2009/09/09 by Joseph A. Wolf, Wolf, Joseph A. · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models
paper · pdf · doi:10.48550/arxiv.0909.1735
In earlier papers we studied direct limits (G,K) = \varinjlim (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\rtimes 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) \hookrightarrow L2(Gm/Km) for m \geqq n and prove that the left regular representation of G on the Hilbert space direct limit L2(G/K) := \varinjlim L2(Gn/Kn) is multiplicity--free. This left open questions concerning the nature of the elements of L2(G/K). Here we define spaces \cA(Gn/Kn) of regular functions on Gn/Kn and injections νm,n : \cA(Gn/Kn) → \cA(Gm/Km) for m \geqq n related to restriction by νm,n(f)|Gn/Kn = f. Thus the direct limit \cA(G/K):= \varinjlim \\cA(Gn/Kn), νm,n\ sits as a particular G--submodule of the much larger inverse limit \varprojlim \\cA(Gn/Kn), restriction\. Further, we define a pre Hilbert space structure on \cA(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 \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.