Theta correspondence via group C*-algebras

TL;DR

This paper interprets the local and global theta correspondence using group C*-algebras, revealing new insights into their structure and connections with Rallis inner product formula, and showing that in key cases, the correspondence arises from a continuous functor.

Contribution

It provides a C*-algebraic interpretation of the theta correspondence, linking it to continuous functors and offering a new perspective on Rallis' formula.

Findings

Theta correspondence can be understood via group C*-algebras.

In key cases, the correspondence arises from a continuous functor.

Rallis inner product formula corresponds to an isometry in this framework.

Abstract

We prove that the well-known explicit construction of the local theta correspondence by Li has a simple interpretation in terms of group C*-algebras. In particular, we deduce that in two standard cases where Li's method work, local theta correspondence arises from a continuous functor. Moreover, using results from a companion paper, we treat global theta correspondence using C*-algebraic methods. As a byproduct, we exhibit that Rallis inner product formula can be interpreted as a certain natural inclusion being an isometry.