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

Abstract

We relate a certain category of sheaves of k k -vector spaces on a complex affine Schubert variety to modules over the k k -Lie algebra (for char ⁡ k > 0 \operatorname {char} k>0 ) or to modules over the small quantum group (for char ⁡ k = 0 \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.