Twisted Crystalline Differential Operators

Motivated by the notion of reduced enveloping algebra, we introduce central reductions of universal enveloping Dalgebras of restricted Lie algebroids. In the case of a tangent Lie algebroid the central reductions are sheaves of twisted crystalline differential operators. For a nilpotent p-character of a classical semisimple Lie algebra, we discover a connection between the reduced enveloping algebra and the sheaf of twisted crystalline differential operators on the Frobenius neighborhood of the Springer fiber of the p-character. In positive characteristic, a relation between g-modules andD-modules on the flag variety X involves more complicated rings of twisted differential operators. The basic ring of differential operators related to g-modules is the ring of crystalline differential operators, i.e., the enveloping D-algebra U(TX) of the Lie algebroid TX of vector fields. We examine its quotients obtained by reducing the center, more precisely the p-part of the center which is the sheaf of functions on the Frobenius twist of T X. A closed subscheme Y of X and a section ω of T X over Y define a subscheme ω(Y ) of T X, hence a quotient D Y of U(TX). For χ ∈ g , using a Springer fiber Y = X and a section ω over Y , one defines a sheaf of twisted differential operators Dχ = D ω Y . This is a localization of the quotient Uχ of U(g) defined by χ. We start by discussing Frobenius twists in section 1. A construction of a central reduction of the universal enveloping D-algebra is introduced in section 2. We study crystalline differential operators in section 3, proving in Theorem 3.3.1 that crystalline differential operators form a locally matrix algebra over the twisted cotangent bundle. We examine the flag variety of a classical semisimple Lie algebra g in section 4. There we prove the main result of the present paper (Theorem 4.1.3), giving a direct link between the Springer fiber of χ ∈ g and the reduced enveloping algebra Uχ(g). Date: 19 September 1999.