Journal of Lie Theory 24 (2014), No. 3, 657--685
Copyright Heldermann Verlag 2014
The Spherical Transform Associated with the Generalized Gelfand Pair (U(p,q),Hn), p+q=n
Silvina Campos
CIEM-FaMAF, Universidad Nacional de Córdoba, Córdoba 5000, Argentina
silcampos@famaf.unc.edu.ar
Linda Saal
CIEM-FaMAF, Universidad Nacional de Córdoba, Córdoba 5000, Argentina
saal@mate.uncor.edu.ar
[Abstract-pdf]
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 ${\cal S}(\mathbb{R})$.
Keywords: Heisenberg group, spherical transform.
MSC: 43A80; 22E25
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