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

Fiebig, Peter

doi:10.1090/S0894-0347-2010-00679-0

Sheaves on affine Schubert varieties, modular representations, and Lusztig’s conjecture
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J. Amer. Math. Soc. 24 (2011), 133-181 Request permission

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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