Let $(K\ltimes G,K)$ be a Gelfand pair, where $K\ltimes G$ is the
semidirect product of a Lie group $G$ with polynomial growth and $K$
a compact group of automorphisms of $G$. Then the Gelfand spectrum
$\Sigma$ of the commutative convolution algebra of $K$-invariant
integrable functions on $G$ admits natural embeddings into $\mathbb{R}^n$
spaces as a closed subset. Let $\mathcal{S}(G)^K$ be the space of
$K$-invariant Schwartz functions on $G$. Defining $\mathcal{S}(\Sigma)$
as the space of restrictions to $\Sigma$ of Schwartz functions on
$\mathbb R^n$, we call *Schwartz correspondence* for $(K\ltimes G,K)$ the
property that the spherical transform is an isomorphism of $\mathcal{S}(G)^K$
onto $\mathcal{S}(\Sigma)$. In all the cases studied so far, the Schwartz
correspondence has been proved to hold true. These include all pairs
$(K\ltimes G,K)$ with $K$ abelian and a large number of pairs with $G$
nilpotent. In this talk we show that the Schwartz correspondence holds for
the pair $(K\ltimes G,K)$, where $G=U_2\ltimes \mathbb{C}^2$ is the complex
motion group and $K={\rm Int}(U_2) $ is the group of inner automorphisms
of $G$ induced by elements of $U_2$. This is one of the simplest pairs with
$G$ non-nilpotent and $K$ non-abelian. This work arises from a collaboration
with Francesca Astengo and Fulvio Ricci.

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Last updated: 03 Feb 2023