# Geometry of Coadjoint Orbits and Multiplicity-one Branching Laws for Symmetric Pairs

Authors
• 1 The University of Tokyo, Graduate School of Mathematical Sciences and Kavli IPMU, Tokyo, Japan , Tokyo (Japan)
• 2 University of Dhaka, Department of Mathematics, Dhaka, Bangladesh , Dhaka (Bangladesh)
Type
Published Article
Journal
Algebras and Representation Theory
Publisher
Springer Netherlands
Publication Date
Jun 27, 2018
Volume
21
Issue
5
Pages
1023–1036
Identifiers
DOI: 10.1007/s10468-018-9810-8
Source
Springer Nature
Keywords
Consider the restriction of an irreducible unitary representation π of a Lie group G to its subgroup H. Kirillov’s revolutionary idea on the orbit method suggests that the multiplicity of an irreducible H-module ν occurring in the restriction π|H could be read from the coadjoint action of H on OG∩pr−1(OH)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {O}^{G} \cap \text {pr}^{-1}({\mathcal {O}}^{H})$\end{document}, provided π and ν are ‘geometric quantizations’ of a G-coadjoint orbit OG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {O}^{G}$\end{document} and an H-coadjoint orbit OH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {O}^{H}$\end{document}, respectively, where pr:−1𝔤∗→−1𝔥∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\text {pr} \colon \sqrt {-1}\mathfrak {g}^{\ast } \to \sqrt {-1}\mathfrak {h}^{\ast }$\end{document} is the projection dual to the inclusion 𝔥⊂𝔤\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathfrak {h} \subset \mathfrak {g}$\end{document} of Lie algebras. Such results were previously established by Kirillov, Corwin and Greenleaf for nilpotent Lie groups. In this article, we highlight specific elliptic orbits OG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {O}^{G}$\end{document} of a semisimple Lie group G corresponding to highest weight modules of scalar type. We prove that the Corwin–Greenleaf number ♯(OG∩pr−1(OH))/H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sharp (\mathcal {O}^{G} \cap \text {pr}^{-1}({\mathcal {O}}^{H}))/H$\end{document} is either zero or one for any H-coadjoint orbit OH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {O}^{H}$\end{document}, whenever (G,H) is a symmetric pair of holomorphic type. Furthermore, we determine the coadjoint orbits OH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {O}^{H}$\end{document} with nonzero Corwin–Greenleaf number. Our results coincide with the prediction of the orbit philosophy, and can be seen as ‘classical limits’ of the multiplicity-free branching laws of holomorphic discrete series representations (Kobayashi [Progr. Math. 2007]).