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

Tyrone Crisp, Ehud Meir, Uri Onn

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.

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

Paper Structure

This paper contains 33 sections, 40 theorems, 186 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview
  3. Main results
  4. Related and future work
  5. Organisation of the paper
  6. Notation
  7. Preliminaries
  8. A variant of Harish-Chandra functors.
  9. Compatibility with Clifford theory and natural operations
  10. Setup for general linear groups
  11. Pontryagin duals
  12. Some linear algebra
  13. Linear algebra over fields
  14. Linear algebra over discrete valuation rings
  15. One ring to rule them all
  16. ...and 18 more sections

Key Result

Theorem 1

The multiplication $\circ$ induces an isomorphism of algebras and coalgebras This isomorphism gives a one-to-one correspondence between tensor products of irreducible elements on the left and irreducible elements in $\mathop{\mathrm{\mathcal{R}}}\nolimits^{\mathfrak o,\ell}$ on the right.

Figures (1)

  • Figure 1: Branching graph $\Gamma$

Theorems & Definitions (79)

  • Conjecture 1.1
  • Theorem 1: Primary decomposition
  • Corollary 1.2
  • Theorem 2: Base change
  • Corollary 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 3: Local PSH
  • Theorem 4: Global PSH
  • Remark 1.6
  • ...and 69 more