Algebraic \({\mathcal D}\)-modules and representation theory of semisimple Lie groups.

*(English)*Zbl 0821.22005
Eastwood, Michael (ed.) et al., The Penrose transform and analytic cohomology in representation theory. AMS-IMS-SIAM summer research conference, June 27 - July 3, 1992, South Hadley, MA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 154, 133-168 (1993).

This is an interesting, readable and fairly self-contained exposition describing a new approach to the classification of the irreducible Harish-Chandra modules using the techniques of algebraic geometry, based in part on the work of H. Hecht, W. Schmid, J. Wolf and the author [cf. e.g. Invent. Math. 90, 297-332 (1987; Zbl 0699.22022)]. In a broad perspective this approach can be viewed as a far-reaching generalization of the classical realization of irreducible representations of compact semi-simple Lie groups described by the theorem of Borel-Weil.

The principal tool employed in the paper is the localization functor of Beilinson and Bernstein which provides means for establishing the equivalence of the category of \({\mathcal U} ({\mathfrak g})\)-modules with an infinitesimal character with the category of \({\mathcal D}\)-modules on the flag variety of \(\mathfrak g\). This in turn induces an equivalence of the category of Harish-Chandra modules with an infinitesimal character with a category of Harish-Chandra sheaves on the flag variety.

Following this scheme the author describes in the first part of the paper the basic notions and constructions of the algebraic theory of \(\mathcal D\)- modules and also presents needed results on the structure of \(K\)-orbits in the flag variety of \(\mathfrak g\). Subsequently a classification of all irreducible Harish-Chandra sheaves and a necessary and sufficient condition for vanishing of cohomology of irreducible Harish-Chandra sheaves is given [cf. the author, Harmonic analysis on reductive groups, Prog. Math. 101, 209-222 (1991; Zbl 0760.22019)]. This leads to a geometric classification of irreducible Harish-Chandra modules in a final part of the paper. The paper closes with a comparison of the given classification with the Langlands classification and a detailed elaboration of the construction for the case of the group \(\text{SU}(2,1)\).

For the entire collection see [Zbl 0780.00026].

The principal tool employed in the paper is the localization functor of Beilinson and Bernstein which provides means for establishing the equivalence of the category of \({\mathcal U} ({\mathfrak g})\)-modules with an infinitesimal character with the category of \({\mathcal D}\)-modules on the flag variety of \(\mathfrak g\). This in turn induces an equivalence of the category of Harish-Chandra modules with an infinitesimal character with a category of Harish-Chandra sheaves on the flag variety.

Following this scheme the author describes in the first part of the paper the basic notions and constructions of the algebraic theory of \(\mathcal D\)- modules and also presents needed results on the structure of \(K\)-orbits in the flag variety of \(\mathfrak g\). Subsequently a classification of all irreducible Harish-Chandra sheaves and a necessary and sufficient condition for vanishing of cohomology of irreducible Harish-Chandra sheaves is given [cf. the author, Harmonic analysis on reductive groups, Prog. Math. 101, 209-222 (1991; Zbl 0760.22019)]. This leads to a geometric classification of irreducible Harish-Chandra modules in a final part of the paper. The paper closes with a comparison of the given classification with the Langlands classification and a detailed elaboration of the construction for the case of the group \(\text{SU}(2,1)\).

For the entire collection see [Zbl 0780.00026].

Reviewer: A.Strasburger (Warszawa)

##### MSC:

22E46 | Semisimple Lie groups and their representations |

32C38 | Sheaves of differential operators and their modules, \(D\)-modules |