For any given bounded symmetric domain, we prove the existence of commutative
C∗-algebras generated by Toeplitz operators acting on any weighted Bergman
space. The symbols of the Toeplitz operators that generate such algebras are
defined by essentially bounded functions invariant under suitable subgroups of
the group of biholomorphisms of the domain. These subgroups include the maximal
compact groups of biholomorphisms. We prove the commutativity of the Toeplitz
operators by considering the Bergman spaces as the underlying space of the
holomorphic discrete series and then applying known multiplicity-free results
for restrictions to certain subgroups of the holomorphic discrete series. In
the compact case we completely characterize the subgroups that define invariant
symbols that yield commuting Toeplitz operators in terms of the
multiplicity-free property
