Corwin-Greenleaf multiplicity function of a class of Lie groups
type: Item
instance of: scholarly article
title: Corwin-Greenleaf multiplicity function of a class of Lie groups
author: Aymen Rahali
published in: International Journal of Mathematics
publication date: 2024-12-16T00:00:00Z
review text: The paper is devoted to the study of the multiplicity problem in the framework of the orbit method. Let \(G= K \ltimes N\) be the semi-direct product of a compact Lie group \(K\) and a nilpotent group \(N\), \(\mathfrak g = \mathrm{Lie}\; G, \mathfrak k = \mathrm{Lie}\; K\) the corresponding Lie algebras, \(\mathfrak g^*, \mathfrak k^*\) the dual vector spaces, \(q : \mathfrak g^* \to \mathfrak k^*\) be the natural projection map, \(\mathcal O^G, \mathcal O^K\) the corresponding coadjoint orbit spaces. Following the orbit method the irreducible representations of \(G\), \(K\) are in a bijective correspondence with the admissible coadjoint orbits and the Mackey induction therefore can be parametrized by some functionals on Lie algebras. The question is for the Mackey induced parameters \((\mu,s)\) how to read out the multiplicity \(m_{\pi(\mu,s)}(\tau_\lambda)\) of \(\tau_\lambda\in \hat K\) in the restriction \(\pi_{(\mu,s)} |_{K} \) of the induced representation \(\mathrm{ind}^G_{N_A \ltimes N}(1_{K_A} \otimes \sigma_s \circ W_s)\) from the Corwin-Greenleaf multiplicity function \(\chi(\mathcal O^G, \mathcal O^K)\) of the number of \(K\)-orbits in the inverse image \(\mathcal O^G \cap q^{-1}(\mathcal O^K)\). \N\NThe main results of the paper are:\N\begin{itemize}\N\item[(1)] for an irreducible representation \(\tau_\lambda \in \hat K\) the multiplicity \(m_{\pi_{(0,s)}}(\tau_\lambda)\) of \(\tau_\lambda\) in the restriction to \(K\) of the induced representation \(\pi_{(0,s)} =\mathrm{ind}^G_{N_A \ltimes N}(1_{K_A} \otimes \sigma_s \circ W_s)\) is non-zero, then the Corwin-Greenleaf multiplicity function \(\chi(\mathcal O^G, \mathcal O^K)\) is non-zero (Theorems 1.3);\N\item[(2)] For any central element \(\varphi \in \mathfrak k_A^*\), Corwin-Greenleaf multiplicity is multiplicity-free such that \(\chi(\mathcal O^{G_A}_{(\varphi,s)}, \mathcal O^{K_A}_\nu) \leq 1\), where \(K_A\) is the stabilizer of \(A \in \mathfrak z\) in \(K\), \(G_A = K_A \ltimes N\) (Theorems …
reviewed by: Do Ngoc Diep
Mathematics Subject Classification ID: 22D10
Mathematics Subject Classification ID: 22E27
Mathematics Subject Classification ID: 22E45
zbMATH DE Number: 7958659
zbMATH Keywords: Gelfand pairs
zbMATH Keywords: orbit method
zbMATH Keywords: Corwin-Greenleaf multiplicity function
zbMATH Keywords: branching laws
MaRDI profile type: Publication
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DOI: 10.1142/S0129167X24500745
description: scientific article; zbMATH DE number 7958659