Mathematics – Representation Theory
Scientific paper
2007-11-06
J. Amer. Math. Soc. 24 (2011), 133-181
Mathematics
Representation Theory
51 pages; main constructions are now carried out over the integers (with 2 inverted)
Scientific paper
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 ch k>0) or to modules over the small quantum group (for ch 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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