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Kazhdan-Lusztig conjecture for a symmetrizable Kac-Moody Lie algebra. (English) Zbl 0727.17013

The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. II, Prog. Math. 87, 407-433 (1990).
[For the entire collection see Zbl 0717.00009.]
The purpose of the paper under review is to generalize the results of J. L. Brylinski and the author [Invent. Math. 64, 387-410 (1981; Zbl 0473.22009)] and A. Beilinson and J. Bernstein [C. R. Acad. Sci., Paris, Sér. I 292, 15-18 (1981; Zbl 0476.14019)] to the symmetrizable Kac-Moody algebra case by using the infinite dimensional flag varieties, established earlier by the author. The main result is that the generalization of the Kazhdan-Lusztig conjecture for symmetrizable Kac-Moody algebras is true. The proof is not much different from the one of Brylinski-Kashiwara. However, since the flag variety is infinite dimensional in general, special care is needed.
Let \(X=G/B_ -\) be the flag variety of a symmetrizable Kac-Moody algebra \({\mathfrak g}\) and \({\mathcal D}_ X\) be the sheaf of differential operators on X. For an integrable dominant weight \(\lambda\), let \({\mathcal O}(\lambda)\) be the corresponding line bundle over X and let \({\mathcal D}_{\lambda}={\mathcal O}(\lambda)\otimes {\mathcal D}_ X\otimes {\mathcal O}(- \lambda)\) be the twisted ring of differential operators on X. For an admissible \({\mathfrak g}\)-module M, the functor from the category of admissible \({\mathcal D}_{\lambda}\)-modules \({\mathcal M}\mapsto Hom_{{\mathfrak g}}(M,\Gamma (X,{\mathcal M}))\) is representable in the category of admissible \({\mathcal D}_{\lambda}\)-modules. Denote by \({\mathcal D}_{\lambda}{\hat \otimes}M\) the admissible \({\mathcal D}_{\lambda}\)- module that represents the functor above. If M is a (\({\mathfrak g},B)\)- module, then \({\mathcal D}_{\lambda}\otimes M\) is a B-equivariant \({\mathcal D}_{\lambda}\)-module. For an admissible B-equivariant \({\mathcal D}_{\lambda}\)-module \({\mathcal M}\), set \(\tilde H^ n(X,{\mathcal M})=\oplus_{\mu \in P}\lim_{\overset\leftarrow U}H^ n(U,{\mathcal M})_{\mu},\) where U ranges over quasi-compact B-stable open subsets. The main theorem says that, for an admissible B-equivariant \({\mathcal D}_{\lambda}\)-module \({\mathcal M}\), (1) \(\tilde H^ n(X,{\mathcal M})=0\) for any \(n\neq 0\). (2) \({\mathcal D}_{\lambda}\otimes {\tilde \Gamma}(X,{\mathcal M})\to {\mathcal M}\) is an isomorphism.
Reviewer: H.Yamada (Tokyo)

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
14M15 Grassmannians, Schubert varieties, flag manifolds
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)