Localization of modules for a semisimple Lie algebra in prime characteristic (with an Appendix by R. Bezrukavnikov and S. Riche: Computation for sl(3))

By R. Bezrukavnikov, I. Mirkovi'c, D. Rumynin. 2002

223 citations recorded by the source corpus.

We show that, on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of finite characteristic with a given (generalized) regular central character are the same as coherent sheaves on the formal neighborhood of the corresponding (generalized) Springer fiber. The first step is to observe that the derived functor of global sections provides an equivalence between the derived category of D-modules (with no divided powers) on the flag variety and the appropriate derived category of modules over the corresponding Lie algebra. Thus the “derived” version of the Beilinson-Bernstein localization theorem holds in sufficiently large positive characteristic. Next, one finds that for any smooth variety this algebra of differential operators is an Azumaya algebra on the cotangent bundle. In the case of the flag variety it splits on Springer fibers, and this allows us to pass from D-modules to coherent sheaves. The argument also generalizes to twisted D-modules. As an application we prove Lusztig’s conjecture on the number of irreducible modules with a fixed central character. We also give a formula for behavior of dimension of a module under translation functors and reprove the Kac-Weisfeiler conjecture. The sequel to this paper [BMR2] treats singular infinitesimal characters.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1f09afc7f67bdbff69b25cefce19588a15d28c2d

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Citations in the corpus

Fedosov quantization in positive characteristic (2005, unknown): “This phenomenon, already observed in [ BMR ], allows for interesting applications (see e.g.”
Dualizing complexes and perverse sheaves on noncommutative ringed schemes (2002, background): “Also A is an Azumaya algebra over its center; see [ BMR , Theorem 2.2.3].”
McKay equivalence for symplectic resolutions of singularities (2004, methodology): “ee e.g. [Ka1]). The proof uses reduction to positive characteristic, and quantization of the symplectic variety X k over a ﬁeld k of characteristic p > 0. Our method is suggested by the results of [BMR]. The key ingredient of the proof is a quantization of X k whose global sections coincide with ”
Nonabelian Hodge theory in characteristic p (2005, methodology): “We begin by reviewing in Theorem 2.1 the Azumaya property of the algebra of differential operators and the canonical fppf splitting module described in [4].”
Cohomology of tilting modules over quantum groups and t-structures on derived categories of coherent sheaves (2004, background): “N appears in some other contexts, e.g., it is related to the t-structures appearing in [ BMR ], and, conjecturally, 2 to modules over the affine Lie algebra on the critical level studied by Frenkel and Gaitsgory, cf. [FG].”