Journal of Lie Theory

no. 3
Gelfand Pairs and Corwin-Greenleaf Multiplicity Function
Journal of Lie Theory, Volume 35 (2025) no. 3, pp. 651-665
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 $G$.
DOI: 10.5802/jolt.1402
Classification: 22D10, 22E27, 22E45
Keywords: Gelfand pairs, orbit method, Corwin-Greenleaf multiplicity function, branching laws 
@article{JOLT_2025_35_3_a10,
     author = {A. Rahali and S. Hamdani},
     title = {Gelfand {Pairs} and {Corwin-Greenleaf} {Multiplicity} {Function}},
     journal = {Journal of Lie Theory},
     pages = {651--665},
     year = {2025},
     volume = {35},
     number = {3},
     doi = {10.5802/jolt.1402},
     zbl = {08103107},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1402/}
}
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A. Rahali; S. Hamdani. Gelfand Pairs and Corwin-Greenleaf Multiplicity Function. Journal of Lie Theory, Volume 35 (2025) no. 3, pp. 651-665. doi: 10.5802/jolt.1402

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