Nilpotent Gelfand pairs and spherical transforms of Schwartz functions   III. Isomorphisms between Schwartz spaces under Vinberg's condition
  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 L^1(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. 
Veronique Fischer, Fulvio Ricci and Oksana Yakimova
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