Nilpotent Gelfand pairs and spherical transforms of Schwartz functions III. Isomorphisms between Schwartz spaces under Vinberg's condition

2012/10/30 by Fischer, Veronique, Ricci, Fulvio, Yakimova, Oksana
#43A32 #43A90 #Commutative Algebra (math.AC) #FOS: Mathematics #Functional Analysis (math.FA) #Primary: 13A50 #Representation Theory (math.RT) #Secondary: 43A85

paper · doi:10.48550/arxiv.1210.7962

Abstract

Let (N,K) be a nilpotent Gelfand pair, i.e., N is a nilpotent Lie group, K a compact group of automorphisms of N, and the algebra D(N)^K of left-invariant and K-invariant differential operators on N is commutative. In these hypotheses, N is necessarily of step at most two. We say that (N,K) satisfies Vinberg's condition if K acts irreducibly on n/[n,n], where n= Lie(N). Fixing a system D of d formally self-adjoint generators of D(N)^K, the Gelfand spectrum of the commutative convolution algebra L¹(N)^K can be canonically identified with a closed subset S_D of R^d. We prove that, on a nilpotent Gelfand pair satisfying Vinberg's condition, the spherical transform establishes an isomorphism from the space of K-invariant Schwartz functions on N and the space of restrictions to S_D of Schwartz functions in R^d.

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