The Spherical Transform Associated with the Generalized Gelfand Pair (U(p,q),Hn), p+q=n
Journal of Lie Theory, Volume 24 (2014) no. 3, pp. 657-685
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 ${\cal S}(\mathbb{R})$.
@article{JOLT_2014_24_3_a2,
author = {S. Campos and L. Saal},
title = {The {Spherical} {Transform} {Associated} with the {Generalized} {Gelfand} {Pair} {(U(p,q),H\protect\textsubscript{n}),} p+q=n},
journal = {Journal of Lie Theory},
pages = {657--685},
year = {2014},
volume = {24},
number = {3},
doi = {10.5802/jolt.800},
zbl = {1309.43004},
url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.800/}
}
TY - JOUR AU - S. Campos AU - L. Saal TI - The Spherical Transform Associated with the Generalized Gelfand Pair (U(p,q),Hn), p+q=n JO - Journal of Lie Theory PY - 2014 SP - 657 EP - 685 VL - 24 IS - 3 UR - https://jolt.centre-mersenne.org/articles/10.5802/jolt.800/ DO - 10.5802/jolt.800 ID - JOLT_2014_24_3_a2 ER -
S. Campos; L. Saal. The Spherical Transform Associated with the Generalized Gelfand Pair (U(p,q),Hn), p+q=n. Journal of Lie Theory, Volume 24 (2014) no. 3, pp. 657-685. doi: 10.5802/jolt.800
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