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

TL;DR

This paper investigates the structure of direct limits of compact Gelfand pairs and symmetric spaces, establishing criteria for multiplicity-free representations and introducing new methods for non-symmetric cases.

Contribution

It develops a criterion for multiplicity-free direct limit representations and introduces two novel methods for non-symmetric Gelfand pairs, expanding understanding of infinite-dimensional homogeneous spaces.

Findings

Regular representation on certain function spaces is multiplicity free for symmetric space limits.

New methods based on parabolic limits and branching rules are effective for non-symmetric cases.

Defined function spaces where multiplicity-free criteria can be applied.

Abstract

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.