2008/01/25 by Joseph A. Wolf, Wolf, Joseph A.
Mathematics · #22E25 #22E45 #22E65 #22G05 #43A85 #43A90 #53C35 #Differential Geometry (math.DG) #FOS: Mathematics #Representation Theory (math.RT) #math.DG #math.RT #msc:22E25 #msc:22E45 #msc:22E65 #msc:22G05 #msc:43A85 #msc:43A90 #msc:53C35
paper · pdf · doi:10.48550/arxiv.0801.3869
23 pages
arxiv created 2008/01/25 · arxiv updated 2009/12/01
We study direct limits (G,K) = \varinjlim (Gn,Kn) of compact Gelfand pairs. First, we develop a criterion for a direct limit representation to be a multiplicity--free discrete direct sum of irreducible representations. Then we look at direct limits G/K = \varinjlim Gn/Kn of compact riemannian symmetric spaces, where we combine our criterion with the Cartan--Helgason Theorem to show in general that the regular representation of G = \varinjlim Gn on a certain function space \varinjlim L2(Gn/Kn) is multiplicity free. That method is not applicable for direct limits of nonsymmetric Gelfand pairs, so we introduce two other methods. The first, based on ``parabolic direct limits'' and ``defining representations'', extends the method used in the symmetric space case. The second uses some (new) branching rules from finite dimensional representation theory. In both cases we define function spaces \cA(G/K), \cC(G/K) and L2(G/K) to which our multiplicity--free criterion applies.