Nilpotent Gelfand pairs and Schwartz extensions of spherical transforms via quotient pairs

It has been shown [1,2,9,10] that for several nilpotent Gelfand pairs (N,K) (i.e., with N a nilpotent Lie group, K a compact group of automorphisms of N and the algebra L ¹(N) ^K commutative) the spherical transform establishes a 1-to-1 correspondence between the space S(N) ^K of K-invariant Schwartz functions on N and the space S(Σ) of functions on the Gelfand spectrum Σ of L ¹(N) ^K which extend to Schwartz functions on R ^d, once Σ is suitably embedded in R ^d. We call this property (S). We present here a general bootstrapping method which allows to establish property (S) to new nilpotent pairs (N,K), once the same property is known for a class of quotient pairs of (N,K) and a K-invariant form of Hadamard formula holds on N. We finally show how our method can be recursively applied to prove property (S) for a significant class of nilpotent Gelfand pairs.