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 $G$.
@article{JOLT_2025_35_3_a10,
author = {Aymen Rahali and Sofien Hamdani},
title = {Gelfand {Pairs} and {Corwin-Greenleaf} {Multiplicity} {Function}},
journal = {Journal of Lie Theory},
pages = {651--665},
year = {2025},
volume = {35},
number = {3},
url = {http://geodesic.mathdoc.fr/item/JOLT_2025_35_3_a10/}
}
TY - JOUR
AU - Aymen Rahali
AU - Sofien Hamdani
TI - Gelfand Pairs and Corwin-Greenleaf Multiplicity Function
JO - Journal of Lie Theory
PY - 2025
SP - 651
EP - 665
VL - 35
IS - 3
UR - http://geodesic.mathdoc.fr/item/JOLT_2025_35_3_a10/
ID - JOLT_2025_35_3_a10
ER -
%0 Journal Article
%A Aymen Rahali
%A Sofien Hamdani
%T Gelfand Pairs and Corwin-Greenleaf Multiplicity Function
%J Journal of Lie Theory
%D 2025
%P 651-665
%V 35
%N 3
%U http://geodesic.mathdoc.fr/item/JOLT_2025_35_3_a10/
%F JOLT_2025_35_3_a10
