KLR and Schur algebras for curves and semi-cuspidal representations

TL;DR

This paper introduces analogues of KLR and Schur algebras for smooth curves using torsion sheaves, providing geometric realizations and revealing Morita equivalences, with implications for the theory of parity sheaves.

Contribution

It develops a geometric framework for KLR and Schur algebras on curves, extending their theory beyond quivers and exploring their categorical properties.

Findings

Curve Schur algebra is Morita equivalent to the imaginary semi-cuspidal category of the Kronecker quiver.

Provides a geometric realization for certain affinized symmetric algebras.

Argues against the existence of a reasonable parity sheaves theory for affine quivers.

Abstract

Given a smooth curve $C$, we define and study analogues of KLR algebras and quiver Schur algebras, where quiver representations are replaced by torsion sheaves on $C$. In particular, they provide a geometric realization for certain affinized symmetric algebras. When $C=\mathbb P^1$, a version of curve Schur algebra turns out to be Morita equivalent to the imaginary semi-cuspidal category of the Kronecker quiver in any characteristic. As a consequence, we argue that one should not expect to have a reasonable theory of parity sheaves for affine quivers.