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Wehrl inequalities for matrix coefficients of holomorphic discrete series

Robin van Haastrecht, Genkai Zhang

TL;DR

The article extends Wehrl-type $L^2-L^{2n}$ inequalities to matrix coefficients of vector-valued holomorphic discrete series on Hermitian symmetric spaces realized as Bergman spaces. It develops a tensor-power/Cartan-component framework, identifies the sharp constants via Harish-Chandra formal degrees, and proves that reproducing kernels are precisely the maximizers. A detailed calculation of the normalization constant $c_G$ is given, including explicit formulas across classical types, with the reproducing kernel $K(z,w)$ furnishing the equality cases. An additional improved inequality is established for the unit disk, featuring a derivative-remainder term, and the work outlines open questions and potential extensions to broader Bergman spaces and quantum-channel contexts.

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.

Wehrl inequalities for matrix coefficients of holomorphic discrete series

TL;DR

The article extends Wehrl-type inequalities to matrix coefficients of vector-valued holomorphic discrete series on Hermitian symmetric spaces realized as Bergman spaces. It develops a tensor-power/Cartan-component framework, identifies the sharp constants via Harish-Chandra formal degrees, and proves that reproducing kernels are precisely the maximizers. A detailed calculation of the normalization constant is given, including explicit formulas across classical types, with the reproducing kernel furnishing the equality cases. An additional improved inequality is established for the unit disk, featuring a derivative-remainder term, and the work outlines open questions and potential extensions to broader Bergman spaces and quantum-channel contexts.

Abstract

We prove Wehrl-type inequalities for matrix coefficients of vector-valued holomorphic discrete series of , for even integers . 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.

Paper Structure

This paper contains 24 sections, 14 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Background and Main Problem
  3. Our Main Results and Methods
  4. Further Questions
  5. Organization of the paper
  6. Notation
  7. Hermitian symmetric spaces realized as bounded domains $D=G/K$
  8. Hermitian Symmetric spaces $G/K$ the Lie algebras $\mathfrak{g}$ of $G$.
  9. The Harish-Chandra factorization of $G$ in $G^{\mathbb C}=P^+K^{\mathbb C} P^-$ and the Bergman operator
  10. Holomorphic discrete series of $G$ realized as Bergman spaces of vector-valued holomorphic functions on $D$
  11. Bergman space of holomorphic functions on $D$. Invariant measure
  12. Bergman space of vector-valued holomorphic functions
  13. The formal degree of the holomorphic discrete series
  14. Definition of the formal degree
  15. Scalar holomorphic discrete series
  16. ...and 9 more sections

Key Result

Theorem 1.1

(Theorem wherlineq and Corollary wehrlineqcor) Let $n\ge 2$ be an integer, $(V_\Lambda, \tau, K)$ be an irreducible representation of $K$ with a unit highest weight vector $v_\Lambda$ and $\mathcal{H}_{\Lambda}$ the holomorphic discrete series realized as the Bergman space of $V_\Lambda$-valued holo and for $f\in \mathcal{H}_\Lambda$ and $F_f(g) :=\langle\pi(g)f, v_\Lambda\rangle$, $g\in G$. The

Theorems & Definitions (29)

  • Theorem 1.1
  • Definition 2.1
  • Definition 3.1
  • Theorem 3.2
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • ...and 19 more