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Realizing semisimple Lie groups as holomorphic automorphism groups of bounded domains

George Shabat, Alexander Tumanov

TL;DR

The paper addresses realizing Lie groups as the full holomorphic automorphism group of a bounded domain. It develops a framework that combines holomorphic group actions on domains of bounded type, admissible covering maps, and polar decomposition to lift realizations from linear, compact, and semisimple groups to their covering groups. The main result shows that a real connected linear self-adjoint group $G$ admits a bounded-type domain $D$ with $Aut(D)\simeq\widehat{G}$ for a suitable covering $\widehat{G}$, and, in particular, every connected real semisimple group can be realized (up to covering) as $Aut(D)$. The approach hinges on constructing a holomorphic map from a neighborhood of $G^0$ to a compact subgroup, ensuring the pullback of a covering yields the desired automorphism group, with detailed analysis of matrices near the real locus via polar decomposition and square roots of positive-definite perturbations.

Abstract

We consider a problem whether a given Lie group can be realized as the group of all biholomorphic automorphisms of a bounded domain in the affine complex space. In an earlier paper of 1990, we proved the result for connected linear Lie groups. In this paper we prove the result for a fairly large class of Lie groups including all connected real semisimple groups. This class contains many non-linear Lie groups.

Realizing semisimple Lie groups as holomorphic automorphism groups of bounded domains

TL;DR

The paper addresses realizing Lie groups as the full holomorphic automorphism group of a bounded domain. It develops a framework that combines holomorphic group actions on domains of bounded type, admissible covering maps, and polar decomposition to lift realizations from linear, compact, and semisimple groups to their covering groups. The main result shows that a real connected linear self-adjoint group admits a bounded-type domain with for a suitable covering , and, in particular, every connected real semisimple group can be realized (up to covering) as . The approach hinges on constructing a holomorphic map from a neighborhood of to a compact subgroup, ensuring the pullback of a covering yields the desired automorphism group, with detailed analysis of matrices near the real locus via polar decomposition and square roots of positive-definite perturbations.

Abstract

We consider a problem whether a given Lie group can be realized as the group of all biholomorphic automorphisms of a bounded domain in the affine complex space. In an earlier paper of 1990, we proved the result for connected linear Lie groups. In this paper we prove the result for a fairly large class of Lie groups including all connected real semisimple groups. This class contains many non-linear Lie groups.

Paper Structure

This paper contains 8 sections, 17 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. General results
  3. Action of a linear Lie group on a domain of bounded type
  4. Covering groups of compact groups
  5. Polar decomposition and main results
  6. Complex numbers, vectors, and matrices close to real ones
  7. Square roots of complex matrices close to positive definite ones
  8. Complex orthogonal matrices close to real ones

Key Result

Proposition 2.1

BedfordSaerensShabatWinkelmann Let $G:\Omega$ be a holomorphic group action of a connected Lie group $G$ on a domain $\Omega\subset\mathbb C^n$, $n\ge 2$. Suppose the action is proper, free, and the orbits are totally real. Then a generic smooth small tubular $G$-invariant neighborhood $D$ of each o

Theorems & Definitions (34)

  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Proposition 4.1
  • Proposition 4.2
  • Lemma 4.3
  • Lemma 4.4
  • ...and 24 more