Deformed W-algebras in Type A for Rectangular Nilpotent

Abstract

For any \(n,r \in {\mathbb {N}}\), we construct an algebra \({\mathcal {P}}_n^r\) via generators and quadratic relations, and show that it deforms the W–algebra of \({\mathfrak {gl}}_{nr}\) with respect to a nilpotent with Jordan block decomposition \(r+ \ldots +r\). We introduce a surjective morphism from half of the quantum toroidal algebra of \({\mathfrak {gl}}_n\) to \({\mathcal {P}}_n^r\), and show that the action of the quantum toroidal algebra on the K–theory groups of the moduli spaces of parabolic sheaves factors through \({\mathcal {P}}_n^r\).