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

TL;DR

This paper proves that for certain nilpotent Gelfand pairs satisfying Vinberg's condition, the spherical transform creates an isomorphism between K-invariant Schwartz functions on the group and Schwartz functions on the Gelfand spectrum.

Contribution

It establishes a precise isomorphism between Schwartz spaces under the spherical transform for a class of nilpotent Gelfand pairs satisfying Vinberg's condition.

Findings

Spherical transform is an isomorphism between Schwartz spaces for these pairs.

The Gelfand spectrum can be identified with a closed subset of R^d.

The result applies to pairs where K acts irreducibly on the quotient of the Lie algebra.

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 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.