Geodesic
Representations of semisimple Lie algebras in prime characteristic and the noncommutative Springer resolution (with an Appendix by Eric Sommers)
Roman Bezrukavnikov1 ; Ivan Mirković2 ; Eric Sommers3
1 Massachusetts Institute of Technology, Cambridge, MA, and Institute for Information Transmission Problems, Moscow, Russia
2 University of Massachusetts, Amherst, MA
3 University of Massachusetts, Amherst, MA
Annals of mathematics, Tome 178 (2013) no. 3, pp. 835-919.

Voir la notice de l'article provenant de la source Annals of Mathematics website

We prove most of Lusztig’s conjectures on the canonical basis in homology of a Springer fiber. The conjectures predict that this basis controls numerics of representations of the Lie algebra of a semisimple algebraic group over an algebraically closed field of positive characteristic. We check this for almost all characteristics. To this end we construct a noncommutative resolution of the nilpotent cone which is derived equivalent to the Springer resolution. On the one hand, this noncommutative resolution is closely related to the positive characteristic derived localization equivalences obtained earlier by the present authors and Rumynin. On the other hand, it is compatible with the $t$-structure arising from an equivalence with the derived category of perverse sheaves on the affine flag variety of the Langlands dual group. This equivalence established by Arkhipov and the first author fits the framework of local geometric Langlands duality. The latter compatibility allows one to apply Frobenius purity theorem to deduce the desired properties of the basis. We expect the noncommutative counterpart of the Springer resolution to be of independent interest from the perspectives of algebraic geometry and geometric Langlands duality.
DOI : 10.4007/annals.2013.178.3.2

Roman Bezrukavnikov 1 ; Ivan Mirković 2 ; Eric Sommers 3

1 Massachusetts Institute of Technology, Cambridge, MA, and Institute for Information Transmission Problems, Moscow, Russia
2 University of Massachusetts, Amherst, MA
3 University of Massachusetts, Amherst, MA 
@article{10_4007_annals_2013_178_3_2,
     author = {Roman Bezrukavnikov and Ivan Mirkovi\'c and Eric Sommers},
     title = {Representations of semisimple {Lie} algebras in prime characteristic and the  noncommutative {Springer} resolution (with an {Appendix} by {Eric} {Sommers)}},
     journal = {Annals of mathematics},
     pages = {835--919},
     publisher = {mathdoc},
     volume = {178},
     number = {3},
     year = {2013},
     doi = {10.4007/annals.2013.178.3.2},
     mrnumber = {3092472},
     zbl = {06220725},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2013.178.3.2/}
}
TY  - JOUR
AU  - Roman Bezrukavnikov
AU  - Ivan Mirković
AU  - Eric Sommers
TI  - Representations of semisimple Lie algebras in prime characteristic and the  noncommutative Springer resolution (with an Appendix by Eric Sommers)
JO  - Annals of mathematics
PY  - 2013
SP  - 835
EP  - 919
VL  - 178
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.4007/annals.2013.178.3.2/
DO  - 10.4007/annals.2013.178.3.2
LA  - en
ID  - 10_4007_annals_2013_178_3_2
ER  - 
%0 Journal Article
%A Roman Bezrukavnikov
%A Ivan Mirković
%A Eric Sommers
%T Representations of semisimple Lie algebras in prime characteristic and the  noncommutative Springer resolution (with an Appendix by Eric Sommers)
%J Annals of mathematics
%D 2013
%P 835-919
%V 178
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.4007/annals.2013.178.3.2/
%R 10.4007/annals.2013.178.3.2
%G en
%F 10_4007_annals_2013_178_3_2
Roman Bezrukavnikov; Ivan Mirković; Eric Sommers. Representations of semisimple Lie algebras in prime characteristic and the  noncommutative Springer resolution (with an Appendix by Eric Sommers). Annals of mathematics, Tome 178 (2013) no. 3, pp. 835-919. doi : 10.4007/annals.2013.178.3.2. http://geodesic.mathdoc.fr/articles/10.4007/annals.2013.178.3.2/

Cité par Sources :