a form of Steinberg representation for groups of parahoric type over finite fields, focusing on their irreducible factors in the special case related to maximal special parahoric subgroups.

Contribution

It defines Steinberg representations for parahoric type groups and determines their irreducible factors in the special case when associated with maximal special parahoric subgroups.

Findings

Identified irreducible factors of Steinberg representations under specific conditions.

Extended the understanding of representations for groups of parahoric type.

Provided explicit descriptions in the special case scenario.

Abstract

In this paper, we define and study a kind of Steinberg representation for linear algebraic groups of a particular kind, called groups of parahoric type, defined overa finite field; in particular, when G is the group of F-points of a connected reductive quasisplit group defined over F which splits over an unramified extension of F, the quotients of parahoric subgroups of G by their congruence subgroups are groups of parahoric type. In particular, under certain conditions on the residual characteristic p of F, we determine the irreducible factors of the Steinberg representation of a group of parahoric type associated to a pseudo-Borel subgroup of this group in the special case, that is when this group os a quotient of a maximal special parahoric subgroup of G.