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

Original · EN

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 SD of Rᵈ. 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 SD of Schwartz functions in Rᵈ.