Mathematics – Representation Theory
Scientific paper
2002-05-14
Mathematics
Representation Theory
Bits and ends cleaned up, assumptions on p improved in some places. Appendix by Simon Riche and R.B. added
Scientific paper
We observe that on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of characteristic $p> h$ (where $h$ is the Coxeter number), with a given (generalized) central character are the same as the coherent sheaves on (generalized) Springer fibers. 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, the algebra of (``crystalline'') differential operators is an Azumaya algebra and its splittings on Springer fibers allow us to pass from D-modules to coherent sheaves. As an application we compute the rank of the Grothendieck group of the category of modules over the Lie algebra with a fixed central character.
Bezrukavnikov Roman
Mirkovic Ivan
Rumynin Dmitriy
No associations
LandOfFree
Localization of modules for a semisimple Lie algebra in prime characteristic does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Localization of modules for a semisimple Lie algebra in prime characteristic, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Localization of modules for a semisimple Lie algebra in prime characteristic will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-652560