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Non-linear Lie groups that can be realized as automorphism groups of bounded domains

George Shabat, Alexander Tumanov

TL;DR

The paper addresses which Lie groups can occur as the automorphism group ${\rm Aut}(D)$ of a bounded domain $D\subset \mathbb{C}^n$, extending prior results that realized connected linear groups on bounded strongly pseudoconvex domains. It develops a general method: if a connected Lie group $G$ acts holomorphically, properly, and freely with totally real orbits on a domain, a small $G$-invariant neighborhood yields ${\rm Aut}(D)\cong G$, and this extends to covering groups via a holomorphic map $\phi:\Omega\to \mathbb{C}^*$ to produce higher-dimensional bounded-type domains. Applying this to groups locally isomorphic to $SL(2,\mathbb{R})$, the authors give an explicit construction realizing any such connected $G$ as ${\rm Aut}(D)$ for a strongly pseudoconvex domain $D$ of bounded type in $\mathbb{C}^4$ (and provide a non-linear example). The key steps involve lifting actions to the complexification $G^c$, selecting a holomorphic $\phi$ with the correct fundamental-group effect, and ensuring a uniform lower bound near the identity to guarantee bounded-type. Together, these results broaden the class of Lie groups realizable as automorphism groups of bounded domains.

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 ${\mathbb C}^n$. In an earlier paper of 1990, we proved the result for connected linear Lie groups. In this paper we give examples of non-linear groups for which the result still holds.

Non-linear Lie groups that can be realized as automorphism groups of bounded domains

TL;DR

The paper addresses which Lie groups can occur as the automorphism group of a bounded domain , extending prior results that realized connected linear groups on bounded strongly pseudoconvex domains. It develops a general method: if a connected Lie group acts holomorphically, properly, and freely with totally real orbits on a domain, a small -invariant neighborhood yields , and this extends to covering groups via a holomorphic map to produce higher-dimensional bounded-type domains. Applying this to groups locally isomorphic to , the authors give an explicit construction realizing any such connected as for a strongly pseudoconvex domain of bounded type in (and provide a non-linear example). The key steps involve lifting actions to the complexification , selecting a holomorphic with the correct fundamental-group effect, and ensuring a uniform lower bound near the identity to guarantee bounded-type. Together, these results broaden the class of Lie groups realizable as automorphism groups of bounded domains.

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 . In an earlier paper of 1990, we proved the result for connected linear Lie groups. In this paper we give examples of non-linear groups for which the result still holds.
Paper Structure (4 sections, 4 theorems, 18 equations)

This paper contains 4 sections, 4 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. General results
  3. Proof of main result
  4. Another example

Key Result

Theorem 1

Let $G$ be a connected Lie group locally isomorphic to $SL(2,\mathbb R)$. Then there exists a strongly pseudoconvex domain $D$ of bounded type in $\mathbb C^4$ such that ${\rm Aut}(D)$ is isomorphic to $G$.

Theorems & Definitions (6)

  • Theorem 1
  • Proposition 2
  • Proposition 3
  • proof
  • Lemma 4
  • proof