Torus Equivariant Cohomology for the $\Delta$-Springer Fiber

TL;DR

This paper introduces a new torus action on $ abla$-Springer varieties and provides an explicit presentation of their equivariant cohomology rings using advanced algebraic techniques.

Contribution

It defines a specific torus action on $ abla$-Springer varieties and derives a Borel-style presentation for their equivariant cohomology rings.

Findings

Established a torus $U$ acting on $ abla$-Springer varieties.

Derived a Borel-style presentation for the equivariant cohomology ring.

Utilized orbit harmonics deformation and advanced algebraic methods.

Abstract

We define a torus $U \subset T = (\mathbb{C}^\times)^K$ which acts on the $\Delta$-Springer varieties $Y_{n,\lambda,s}$ defined by Griffin-Levinson-Woo and give a Borel-style presentation for the equivariant cohomology ring $H^*_U(Y_{n,\lambda,s})$. Our presentation arises from the orbit harmonics deformation technique, and uses methods of Chou-Matsumura-Rhoades and Abe-Horiguchi.