Orbits in the flag variety and images of the moment map for \(U(p, q)\)
type: Item
instance of: scholarly article
title: Orbits in the flag variety and images of the moment map for \(U(p, q)\)
zbMATH Open document ID: 0873.22008
author: A. Yamamoto
published in: Proceedings of the Japan Academy. Series A
publication date: 1997-10-20T00:00:00Z
review text: Let \(G_{\mathbb{R}}\) be a real classical group. Assume that the Cartan subgroups of \(G_{\mathbb{R}}\) are connected. There exists a correspondence between irreducible representations of \(G_{\mathbb{R}}\) with a given infinitesimal character and the orbits of a flag of \(G_{\mathbb{C}}\) -- a complexification of \(G_{\mathbb{R}}\) on which \(K_{\mathbb{C}}\) operates, where \(K_{\mathbb{C}}\) is a complexification of a maximal compact subgroup \(K\) of \(G_{\mathbb{R}}\). One is therefore interested in finding an algorithm which can yield representatives of orbits for a suitable parametrization. The author considers the case \(G=U(p,q)\), \(n=p+q\), \(U(p,q)= \{g\in GL(n,\mathbb{C})\), \(^tgJg=J\}\), \(J= \left(\begin{smallmatrix} I_p &0\\ 0&-I_q \end{smallmatrix}\right)\), \(\theta\) the Cartan involution of \(G_{\mathbb{R}}\); \(\theta(x)= JxJ\). In Definition 2.1 the notion of clan is introduced as an equivalence class of indications and an earlier result of Matsuki-Oshima (Theorem 2.2) is stated and applied to show that clans parametrize \(K\)-orbits in the flag variety \(X\). In Theorem 3.7 for each clan \(\gamma\), and signed class \(\delta\) of \(\gamma\), \(\delta= (d_1,\dots, d_n)\) the representative matrix \(g(\delta)= (g_1,g_2,\dots, g_n)\) is defined by explicitly exhibiting its column vectors. In Proposition 3.9 the dimension and co-dimension of the orbit \(Q_\gamma= Kg(\delta)B: \dim Q_\gamma= \ell(\gamma)+ \frac{1}{2} p(p-1)+ \frac{1}{2} q(q-1)\), \(\text{co-dim}= pq-\ell(\gamma)\), is given, where \(\ell(\gamma)\) is a certain integer introduced in Definition 3.8. Let \(\gamma=(c_1,\dots,c_n)\) be a clan, \(\delta\) a signed clan of \(\gamma\), \(g(\delta)\) the matrix associated to \(\delta\) in Theorem 3.7. An algorithm is described to obtain the image \(g(\delta){\mathfrak E}^\perp g(\delta)^{-1}\cap{\mathfrak P}\) of the moment map of a fiber at \(g(\delta)\) of the conormal bundle of \(Q_\gamma= Kg(\delta)B\) (Theorem 4.1). In Section 5 tabl…
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 14M15
zbMATH DE Number: 952358
zbMATH Keywords: real classical group
zbMATH Keywords: representations
zbMATH Keywords: character
zbMATH Keywords: flag variety
zbMATH Keywords: moment map
zbMATH Keywords: Young diagrams
MaRDI profile type: Publication
full work available at URL: pjaa.72.114
OpenAlex ID: W2050669378
cites work: Differential operators on homogeneous spaces. III: Characteristic varieties of Harish Chandra modules and of primitive ideals
cites work: Q4863383
cites work: The Annihilators of Irreducible Harish-Chandra Modules for SU(p,q) and Other Type A n-1 Groups
cites work: The orbits of affine symmetric spaces under the action of minimal parabolic subgroups
cites work: Q3973974
cites work: Q3353273
cites work: Irreducible characters of semisimple Lie groups. III: Proof of Kazhdan-Lusztig conjecture in the integral case
cites work: A path-following procedure to find a proper equilibrium of finite games
DOI: 10.3792/PJAA.72.114
Recommended article: Q4382855
Recommended article: Closures of \(K\)-orbits in the flag variety for \(U(p,q)\)
Recommended article: Polynomials for \(\mathrm{GL}_p\times \mathrm{GL}_q\) orbit closures in the flag variety
Recommended article: \(B_{n-1}\)-orbits on the flag variety \(\text{GL}_n/B_n\)
Recommended article: Relationship between orbit decomposition on the flag varieties and multiplicities of induced representations
Recommended article: On orbits in double flag varieties for symmetric pairs
Recommended article: ON THE NUMBER OF TRIANGULATION SIMPLEXES
Recommended article: \(K\)-orbits on the flag variety and strongly regular nilpotent matrices
Recommended article: On Orbit Closures of Symmetric Subgroups in Flag Varieties
Recommended article: Orbit embedding for double flag varieties and Steinberg maps
description: scientific article; zbMATH DE number 952358