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

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