2008/01/25 by Joseph A. Wolf, Wolf, Joseph A.
Mathematics · #22E25 #22E45 #22E65 #53C35 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Representation Theory (math.RT) #math.DG #math.RT #msc:22E25 #msc:22E45 #msc:22E65 #msc:53C35
paper · pdf · doi:10.48550/arxiv.0801.3866
31 pages
arxiv created 2008/01/25 · openalex publication_date 2008/01/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study direct limits (G,K) = \varinjlim (Gn,Kn) of Gelfand pairs of the form Gn = Nn\rtimes Kn with Nn nilpotent, in other words pairs (Gn,Kn) for which Gn/Kn is a commutative nilmanifold. First, we extend the criterion of \citeW3 for a direct limit representation to be multiplicity free. Then we study direct limits G/K = \varinjlim Gn/Kn of commutative nilmanifolds and look to see when the regular representation of G = \varinjlim Gn on an appropriate Hilbert space \varinjlim L2(Gn/Kn) is multiplicity free. One knows that the Nn are commutative or 2--step nilpotent. In many cases where the derived algebras [\gnn,\gnn] are of bounded dimension we construct Gn--equivariant isometric maps ζn : L2(Gn/Kn) → L2(Gn+1/Kn+1) and prove that the left regular representation of G on the Hilbert space L2(G/K) := \varinjlim \L2(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.