Double cells for unitary groups
The main result of this paper is the fact that the associated variety of an irreducible Harish-Chandra module \(\pi\) for \(G_{\mathbb R}=U(p,q)\) with trivial infinitesimal character is the closure of a single nilpotent orbit \(\mathcal O\) for the complexified maximal compact subgroup \(K\). The method employed is to first pass to the category of \(K\)-equivariant D-modules on the flag variety \(X\) via Beilinson-Bernstein localization. After extending scalars to \(\mathbb Q\), the Grothendieck group of this category is shown to be isomorphic to the top Borel-Moore homology of the union of conormal bundles of \(K\)-orbits in \(X\). This step uses the characteristic cycle map. The isomorphism respects the Weyl group actions. This leads to the main result. It is also shown that the corresponding LR-cell, defined by \textit{D. Vogan} [Duke Math. J. 49, 943--1073 (1982; Zbl 0536.22022)], is isomorphic to the top Borel-Moore homology of the preimage of the corresponding orbit under the moment map. The author announces generalizations to other classical groups.
scientific article; zbMATH DE number 1867972