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Construction and classification of irreducible Harish-Chandra modules. (English) Zbl 0751.22003
Harmonic analysis on reductive groups, Proc. Conf., Brunswick/ME (USA) 1989, Prog. Math. 101, 235-275 (1991).
[For the entire collection see Zbl 0742.00061.]
These expository lectures provided a thorough introduction into the ways in which standard modules have been used to classify irreducible Harish- Chandra modules. The first part discusses infinitesimally equivariant line bundles and twisted sheaves of differential operators. It is followed by an outline of the \({\mathcal D}\)-module construction of standard Harish-Chandra modules due to Beilinson and Bernstein and touches on local cohomology. The next subject is \(G\)-orbits from which standard modules arise. Here duality bridges between modules from \(G\)-orbits and \(K_ c\)-orbits with \(K_ c\) a complexified maximal compact subgroup of the semisimple Lie group \(G\). Different types of standard modules are identified with each other and, finally, the example, where \(G=SL(2,C)\), is examined in greater detail. Though the subject of “geometric quantization” is mentioned once in these lectures, no hint has been given as to how the theory could possibly be applied to physics.

22E30 Analysis on real and complex Lie groups
22E46 Semisimple Lie groups and their representations