Infinite Dimensional Multiplicity Free Spaces III: Matrix Coefficients and Regular Functions

In earlier papers we studied direct limits $(G,K) = \varinjlim (G_n,K_n)$ of two types of Gelfand pairs. The first type was that in which the $G_n/K_n$ are compact Riemannian symmetric spaces. The second type was that in which $G_n = N_n\rtimes K_n$ with $N_n$ nilpotent, in other words pairs $(G_n,K_n)$ for which $G_n/K_n$ is a commutative nilmanifold. In each we worked out a method inspired by the Frobenius--Schur Orthogonality Relations to define isometric injections $\zeta_{m,n}: L^2(G_n/K_n) \hookrightarrow L^2(G_m/K_m)$ for $m \geqq n$ and prove that the left regular representation of $G$ on the Hilbert space direct limit $L^2(G/K) := \varinjlim L^2(G_n/K_n)$ is multiplicity--free. This left open questions concerning the nature of the elements of $L^2(G/K)$. Here we define spaces $\cA(G_n/K_n)$ of regular functions on $G_n/K_n$ and injections $\nu_{m,n} : \cA(G_n/K_n) \to \cA(G_m/K_m)$ for $m \geqq n$ related to restriction by $\nu_{m,n}(f)|_{G_n/K_n} = f$. Thus the direct limit $\cA(G/K):= \varinjlim \{\cA(G_n/K_n), \nu_{m,n}\}$ sits as a particular $G$--submodule of the much larger inverse limit $\varprojlim \{\cA(G_n/K_n), \text{restriction}\}$. Further, we define a pre Hilbert space structure on $\cA(G/K)$ derived from that of $L^2(G/K)$. This allows an interpretation of $L^2(G/K)$ as the Hilbert space completion of the concretely defined function space $\cA(G/K)$, and also defines a $G$--invariant inner product on $\cA(G/K)$ for which the left regular representation of $G$ is multiplicity--free.