Associated varieties and Gelfand-Kirillov dimensions for the discrete series of a semisimple Lie group
In this article the author describes the associated variety of the Harish-Chandra \(({\mathfrak g},K)\)-module of discrete series as the cone \(K_\mathbb{C} {\mathfrak p}_-\). Here \(K_\mathbb{C}\) denotes the complexification of a maximal compact subgroup of a connected semisimple Lie group \(G\) with finite center, and \({\mathfrak p}_-\) the sum of the root spaces associated with a negative system of noncompact roots. One of the tools used in the outline of the proof the author gives is a result of \textit{R. Hotta} and \textit{R. Parthasarathy} [Invent. Math. 26, 133-178 (1974; Zbl 0298.22013)] that shows a graduation module of a discrete series as the kernel of an operator graduation of the Schmid operator. As the author mentions, the result can be also obtained using \(\mathcal D\)- module theory combining some results of \textit{W. Borho} and \textit{J.-L. Brylinski} [Invent. Math. 80, 1-68 (1985; Zbl 0577.22014)] and \textit{W. Schmid} [Harmonic Analysis on Reductive Groups, Proc. Conf., Brunswick/ME (USA) 1989, Prog. Math. 101, 235-275 (1991; Zbl 0751.22003)]. As consequences, the author shows that the associated ideal coincides with its radical, and that the associated variety of a discrete series is presented in terms of the symbol mapping of the Schmid operator. Finally, the author gives a formula for the Gelfand-Kirillov dimension for discrete series of \(\text{SU} (p,q)\).
Harish- Chandra \(({\mathfrak g},K)\)-module
scientific article; zbMATH DE number 605461