Rotational symmetries of domains and orthogonality relations

Abstract

Let $\Omega \subset C^n$ be a domain whose Bergman space contains all holomorphic monomials. We derive sufficient conditions for Ω 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 Ω to be invariant under a linear group action of an r-dimensional torus, where $r\in \{1,\dots ,n\}$.