An inductive approach to representations of general linear groups over compact discrete valuation rings

TL;DR

This paper develops a new algebraic framework to study representations of general linear groups over compact discrete valuation rings, extending Zelevinsky's PSH-algebras from finite fields.

Contribution

It constructs a bialgebra extending Zelevinsky's PSH-algebra to general linear groups over compact discrete valuation rings and proves base change results.

Findings

Constructed an analogous bialgebra for these groups

Extended Zelevinsky's PSH-algebra to a broader setting

Proved base change theorems relating different rings

Abstract

In his seminal Lecture Notes in Mathematics published in 1981, Andrey Zelevinsky introduced a new family of Hopf algebras which he called {\em PSH-algebras}. These algebras were designed to capture the representation theory of the symmetric groups and of classical groups over finite fields. The gist of this construction is to translate representation-theoretic operations such as induction and restriction and their parabolic variants to algebra and coalgebra operations such as multiplication and comultiplication. The Mackey formula, for example, is then reincarnated as the Hopf axiom on the algebra side. In this paper we take substantial steps to adapt these ideas for general linear groups over compact discrete valuation rings. We construct an analogous bialgebra that contains a large PSH-algebra that extends Zelevinsky's algebra for the case of general linear groups over finite fields. We prove several base change results relating algebras over extensions of discrete valuation rings.