Infinite Dimensional Multiplicity Free Spaces II: Limits of Commutative Nilmanifolds

We study direct limits (G,K) = lim −→ (Gn, Kn) of Gelfand pairs of the form Gn = Nn Kn with Nn nilpotent, in other words pairs (Gn, Kn) for which Gn/Kn is a commutative nilmanifold. First, we extend the criterion of [W4] for a direct limit representation to be multiplicity free. Then we study direct limits G/K = lim −→ Gn/Kn of commutative nilmanifolds and look to see when the regular representation of G = lim −→n on an appropriate Hilbert space lim −→ (Gn/Kn) is multiplicity free. One knows that the Nn are commutative or 2–step nilpotent. In many cases where the derived algebras [nn,nn] are of bounded dimension we construct Gn–equivariant isometric maps ζn : L(Gn/Kn) → L(Gn+1/Kn+1) and prove that the left regular representation of G on the Hilbert space L2(G/K) := lim −→{L (Gn/Kn), ζn} 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.