Infinite Dimensional Multiplicity Free Spaces III: Matrix Coefficients and Regular Functions
Abstract
In earlier papers we studied direct limits (G,K) = (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: L2(Gn/Kn) L2(Gm/Km) for m ≥q n and prove that the left regular representation of G on the Hilbert space direct limit L2(G/K) := L2(Gn/Kn) is multiplicity--free. This left open questions concerning the nature of the elements of L2(G/K). Here we define spaces (Gn/Kn) of regular functions on Gn/Kn and injections m,n : (Gn/Kn) (Gm/Km) for m ≥q n related to restriction by m,n(f)|Gn/Kn = f. Thus the direct limit (G/K):= \(Gn/Kn), m,n\ sits as a particular G--submodule of the much larger inverse limit \(Gn/Kn), restriction\. Further, we define a pre Hilbert space structure on (G/K) derived from that of L2(G/K). This allows an interpretation of L2(G/K) as the Hilbert space completion of the concretely defined function space (G/K), and also defines a G--invariant inner product on (G/K) for which the left regular representation of G is multiplicity--free.