D-modules in various characteristics and Langlands duality
 Ivan Mirkovic (Massachusettes)

Representation theory of a real reductive group largely reduces to the corresponding theory for the Lie algebra. A formalism developed by Beilinson and Bernstein then provides a powerful approach to representations of Lie algebras in a geometric setting of the flag variety in the language of D-modules (modules for the ring of differential operators) and perverse sheaves (essential sheaves of solutions of nice differential equations). It turns out that the the Beilinson-Bernstein formalism also works for Lie algebras over fields of positive characteristic, however in this case perverse sheaves are replaced by something more traditional (a twisted version of) coherent sheaves on flag variety. This reduces the the study of Lie algebra representations (and in particular of algebraic representations of the group) to the questions in the geometry of Springer fibers -- the well know and much studied subvarieties of the flag variety. Finally, verification of the detailed numerical description of this representation theory that was predicted by Lusztig, requires one more step which can be handled by passing to the (complex) Langlands dual group.

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