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
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