Infinite Dimensional Multiplicity Free Spaces II: Limits of Commutative Nilmanifolds

We study direct limits $(G,K) = \varinjlim (G_n,K_n)$ of Gelfand pairs of the form $G_n = N_n\rtimes K_n$ with $N_n$ nilpotent, in other words pairs $(G_n,K_n)$ for which $G_n/K_n$ is a commutative nilmanifold. First, we extend the criterion of \cite{W3} for a direct limit representation to be multiplicity free. Then we study direct limits $G/K = \varinjlim G_n/K_n$ of commutative nilmanifolds and look to see when the regular representation of $G = \varinjlim G_n$ on an appropriate Hilbert space $\varinjlim L^2(G_n/K_n)$ is multiplicity free. One knows that the $N_n$ are commutative or 2--step nilpotent. In many cases where the derived algebras $[\gn_n,\gn_n]$ are of bounded dimension we construct $G_n$--equivariant isometric maps $\zeta_n : L^2(G_n/K_n) \to L^2(G_{n+1}/K_{n+1})$ and prove that the left regular representation of $G$ on the Hilbert space $L^2(G/K) := \varinjlim \{L^2(G_n/K_n),\zeta_n\}$ is a multiplicity free direct integral of irreducible unitary representations. The direct integral and its irreducible constituents are described explicitly. One constituent of our argument is an extension of the classical Peter--Weyl Theorem to parabolic direct limits of compact groups.