Yangians and cohomology rings of Laumon spaces

TL;DR

The work builds a bridge between the geometry of Laumon and affine Laumon spaces and the representation theory of Yangians and affine Yangians by constructing actions through natural correspondences and fixed-point bases. It identifies an affine Gelfand-Tsetlin basis and shows the affine GT subalgebra surjects onto the localized cohomology rings, while the center of $Y({\mathfrak{gl}}_n)$ maps into this structure to realize $H^{\bullet}(M_{n,d})$ as its image; notably, $c_1(\mathcal D_0)$ is captured by a noncommutative power sum. The paper also develops a parallel affine theory via parabolic/ orbifold realizations, proves explicit matrix coefficients for the affine Yangian action in fixed-point bases, and connects cohomology ring generation to Künneth components and Okounkov’s vector bundle $E$ through Beilinson-type arguments. Together, these results illuminate how geometric correspondences encode rich algebraic structures and furnish concrete descriptions of central and tautological operators on cohomology, with potential links to integrable representations and crystal bases in the affine setting.

Abstract

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of $GL_n$. We construct the action of the Yangian of $sl_n$ in the cohomology of Laumon spaces by certain natural correspondences. We construct the action of the affine Yangian (two-parametric deformation of the universal enveloping algebra of the universal central extension of $sl_n[s^{\pm1},t]$) in the cohomology of the affine version of Laumon spaces. We compute the matrix coefficients of the generators of the affine Yangian in the fixed point basis of cohomology. This basis is an affine analogue of the Gelfand-Tsetlin basis. The affine analogue of the Gelfand-Tsetlin algebra surjects onto the equivariant cohomology rings of the affine Laumon spaces. The cohomology ring of the moduli space $M_{n,d}$ of torsion free sheaves on the plane, of rank $n$ and second Chern class $d$, trivialized at infinity, is naturally embedded into the cohomology ring of certain affine Laumon space. It is the image of the center $Z$ of the Yangian of $gl_n$ naturally embedded into the affine Yangian. In particular, the first Chern class of the determinant line bundle on $M_{n,d}$ is the image of a noncommutative power sum in $Z$.

Theorems & Definitions (42)

• Theorem 2.7
• Theorem 2.9
• Theorem 2.12
• Corollary 2.14
• Lemma 3.3
• Lemma 3.5
• Theorem 3.9
• Remark 3.10
• Theorem 3.11
• Conjecture 3.13