Wehrl inequalities for matrix coefficients of holomorphic discrete series

TL;DR

This paper establishes Wehrl-type inequalities for matrix coefficients of holomorphic discrete series representations of a group, identifying optimal constants and maximizers, thus advancing understanding of harmonic analysis on these representations.

Contribution

It introduces Wehrl inequalities for matrix coefficients of holomorphic discrete series and characterizes the maximizers as reproducing kernels, with constants linked to Harish-Chandra degrees.

Findings

Proved Wehrl inequalities for matrix coefficients in holomorphic discrete series.

Identified reproducing kernels as the unique maximizers.

Expressed optimal constants via Harish-Chandra formal degrees.

Abstract

We prove Wehrl-type $L^2(G)-L^{p}(G)$ inequalities for matrix coefficients of vector-valued holomorphic discrete series of $G$, for even integers $p=2n$. The optimal constant is expressed in terms of Harish-Chandra formal degrees for the discrete series. We prove the maximizers are precisely the reproducing kernels.