Journal of Lie Theory 35 (2025), No. 3, 651--665
Copyright Heldermann Verlag 2025
Gelfand Pairs and Corwin-Greenleaf Multiplicity Function
Aymen Rahali
Faculté des Sciences, Université de Sfax, Tunisia
aymenrahali@yahoo.fr
Sofien Hamdani
Faculté des Sciences, Université de Sfax, Tunisia
sof.hamdani@gmail.com
[Abstract-pdf]
Let $(K,N)$ be a nilpotent Gelfand pair and let $G:=K\ltimes N$ be the semidirect product associated with $(K,N)$. Let $\pi\in\widehat{G}$ be a generic representation of $G$ and let $\tau\in\widehat{K}$. The Kirillov-Lipsman's orbit method suggests that the multiplicity $m_\pi(\tau)$ of an irreducible $K$-module $\tau$ occurring in the restriction of $\pi|_K$ can be linked to (the number of $K$-orbits) the Corwin-Greenleaf multiplicity function (C.G.M.F for short). Under some assumptions on the pair $(K,N),$ in this work we focus on the connection between the geometric number C.G.M.F and the multiplicity ($m_\pi(.)$). In the geometric counterpart we give a necessary and sufficient conditions associated with the C.G.M.F. Moreover, we prove that this function is bounded for a special class of subgroups of $G$.
Keywords: Gelfand pairs, orbit method, Corwin-Greenleaf multiplicity function, branching laws.
MSC: 22D10, 22E27, 22E45.
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