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.1991 Mathematics Subject Classification. Primary: 47B35, 22E46, Secondary: 32A36, 32M15, 22E45.Key words and phrases. Toeplitz operator, Bergman space, holomorphic discrete series, multiplicity-free representation.M. Dawson would like to thank the VIGRE program at LSU, DMS-0739382, for support. He would also like to thank CIMAT for its hospitality and support during a visit in September 2013.

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On the unit ball B n we consider the weighted Bergman spaces Centro de Investigación en Matemáticas, Jalisco s/n,

We extend the definition of conical representations for Riemannian symmetric spaces to a certain class of infinite-dimensional Riemannian symmetric spaces. Using an infinite-dimensional version of Weyl's Unitary Trick, there is a correspondence between smooth representations of infinite-dimensional noncompact-type Riemannian symmetric spaces and smooth representations of infinite-dimensional compact-type symmetric spaces. We classify all smooth conical representations which are unitary on the compacttype side. Finally, a new class of non-smooth unitary conical representations appears on the compact-type side which has no analogue in the finite-dimensional case. We classify these representations and show how to decompose them into direct integrals of irreducible conical representations.2000 Mathematics Subject Classification. 43A85, 53C35, 22E46.

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