The Spherical Transform Associated with the Generalized Gelfand Pair (U(p,q),Hn), p+q=n

Abstract

We denote by $H_{n}$ the $2n+1$-dimensional Heisenberg group and study the spherical transform associated with the generalized Gelfand pair $(U(p,q) \rtimes H_{n},U(p,q))$, $p+q=n$, which is defined on the space of Schwartz functions on $H_{n}$, and we characterize its image. In order to do that, since the spectrum associated to this pair can be identified with a subset $\Sigma$ of the plane, we introduce a space ${\cal H}_{n}$ of functions defined on $\mathbb{R}^2$ and we prove that a function defined on $\Sigma$ lies in the image if and only if it can be extended to a function in ${\cal H}_{n}$. In particular, the spherical transform of a Schwartz function $f$ on $H_{n}$ admits a Schwartz extension on the plane if and only if its restriction to the vertical axis lies in ${\cal S}(\mathbb{R})$.