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Journal of the American Mathematical Society

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Sheaves on affine Schubert varieties, modular representations, and Lusztig’s conjecture

Author: Peter Fiebig
Journal: J. Amer. Math. Soc. 24 (2011), 133-181
MSC (2010): Primary 20C20; Secondary 55N30
Published electronically: September 23, 2010
MathSciNet review: 2726602
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Abstract: We relate a certain category of sheaves of $k$-vector spaces on a complex affine Schubert variety to modules over the $k$-Lie algebra (for $\operatorname {char} k>0$) or to modules over the small quantum group (for $\operatorname {char} k=0$) associated to the Langlands dual root datum. As an application we give a new proof of Lusztig’s conjecture on quantum characters and on modular characters for almost all characteristics. Moreover, we relate the geometric and representation-theoretic sides to sheaves on the underlying moment graph, which allows us to extend the known instances of Lusztig’s modular conjecture in two directions: We give an upper bound on the exceptional characteristics and verify its multiplicity-one case for all relevant primes.

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Additional Information

Peter Fiebig
Affiliation: Department Mathematik, Universität Erlangen-Nürnberg, Bismarckstr. $1\frac {1}{2}$, 91054 Erlangen, Germany

Received by editor(s): June 24, 2008
Received by editor(s) in revised form: November 26, 2009, and July 16, 2010
Published electronically: September 23, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.