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Coherent state representations of the holomorphic automorphism group of a quasi-symmetric Siegel domain of type III

Koichi Arashi

TL;DR

The paper develops a framework to classify generic coherent state (CS) representations of the universal cover $G$ of the holomorphic automorphism group of a quasi-symmetric Siegel domain of type III$_{ν,r}$ (with $ν≥3$, $r≥2$). It combines Lie-algebraic structure via Satake and DN theory with holomorphic realization on CS orbits $M\cong G/K$ and $G$-equivariant holomorphic vector bundles, yielding explicit constructions of unitarizable CS representations in terms of central characters $(c_1,c_2)$, holomorphic data $(\Theta_1,\Theta_2)$, and a maximal triangular subgroup $B$. A key outcome is that unitarizability and equivalence of CS representations are governed by the restriction of a one-parameter character $\tau_{c_1}$ to $B$ and by discrete parameters $(c_2,\Theta_2)$, reducing classification to a finite-dimensional accessory data. The results extend the coherent-state paradigm to a broad class of non-symmetric Siegel domains, linking Kähler algebra structure, holomorphic vector bundles, and generalized Iwasawa decompositions in a cohesive classification scheme.

Abstract

The aim of this paper is to study the classification of generic coherent state representations of a Lie group and its connection to the structure theory of Kähler algebras. The group we deal with is the holomorphic automorphism group of a quasi-symmetric Siegel domain of type III. A method using the generalized Iwasawa decomposition is provided.

Coherent state representations of the holomorphic automorphism group of a quasi-symmetric Siegel domain of type III

TL;DR

The paper develops a framework to classify generic coherent state (CS) representations of the universal cover of the holomorphic automorphism group of a quasi-symmetric Siegel domain of type III (with , ). It combines Lie-algebraic structure via Satake and DN theory with holomorphic realization on CS orbits and -equivariant holomorphic vector bundles, yielding explicit constructions of unitarizable CS representations in terms of central characters , holomorphic data , and a maximal triangular subgroup . A key outcome is that unitarizability and equivalence of CS representations are governed by the restriction of a one-parameter character to and by discrete parameters , reducing classification to a finite-dimensional accessory data. The results extend the coherent-state paradigm to a broad class of non-symmetric Siegel domains, linking Kähler algebra structure, holomorphic vector bundles, and generalized Iwasawa decompositions in a cohesive classification scheme.

Abstract

The aim of this paper is to study the classification of generic coherent state representations of a Lie group and its connection to the structure theory of Kähler algebras. The group we deal with is the holomorphic automorphism group of a quasi-symmetric Siegel domain of type III. A method using the generalized Iwasawa decomposition is provided.
Paper Structure (6 sections, 8 theorems, 15 equations)

This paper contains 6 sections, 8 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Siegel domain and holomorphic complete vector fields
  3. Realization of a CS representation
  4. Algebraic structures of a CS orbit
  5. Holomorphic vector bundles
  6. Equivalence classes of CS representations

Key Result

proposition 1

The Lie algebra $\mathfrak{g}$ is isomorphic to the linear Lie algebra $\widetilde{\mathfrak{g}}$ consists of all matrices $\iota(T,K,V,U)$.

Theorems & Definitions (15)

  • proposition 1
  • proof
  • remark 1
  • lemma 1
  • proof
  • proposition 2
  • proof
  • proposition 3
  • proof
  • proposition 4
  • ...and 5 more