Quantization of Drinfeld Zastava in type A

TL;DR

This paper constructs a quantization of the Drinfeld Zastava space in type A using Hamiltonian reduction and shows it is a quotient of the affine Borel Yangian for generic parameters.

Contribution

It introduces a new affine quiver variety related to Zastava spaces and establishes its quantization as a quotient of the affine Borel Yangian.

Findings

Constructed a new affine quiver variety $Z$ mapping bijectively to Zastava space.

Described the Poisson structure on $Z$ via Hamiltonian reduction.

Proved the quantization $Y$ is a quotient of the affine Borel Yangian for generic parameters.

Abstract

Drinfeld Zastava is a certain closure of the moduli space of maps from the projective line to the Kashiwara flag scheme of the affine Lie algebra $\hat{sl}_n$. We introduce an affine, reduced, irreducible, normal quiver variety $Z$ which maps to the Zastava space bijectively at the level of complex points. The natural Poisson structure on the Zastava space can be described on $Z$ in terms of Hamiltonian reduction of a certain Poisson subvariety of the dual space of a (nonsemisimple) Lie algebra. The quantum Hamiltonian reduction of the corresponding quotient of its universal enveloping algebra produces a quantization $Y$ of the coordinate ring of $Z$. The same quantization was obtained in the finite (as opposed to the affine) case generically in arXiv:math/0409031. We prove that, for generic values of quantization parameters, $Y$ is a quotient of the affine Borel Yangian.