Orbits in the flag variety and images of the moment map for \(U(p, q)\)
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 tables of signed Young diagrams are displayed for clans of \(U(2,1)\) and \(U(2,2)\). \textit{D. Garfinkle} [Am. J. Math. 115, 305-369 (1993; Zbl 0786.22023)] has given a different algorithm for the same purpose.
scientific article; zbMATH DE number 952358