Functorial properties of generalised Steinberg representations

TL;DR

This paper investigates the functorial properties of generalized Steinberg representations for reductive groups over non-archimedean local fields, focusing on how these representations behave under certain functors.

Contribution

It provides a detailed study of the functor $ ext{St}_Q^G$ acting on smooth representations, extending understanding of their structure and properties in the context of reductive groups over local fields.

Findings

Characterization of the functor $ ext{St}_Q^G$ on smooth $R$-representations.

Analysis of the extension of representations from Levi subgroups to the whole group.

Insights into the tensor product structure with generalized Steinberg representations.

Abstract

Let $G$ be the $F$-points of a connected reductive group over a non-archimedean local field $F$ of residue characteristic $p$ and $R$ be a commutative ring. Let $P=LU$ be a parabolic subgroup of $G$ and $Q$ be a parabolic subgroup of $G$ containing $P$. We study the functor $\mathrm{St}_Q^G$ taking a smooth $R$-representation $\sigma$ of $L$ which extends to a representation $\mathrm{e}_G(\sigma)$ of $G$ trivial on $U$ to the smooth $R$-representation $\mathrm{e}_G(\sigma) \otimes_R \mathrm{St}_Q^G(R)$ of $G$ where $\mathrm{St}_Q^G(R)$ is the generalised Steinberg representation.