A reduction of the globalization and \(U(1)\)-covering (Q1376665)

Based on the one-dimensional induced bundle, the well-known orbit method gives a construction of a lot of known irreducible unitary representations of Lie groups. There are irreducible representations of (finite-dimensional) semisimple Lie groups and infinite-dimensional Lie groups, which are based on higher-dimensional induced bundles. \textit{Do Ngoc Diep} [Acta Math. Vietnam. 5, 42-55 (1980; Zbl 0502.58019)] proposed a multidimensional quantization procedure which is a multidimensional generalization of the orbit method. \textit{M. Duflo} [Monogr. Stud. Math. 17, 147-156 (1984; Zbl 0522.22011)] proposed an analytic version of this generalization and also two reductions to the solvable and nilpotent radicals, in the language of the Mackey method of small subgroups. \textit{W. Schmid} and \textit{J. A. Wolf} [J. Funct. Anal. 90, 48-112 (1990; Zbl 0781.22009)] described globalizations of Harish-Chandra modules by using the derived Zuckerman functors. \textit{P. L. Robinson} and \textit{J. H. Rawnsley} [Mem. Am. Math. Soc. 410, 92 p. (1989; Zbl 0681.22022)] suggested a quantization over the \(U(1)\)-covering in place of the \({\mathbf Z}/2\)-covering considered by M. Duflo in applying the Mackey machinery. In a previous paper, the author modified the construction, suggested by W. Schmid and J. A. Wolf, for the case of the \(U(1)\)-covering. In the paper under review the author suggests a reduction (in the sense of M. Duflo) of the globalization (in the sense of W. Schmid and J. A. Wolf) over the \(U(1)\)-covering (in the sense of P. L. Robinson and J. H. Rawnsley), in particular the author considers the geometric complexes with respect to the \(U(1)\)-covering and then describes the globalization of the Harish-Chandra modules by local cohomology associated with reductions of the multidimensional quantization over the \(U(1)\)-coverings.