Infinite-Dimensional Multiplicity-Free Spaces I: Limits of Compact Commutative Spaces

We study direct limits (G,K) = lim −→ (Gn,Kn) of compact Gelfand pairs. First, we develop a criterion for a direct limit representation to be a multiplicityfree discrete direct sum of irreducible representations. Then we look at direct limits G/K = lim −→ Gn/Kn of compact riemannian symmetric spaces, where we combine our criterion with the Cartan–Helgason theorem to show in general that the regular representation of G = lim −→n on a certain function space lim −→ (Gn/Kn) is multiplicity-free. That method is not applicable for direct limits of nonsymmetric Gelfand pairs, so we introduce two other methods. The first, based on “parabolic direct limits” and “defining representations”, extends the method used in the symmetric space case. The second uses some (new) branching rules from finite-dimensional representation theory. In both cases we define function spaces A(G/K), C(G/K), and L(G/K) to which our multiplicity-free criterion applies.