#### APRG Seminar

##### Venue: Microsoft Teams (online)

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.

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 25 Jan 2023