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Holomorphic multiplier representations for bounded homogeneous domains

K. Arashi

Abstract

In this paper, we study the unitarizations in the spaces of holomorphic sections of equivariant holomorphic line bundles over a bounded homogeneous domain under the action of a connected algebraic group acting transitively on the domain. We give a complete classification of unitary representations arising from such unitarizations. As an application, we classify all such unitary representations for a specific five-dimensional non-symmetric bounded homogeneous domain.

Holomorphic multiplier representations for bounded homogeneous domains

Abstract

In this paper, we study the unitarizations in the spaces of holomorphic sections of equivariant holomorphic line bundles over a bounded homogeneous domain under the action of a connected algebraic group acting transitively on the domain. We give a complete classification of unitary representations arising from such unitarizations. As an application, we classify all such unitary representations for a specific five-dimensional non-symmetric bounded homogeneous domain.

Paper Structure

This paper contains 16 sections, 54 theorems, 318 equations.

Table of Contents

  1. Introduction
  2. Existence of unitarizations
  3. General theory of holomorphic multiplier representations
  4. The case of bounded homogeneous domains
  5. Normal $j$-algebras and bounded homogeneous domains
  6. Algebraic properties of $\mathfrak{g}$
  7. Holomorphic complete vector fields on Siegel domains
  8. Relationship between $\mathfrak{X}(1/2)$ and $\mathfrak{X}(1)$
  9. Vector fields on homogeneous Siegel domains
  10. Unitary equivalences among the unitarizable representations
  11. Unitary equivalences among representations of $B$
  12. Unitary equivalences among representations of $G$
  13. Isotropy representation
  14. Actions of the isotropy subgroup on holomorphic vector bundles
  15. Unitary equivalences and the actions of the isotropy subgroup
  16. ...and 1 more sections

Key Result

Theorem 1.2

A Hilbert space giving a unitarization of $l$ is unique if it exists. In particular, the unitarization is irreducible.

Theorems & Definitions (98)

  • Definition 1.1
  • Theorem 1.2: ishi 2011, kobayashi, kunze
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5: Ishi, ishi 2011
  • Theorem 1.6: Ishi, ishi 2013
  • Theorem 1.7: see Theorem \ref{['LetmGtimes']}
  • Theorem 1.8: see Theorem \ref{['main']}
  • Theorem 1.9: see Corollary \ref{['LetEandEbe']}
  • Definition 2.1
  • ...and 88 more