Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multiplicity-free representations and coisotropic actions of certain nilpotent Lie groups over quasi-symmetric Siegel domains

Koichi Arashi

TL;DR

This work characterizes when nilpotent Lie groups act multiplicity-freely on Bergman spaces over quasi-symmetric Siegel domains by linking representation-theoretic multiplicity to geometric coisotropicity. It blends the orbit method for nilpotent groups, Jordan-algebra spectral theory, and invariant reproducing-kernel analysis to prove an equivalence between four conditions: (i) ext(\Gamma_{G^W})/\mathbb{R}^× embedding into the unitary dual, (ii) multiplicity-free unitary action on L_a^2(\mathcal{S}(\Omega,Q)), (iii) the vanishing Im Q on a subspace S, and (iv) coisotropic G^W-orbits in the Bergman-symplectic structure. The paper provides an explicit admissible parametrization of extremal G^W-invariant kernels, constructs intertwining operators via holomorphic induction, and derives a multiplicity-free direct-integral decomposition of the Bergman space in the W = V case, thereby clarifying when analytic and geometric criteria align. These results advance understanding of multiplicity-freeness in Siegel-domain settings, offering tools for harmonic analysis on quasi-symmetric domains and potential extensions to visible/coisotropic actions in nilpotent contexts.

Abstract

We study multiplicity-free representations of Lie groups over a quasi-symmetric Siegel domain, with a focus on certain two-step nilpotent Lie groups. We provide necessary and sufficient conditions for the multiplicity-freeness property to hold. Specifically, we establish the equivalence between the disjointness of irreducible unitary representations realized over the domain, the multiplicity-freeness of the unitary representation on the Bergman space, and the coisotropicity of the group action.

Multiplicity-free representations and coisotropic actions of certain nilpotent Lie groups over quasi-symmetric Siegel domains

TL;DR

This work characterizes when nilpotent Lie groups act multiplicity-freely on Bergman spaces over quasi-symmetric Siegel domains by linking representation-theoretic multiplicity to geometric coisotropicity. It blends the orbit method for nilpotent groups, Jordan-algebra spectral theory, and invariant reproducing-kernel analysis to prove an equivalence between four conditions: (i) ext(\Gamma_{G^W})/\mathbb{R}^× embedding into the unitary dual, (ii) multiplicity-free unitary action on L_a^2(\mathcal{S}(\Omega,Q)), (iii) the vanishing Im Q on a subspace S, and (iv) coisotropic G^W-orbits in the Bergman-symplectic structure. The paper provides an explicit admissible parametrization of extremal G^W-invariant kernels, constructs intertwining operators via holomorphic induction, and derives a multiplicity-free direct-integral decomposition of the Bergman space in the W = V case, thereby clarifying when analytic and geometric criteria align. These results advance understanding of multiplicity-freeness in Siegel-domain settings, offering tools for harmonic analysis on quasi-symmetric domains and potential extensions to visible/coisotropic actions in nilpotent contexts.

Abstract

We study multiplicity-free representations of Lie groups over a quasi-symmetric Siegel domain, with a focus on certain two-step nilpotent Lie groups. We provide necessary and sufficient conditions for the multiplicity-freeness property to hold. Specifically, we establish the equivalence between the disjointness of irreducible unitary representations realized over the domain, the multiplicity-freeness of the unitary representation on the Bergman space, and the coisotropicity of the group action.
Paper Structure (12 sections, 42 theorems, 220 equations)

This paper contains 12 sections, 42 theorems, 220 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Invariant Hilbert subspaces and the multiplicity-freeness property
  4. Orbit method
  5. Self-dual homogeneous cone
  6. Quasi-symmetric Siegel domain
  7. Multiplicity-freeness and the vanishing condition for Im Q
  8. Description of extremal invariant reproducing kernels
  9. Construction of intertwining operators
  10. Specific form of an admissible parametrization
  11. Multiplicity-free unitary representation on the Bergman space
  12. Coisotropic action and the multiplicity-freeness property

Key Result

Theorem 1.1

For a real subspace $W\subset V$, the following conditions are equivalent:

Theorems & Definitions (84)

  • Theorem 1.1: see Corollary \ref{['cor:112']} and Theorems \ref{['th:220']}, \ref{['th:5.11']}, \ref{['th:5.14']}
  • Corollary 1.1
  • Theorem 1.2: see Corollary \ref{['cor:115']} and Sect. \ref{['subsect:specificform']}
  • Definition 2.1
  • Definition 2.2: schwartz_sousespaces_1964, kobayashi_propagation_2013a
  • Theorem 2.1: schwartz_sousespaces_1964,faraut_invariant_1999
  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4: corwin_spectrum_1988
  • ...and 74 more