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Smooth representations of involutive algebra groups over non-archimedean local fields

Carlos A. M. André, João Dias

Abstract

An algebra group over a field $F$ is a group of the form $G = 1+J$ where $J$ is a finite-dimensional nilpotent associative $F$-algebra. A theorem of M. Boyarchenko asserts that, in the case where $F$ is a non-archimedean local field, every irreducible smooth representation of $G$ is admissible and smoothly induced by a one-dimensional smooth representation of some algebra subgroup of $G$. If $J$ is a nilpotent algebra endowed with an involution $σ:J\to J$, then $σ$ naturally defines a group automorphism of $G$, and we may consider the fixed point subgroup $C_{G}(σ)$. Assuming that $F$ has characteristic different from $2$, we extend Boyarchenko's result and show that every irreducible smooth representation of $C_{G}(σ)$ is admissible and smoothly induced by a one-dimensional smooth representation of a subgroup of the form $C_{H}(σ)$ where $H$ is an $σ$-invariant algebra subgroup of $G$. As a particular case, the result holds for maximal unipotent subgroups of the classical Chevalley groups defined over $F$.

Smooth representations of involutive algebra groups over non-archimedean local fields

Abstract

An algebra group over a field is a group of the form where is a finite-dimensional nilpotent associative -algebra. A theorem of M. Boyarchenko asserts that, in the case where is a non-archimedean local field, every irreducible smooth representation of is admissible and smoothly induced by a one-dimensional smooth representation of some algebra subgroup of . If is a nilpotent algebra endowed with an involution , then naturally defines a group automorphism of , and we may consider the fixed point subgroup . Assuming that has characteristic different from , we extend Boyarchenko's result and show that every irreducible smooth representation of is admissible and smoothly induced by a one-dimensional smooth representation of a subgroup of the form where is an -invariant algebra subgroup of . As a particular case, the result holds for maximal unipotent subgroups of the classical Chevalley groups defined over .
Paper Structure (3 sections, 9 theorems, 45 equations)

This paper contains 3 sections, 9 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. The inductive step
  3. Proof of Theorem

Key Result

Theorem 1

Let $F$ be a non-archimedean local field, let $G$ be an algebra group over $F$, and let $\mathcal{V}$ be an irreducible smooth $\mathbb{C}[G]$-module. Then, $\mathcal{V}$ is admissible, there exist an algebra subgroup $H$ of $G$ and a smooth character $\xi \colon H \to \mathbb{C}^{\times}$ such that

Theorems & Definitions (17)

  • Theorem 1: Boyarchenko
  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Proposition 1
  • proof
  • ...and 7 more