Generalized matrix coefficients of Unitary Representations

TL;DR

This paper introduces a framework for generalized matrix coefficients in unitary Lie group representations, focusing on distribution vectors and their computation, filling gaps in existing literature.

Contribution

It provides a comprehensive definition and computational approach for matrix coefficients associated with distribution vectors, which was previously not well-documented.

Findings

Defines matrix coefficients for distribution vectors

Provides methods to compute these coefficients

Clarifies the theoretical foundation for generalized matrix coefficients

Abstract

This is an introductory note concerning the distribution vectors in a unitary representation of a Lie group. We discuss the definition of matrix coefficients associated with a pair of distributions and how one can compute them. Most of the results are known to some expert, but missing in the literature.