Homogeneous distributions on finite dimensional vector spaces

TL;DR

This paper characterizes the structure of the space of distributions on a finite dimensional vector space over a local field that are invariant under a character of the multiplicative group, within the context of representation theory.

Contribution

It provides a detailed analysis of the structure of $ ext{GL}(V)$-representations on $ ext{chi}$-invariant distributions, extending understanding of invariant distribution spaces.

Findings

Explicit description of the structure of $ ext{GL}(V)$-modules on $ ext{chi}$-invariant distributions

Identification of conditions under which these distributions are non-trivial

Connections to harmonic analysis on local fields

Abstract

Let $V$ be a finite dimensional vector space over a local field $F$. Let $\chi: F^\times \rightarrow \mathbb C^\times$ be an arbitrary character of $F^\times$. We determine the structure of the natural representation of $\mathrm{GL}(V)$ on the space $\mathcal{S}^*(V)^\chi$ of $\chi$-invariant distributions on $V$.