Alexandre Minets

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.

Abstract: For any free oriented Borel–Moore homology theory \(A\), we construct an associative product on the \(A\)-theory of the stack of Higgs torsion sheaves over a projective curve \(C\). We show that the resulting algebra \(A\mathbf{Ha}_C^0\) admits a natural shuffle presentation, and prove it is faithful when \(A\) is replaced with usual Borel–Moore homology groups. We also introduce moduli spaces of stable triples, heavily inspired by Nakajima quiver varieties, whose \(A\)-theory admits an \(A\mathbf{Ha}_C^0\)-action. These triples can be interpreted as certain sheaves on \(\mathbb P(\omega_C\oplus \mathcal O_C)\). In particular, we obtain an action of \(A\mathbf{Ha}_C^0\) on the cohomology of Hilbert schemes of points on \(T^*C\).