Nilpotent Gelfand pairs and spherical transforms of Schwartz functions II. Taylor expansion on singular sets

Details Nilpotent Gelfand pairs and spherical transforms of Schwartz functions II. Taylor expansion on singular sets Nilpotent Gelfand pairs and spherical transforms of Schwartz functions II. Taylor expansion on singular sets

2010-08-27

arxiv.org/abs/1008.4699v2

Mathematics

Commutative Algebra

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Scientific paper

This paper is a continuation of [8], in the direction of proving the conjecture that the spherical transform on a nilpotent Gelfand pair (N,K) establishes an isomorphism between the space of K-invariant Schwartz functions on N and the space of Schwartz functions restricted to the Gelfand spectrum properly embedded in a Euclidean space. We prove a result, of independent interest for the representation theoretical problems that are involved, which can be viewed as a generalised Hadamard lemma for K-invariant functions on N. The context is that of nilpotent Gelfand pairs satisfying Vinberg's condition. This means that the Lie algebra n of N (which is step 2) decomposes as a direct sum of [n,n] and a K-invariant irreducible subspace.

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