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Coherent state representations of the holomorphic automorphism group of the tube domain over the dual of the Vinberg cone

K. Arashi

Abstract

We classify all irreducible coherent state representations of the holomorphic automorphism group of the tube domain over the dual of the Vinberg cone.

Coherent state representations of the holomorphic automorphism group of the tube domain over the dual of the Vinberg cone

Abstract

We classify all irreducible coherent state representations of the holomorphic automorphism group of the tube domain over the dual of the Vinberg cone.

Paper Structure

This paper contains 8 sections, 15 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. General theory of CS representations
  3. The holomorphic automorphism group of the tube domain over the dual of the Vinberg cone
  4. CS orbits of generic CS representations
  5. Holomorphic multiplier representations over $\mathcal{D}_5$
  6. Generic CS representations
  7. Irreducible non-generic CS representations
  8. Intertwining operators

Key Result

Proposition 2.2

Suppose that $\pi$ is irreducible, and let $M\subset\mathbb{P}(\mathcal{H})$ be a CS orbit. Then the map $\mathcal{H}^*\rightarrow \Gamma^{hol}(M,L)$ given by the composition of the map $\mathcal{H}^*\rightarrow\Gamma^{hol}(\mathbb{P}(\mathcal{H}),L)$ and the restriction map $\Gamma^{hol}(\mathbb{P}

Figures (1)

  • Figure 1: The Hasse diagram of the set of all nontrivial ideals of $\mathfrak{g}$

Theorems & Definitions (23)

  • Definition 2.1
  • Proposition 2.2: Lisiecki91
  • Theorem 2.3: RV85
  • Theorem 3.1: Geatti87, IK20
  • Theorem 3.2: Geatti87
  • Theorem 4.1: encyclopedia
  • Proposition 4.2
  • Theorem 4.3
  • Theorem 5.1: Arashi20, Ishi11
  • Proposition 6.1
  • ...and 13 more