Abstract
In this paper we make another step in the direction of proving a certain conjecture on the spherical transform of a nilpotent spherical pair (N,K). We obtain a result of an independent interest which can be viewed as a generalised Hadamard lemma for K-invariant functions on N. Nilpotent Gelfand pairs here are assumed to satisfy Vinberg’s condition, meaning that K acts irreducibly on the quotient of n = Lie N by its derived subalgebra.
Dedicated to Joe Wolf
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Acknowledgments
Parts of this work were carried out during the third author’s stay at the Max-Planck-Institut für Mathematik (Bonn) and Centro di Ricerca Matematica Ennio De Giorgi (SNS, Pisa). She would like to thank these institutions for warm hospitality and support.
The first author acknowledges the support of the London Mathematical Society via the Grace Chisholm Fellowship held at King’s College London in 2011.
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Fischer, V., Ricci, F., Yakimova, O. (2013). Nilpotent Gelfand Pairs and Spherical Transforms of Schwartz Functions II: Taylor Expansions on Singular Sets. In: Huckleberry, A., Penkov, I., Zuckerman, G. (eds) Lie Groups: Structure, Actions, and Representations. Progress in Mathematics, vol 306. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-7193-6_5
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