Berezin symbols and spectral measures of representation operators

TL;DR

This paper investigates spectral measures of representation operators using Berezin quantization, demonstrating how contraction limits of Lie group representations influence spectral measure convergence, with examples involving $SU(1,1)$, $SU(2)$, and the Heisenberg group.

Contribution

It introduces a novel approach linking Berezin quantization with spectral measures and contraction limits in Lie group representations.

Findings

Spectral measures are analyzed via Berezin symbols.

Contraction of Lie group representations affects spectral measure convergence.

Examples include contractions from $SU(1,1)$ and $SU(2)$ to the Heisenberg group.

Abstract

Let $G$ be a Lie group with Lie algebra $\mathfrak g$ and let $\pi$ be a unitary representation of $G$ realized on a reproducing kernel Hilbert space. We use Berezin quantization to study spectral measures associated with operators $-id\pi(X)$ for $X\in {\mathfrak g}$. As an application, we show how results about contractions of Lie group representations give rise to results on convergence of sequences of spectral measures. We give some examples including contractions of $SU(1,1)$ and $SU(2)$ to the Heisenberg group.