A Stone-von Neumann equivalence of categories for smooth representations of the Heisenberg group

TL;DR

This paper extends the classical Stone-von Neumann theorem to a smooth, categorical setting for the Heisenberg group, connecting representations of its center to those of the group itself, with applications to degenerate Whittaker models.

Contribution

It establishes a smooth categorical equivalence for representations of the Heisenberg group, generalizing classical results to non-unitary, non-irreducible cases.

Findings

Established a smooth equivalence of categories for Heisenberg group representations.

Extended the oscillator representation to the smooth setting.

Applied results to degenerate Whittaker models for reductive groups.

Abstract

The classical Stone-von Neuman theorem relates the irreducible unitary representations of the Heisenberg group $H_n$ to non-trivial unitary characters of its center $Z$, and plays a crucial role in the construction of the oscillator representation for the metaplectic group. In this paper we extend these ideas to non-unitary and non-irreducible representations, thereby obtaining an equivalence of categories between certain representations of $Z$ and those of $H_n$. Our main result is a smooth equivalence, which involves the fundamental ideas of du Cloux on differentiable representations and smooth imprimitivity systems for Nash groups. We show how to extend the oscillator representation to the smooth setting and give an application to degenerate Whittaker models for representations of reductive groups. We also include an algebraic equivalence, which can be regarded as a generalization of Kashiwara's lemma from the theory of $D$-modules.