Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Commutativity of invariant differential operators on vector bundles on Hermitian symmetric spaces

Robin van Haastrecht, Genkai Zhang, Yufeng Zhao

TL;DR

This work classifies when the ring of G-invariant differential operators on homogeneous vector bundles over Hermitian symmetric spaces G/K is commutative by identifying all K-representations τ for which τ|_M is multiplicity-free. It achieves this through a case-by-case analysis of the relevant symmetric pairs, leveraging branching rules, Harish-Chandra theory, and the Szegö transform to realize finite-dimensional G-representations inside induced models and to construct eigenfunctions. The paper then proves a robust invariance property of eigenvalues under the Weyl group and provides a canonical factorization of embeddings via the Poisson-Szegő transform, yielding a clear multiplicity-one behavior for embeddings and a polynomial extension of eigenvalues in the restricted parameter space. These results deepen the understanding of invariant differential operators on G/K, connect representation-theoretic multiplicity phenomena to analytic eigenfunctions, and inform discrete-series characterizations in L^2(G/K,V_τ).

Abstract

Let $G/K$ be a Hermitian symmetric space and $V_τ$ an irreducible representation of $K$. We study the ring $\mathcal D^G(G/K, V_τ)$ of $G$-invariant differential operators on sections of vector bundles $G\times_{(K, τ)} V_τ$ over $G/K$ defined by a finite-dimensional representation $(V_τ, τ)$ of $K$. We classify irreducible representations $(V_τ, τ)$ such that $\mathcal D^G(G/K, V_τ)$ is commutative. We construct eigenfunctions for the differential operators and study the invariance property of the eigenvalues under the Weyl group for the restricted real root system of $G$.

Commutativity of invariant differential operators on vector bundles on Hermitian symmetric spaces

TL;DR

This work classifies when the ring of G-invariant differential operators on homogeneous vector bundles over Hermitian symmetric spaces G/K is commutative by identifying all K-representations τ for which τ|_M is multiplicity-free. It achieves this through a case-by-case analysis of the relevant symmetric pairs, leveraging branching rules, Harish-Chandra theory, and the Szegö transform to realize finite-dimensional G-representations inside induced models and to construct eigenfunctions. The paper then proves a robust invariance property of eigenvalues under the Weyl group and provides a canonical factorization of embeddings via the Poisson-Szegő transform, yielding a clear multiplicity-one behavior for embeddings and a polynomial extension of eigenvalues in the restricted parameter space. These results deepen the understanding of invariant differential operators on G/K, connect representation-theoretic multiplicity phenomena to analytic eigenfunctions, and inform discrete-series characterizations in L^2(G/K,V_τ).

Abstract

Let be a Hermitian symmetric space and an irreducible representation of . We study the ring of -invariant differential operators on sections of vector bundles over defined by a finite-dimensional representation of . We classify irreducible representations such that is commutative. We construct eigenfunctions for the differential operators and study the invariance property of the eigenvalues under the Weyl group for the restricted real root system of .
Paper Structure (20 sections, 9 theorems, 109 equations)

This paper contains 20 sections, 9 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Main results and methods
  3. Related results and questions
  4. List of notation and symbols
  5. Preliminaries
  6. Hermitian symmetric spaces
  7. Cayley transform between Cartan subalgebras
  8. Induced representations from the minimal parabolic subgroup $MAN$
  9. Realization of irreducible finite-dimensional representations $(W_\lambda, G^{\mathbb{C}})$ in the induced representation $( I_{\sigma, \nu}^0, \mathfrak g^{\mathbb{C}})$.
  10. The algebra of invariant differential operators on homogeneous vector bundles
  11. Commutativity of $\mathcal{D}^G(G/K,V_{\tau})$ and the corresponding classification of $(K, \tau)$
  12. $(G,K) = (SU(r+b,r), S(U(r+b) \times U(r))$
  13. $(G, K)=(Sp(n,\mathbb{R}), U(n))$, $(Mp(n, \mathbb{R}), \widehat{U(n)})$
  14. $(G,K) = (SO^*(2n), U(n))$
  15. $(G, K) = (SO_{0}(n,2), SO(n)\times SO(2))$, $(Spin_0(n,2), Spin(n)\times SO(2)/\{(1,1), (-1,-1)\})$
  16. ...and 5 more sections

Key Result

Theorem 1.1

(Theorem KresThm, Proposition finiterepprop, Theorem egvaluethm) (1) The representation $\tau|_M$ is multiplicity-free if and only if $(K, \tau)$ is in the list in Theorem KresThm. (2) Let $(K, \tau)$ be as in (1) acting on $V_\tau$. For any finite-dimensional irreducible representation $(W_\lambda,

Theorems & Definitions (20)

  • Theorem 1.1
  • Proposition 2.1
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • proof
  • ...and 10 more