## Localization of modules for a semisimple Lie algebra in prime characteristic (with an appendix by R. Bezrukavnikov and S. Riche, Computation for $$\text{sl}(3))$$.(English)Zbl 1220.17009

In the paper the geometrical technique for the representation theory of Lie algebra $$\mathfrak{g}$$ of semisimple group $$G$$ over an algebraically closed field of characteristic $$p$$ is developed. It is supposed that $$p$$ is greater than the Coxeter number of $$G$$ though some intermediate statements are proved under weaker assumptions.
The work concentrates around the main result which is an analog of the Beilinson-Bernstein localization theorem in characteristic zero and states that the bounded derived category of $$\mathfrak{g}$$-modules with a given regular generalized Harish-Chandra central character is canonically equivalent under the derived functor of global sections to the bounded derived category of the appropriately twisted $$\mathcal{D}$$-modules on the flag variety $$\mathcal{B}$$, where $$\mathcal{D}=\mathcal{D}_{\mathcal{B}}$$ is the sheaf of crystalline differential operators (i.e. operators without divided powers).
Let $$\chi \in \mathfrak{g}^*, ~\mathcal{B}_{\chi}^{(1)}$$ be the Frobenius twist of generalized Springer fiber. It is shown that the Azumaya algebra of twisted differential operators splits on the formal neighborhood of $$\mathcal{B}_{\chi}$$ in the twisted cotangent bundle. It follows that the category of twisted $$\mathcal{D}$$-modules and that of coherent sheaves both supported on $$\mathcal{B}_{\chi}^{(1)}$$ are equivalent.
The authors demonstrate the potential of this approach by proving Lusztig’s conjecture on the number of irreducible modules with a fixed central character $$\chi$$ for the case when $$\chi$$ is a regular Harish-Chandra central character. In particular, a canonical isomorphism of Grothendieck groups of $$\mathfrak{g}$$-modules with a character like that and of coherent sheaves on the Springer fiber $$\mathcal{B}_{\chi}$$ is found. It is shown that this $$K$$-group has no torsion and its rank equals the dimension of the cohomology of the corresponding Springer fiber over a field of characteristic zero. As another application the authors investigate an interaction between the localization functor and the translation functors and reprove the Kac-Weisfeiler conjecture which was proved by A. Premet under less rigid assumptions on characteristic of the ground field.

### MSC:

 17B50 Modular Lie (super)algebras 18E30 Derived categories, triangulated categories (MSC2010) 18F30 Grothendieck groups (category-theoretic aspects) 16S32 Rings of differential operators (associative algebraic aspects) 20G05 Representation theory for linear algebraic groups
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