##
**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.

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.

Reviewer: Michael Kuznetsov (Nizhnij Novgorod)

### 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 |