Quasi-equivariant \({\mathcal D}\)-modules, equivariant derived category, and representations of reductive Lie groups. (English) Zbl 0854.22014

Kashiwara, Masaki; Schmid, Wilfried

Brylinski, Jean-Luc (ed.) et al., Lie theory and geometry: in honor of Bertram Kostant on the occasion of his 65th birthday. Invited papers, some originated at a symposium held at MIT, Cambridge, MA, USA in May 1993. Boston, MA: Birkhäuser. Prog. Math. 123, 457-488 (1994).

The main theme of this paper is the construction of a real reductive Lie group \({\mathbf G}_\mathbb{R}\) through the theory of \({\mathcal D}_X\)-module analysis on the flag variety \(X\) of the complexification \(\mathbf G\) of \({\mathbf G}_\mathbb{R}\). The authors realize the representation of \({\mathbf G}_\mathbb{R}\) on some kind of cohomology spaces on \(X\), especially using the derived category of \({\mathcal D}_X\)-modules and Riemann-Hilbert correspondence. They propose six conjectures on these representations and their properties, and give an explanation concerning the proofs of these conjectures. Their results, in a more detailed form including complete proofs, will be published in the future.
For the entire collection see [Zbl 0807.00014].

Reviewer: M.Muro (Yanagido)

MSC:

22E45

Representations of Lie and linear algebraic groups over real fields: analytic methods

Keywords:

real reductive Lie group; \({\mathcal D}_ X\)-module; flag variety; representation; Riemann-Hilbert correspondence