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

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:

 2.2e+46 Representations of Lie and linear algebraic groups over real fields: analytic methods