Visible actions on symmetric spaces

Abstract

A visible action on a complex manifold is a holomorphic action that admits a $J$-transversal totally real submanifold $S$. It is said to be strongly visible if there exists an orbit-preserving anti-holomorphic diffeomorphism $σ$ such that $σ|_S = \mathrm{id}$. In this paper, we prove that for any Hermitian symmetric space $D = G/K$ the action of any symmetric subgroup $H$ is strongly visible. The proof is carried out by finding explicitly an orbit-preserving anti-holomorphic involution and a totally real submanifold $S$. Our geometric results provide a uniform proof of various multiplicity-free theorems of irreducible highest weight modules when restricted to reductive symmetric pairs, for both classical and exceptional cases, for both finite and infinite dimensional cases, and for both discrete and continuous spectra.