Rotational symmetries of domains and orthogonality relations

TL;DR

This paper investigates the geometric symmetries of complex domains by analyzing orthogonality relations of monomials in their Bergman spaces, providing conditions for various symmetry types including Reinhardt and circular domains.

Contribution

It introduces new sufficient conditions based on orthogonality and norms of monomials for a domain to exhibit specific rotational symmetries in complex analysis.

Findings

Conditions for domains to be Reinhardt, complete Reinhardt, circular, or Hartogs.

General criteria for invariance under linear torus group actions.

Abstract

Let $\Omega \subset \mathbb{C}^n$ be a domain whose Bergman space contains all holomorphic monomials. We derive sufficient conditions for $\Omega$ to be Reinhardt, complete Reinhardt, circular or Hartogs in terms of the orthogonality relations of the monomials with respect to their $L^2$-inner products and their $L^2$-norms. More generally, we give sufficient conditions for $\Omega$ to be invariant under a linear group action of an $r$-dimensional torus, where $r \in \{1,\ldots, n\}$.