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

In earlier papers we studied direct limits (G,K) = lim −→ (Gn,Kn) of two types of Gelfand pairs. The first type was that in which the Gn/Kn are compact riemannian symmetric spaces. The second type was that in which Gn = Nn ⋊ Kn with Nn nilpotent, in other words pairs (Gn,Kn) for which Gn/Kn is a commutative nilmanifold. In each we worked out a method inspired by the Frobenius–Schur Orthogonality Relations to define isometric injections ζm,n : L(Gn/Kn) →֒ L(Gm/Km) for m ≧ n and prove that the left regular representation of G on the Hilbert space direct limit L(G/K) := lim −→ (Gn/Kn) is multiplicity–free. This left open questions concerning the nature of the elements of L(G/K). Here we define spacesA(Gn/Kn) of regular functions on Gn/Kn and injections νm,n : A(Gn/Kn) → A(Gm/Km) for m ≧ n related to restriction by νm,n(f)|Gn/Kn = f . Thus the direct limit A(G/K) := lim −→{A(Gn/Kn), νm,n} sits as a particular G–submodule of the much larger inverse limit lim ←−{A(Gn/Kn), restriction}. Further, we define a pre Hilbert space structure on A(G/K) derived from that of L(G/K). This allows an interpretation of L(G/K) as the Hilbert space completion of the concretely defined function space A(G/K), and also defines a G–invariant inner product on A(G/K) for which the left regular representation of G is multiplicity–free.