Schwartz correspondence for the complex motion group on C-2

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 & sigma; of the bi-K-invariant algebra L1(K\G/K) admits natural embeddings into Rt spaces as a closed subset.For any such embedding, define S(& sigma;) as the space of restrictions to & sigma; of Schwartz functions on Rt. We call Schwartz correspondence for (G, K) the property that the spherical transform is an isomorphism of S(K\G/K) onto S(& sigma;).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 & alpha; H and H nilpotent.We prove Schwartz correspondence for the pair (U2 IX 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 & tau; & ISIN; K ⠂ with appropriate control of the estimates in terms of & tau;.