Asymptotic Schur orthogonality relations for Heisenberg groups over local fields

TL;DR

This paper establishes asymptotic Schur orthogonality relations for irreducible unitary representations of Heisenberg groups over local fields, extending classical results to a broader class of groups using a limit-based approach.

Contribution

It introduces a framework for asymptotic orthogonality relations for non-discrete groups, specifically Heisenberg groups over local fields, utilizing c-temperedness and convergence to braiding operators.

Findings

Orthogonality relations hold for Heisenberg groups over local fields.

The framework extends classical orthogonality to non-discrete groups.

Convergence to braiding operators is demonstrated.

Abstract

Asymptotic Schur orthogonality relations are for irreducible unitary representations of locally compact groups that need not be discrete series, where $L^2$ pairing of matrix coefficients with respect to Haar measure is replaced by a limit of that with respect to a sequence of bounded measures. We show that such relations hold for Heisenberg groups over local fields. This is achieved in the framework of c-temperedness introduced by Kazhdan and Yom Din. The related condition of convergence to braiding operator is also shown.