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

TL;DR

This paper connects sheaves on affine Schubert varieties to modular and quantum group representations, providing a new proof of Lusztig's conjecture and extending its known validity to more characteristics.

Contribution

It establishes a novel relationship between sheaves on affine Schubert varieties and representation theory, leading to new proofs and extensions of Lusztig's conjecture.

Findings

New proof of Lusztig's conjecture for almost all characteristics.

Extended the range of characteristics where Lusztig's conjecture holds.

Provided bounds on exceptional characteristics and verified multiplicity one cases.

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 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.