Schwartz correspondence for the complex motion group on C2

If (G,K) is a Gelfand pair, with G a Lie group of polynomial growth and K a compact subgroup of G, the Gelfand spectrum Σ of the bi-K-invariant algebra L1(K﹨G/K) admits natural embeddings into Rℓ spaces as a closed subset. For any such embedding, define S(Σ) as the space of restrictions to Σ of Schwartz functions on Rℓ. We call Schwartz correspondence for (G,K) the property that the spherical transform is an isomorphism of S(K﹨G/K) onto S(Σ). In all the cases studied so far, Schwartz correspondence has been proved to hold true. These include all pairs with G=K⋉H and K abelian and a large number of pairs with G=K⋉H and H nilpotent. We prove Schwartz correspondence for the pair (U2⋉M2(C),U2), where M2(C) is the complex motion group and U2=K acts on it by conjugation. Our proof goes through a detailed analysis of (M2(C),U2) as a strong Gelfand pair and reduction of the problem to Schwartz correspondence for each K-type τ∈Kˆ with appropriate control of the estimates in terms of τ.