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

We study direct limits $(G,K) = \varinjlim (G_n,K_n)$ of compact Gelfand pairs. First, we develop a criterion for a direct limit representation to be a multiplicity--free discrete direct sum of irreducible representations. Then we look at direct limits $G/K = \varinjlim G_n/K_n$ of compact riemannian symmetric spaces, where we combine our criterion with the Cartan--Helgason Theorem to show in general that the regular representation of $G = \varinjlim G_n$ on a certain function space $\varinjlim L^2(G_n/K_n)$ 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 $\cA(G/K)$, $\cC(G/K)$ and $L^2(G/K)$ to which our multiplicity--free criterion applies.