On representations of real Jacobi groups

TL;DR

This paper explores the representation theory of real Jacobi groups, establishing equivalences with real reductive groups, and introduces tools like matrix coefficients and criteria for multiplicity one theorems.

Contribution

It demonstrates the equivalence of categories of representations and develops new methods for analyzing Jacobi group representations using reductive group theory.

Findings

Categories of Hilbert and Frechet representations are equivalent to those of a reductive group.

Defined matrix coefficients for distributional vectors of Jacobi group representations.

Formulated Gelfand-Kazhdan criteria for Jacobi groups to aid in proving multiplicity one theorems.

Abstract

We consider a category of continuous Hilbert space representations and a category of smooth Frechet representations, of a real Jacobi group $G$. By Mackey's theory, they are respectively equivalent to certain categories of representations of a real reductive group $\widetilde L$. Within these categories, we show that the two functors of taking smooth vectors for $G$, and for $\widetilde L$, are consistent with each other. By using Casselman-Wallach's theory of smooth representations of real reductive groups, we define matrix coefficients for distributional vectors of certain representations of $G$. We also formulate Gelfand-Kazhdan criteria for Jacobi groups which could be used to prove the multiplicity one theorem for Fourier-Jacobi models.