Springer fibers and the Delta Conjecture at $t=0$

TL;DR

This paper introduces elta Springer varieties, generalizes Springer fibers, and connects their cohomology to the Delta Conjecture at t=0, providing new geometric and algebraic insights.

Contribution

It defines elta Springer varieties, describes their cohomology rings, and relates them to the Delta Conjecture and schemes of diagonal nilpotent matrices.

Findings

Cohomology rings have symmetric group actions generalizing Springer actions.

Top cohomology groups are induction products of Specht modules.

Provides geometric realization for the Delta Conjecture at t=0.

Abstract

We introduce a family of varieties $Y_{n,\lambda,s}$, which we call the \emph{$\Delta$-Springer varieties}, that generalize the type A Springer fibers. We give an explicit presentation of the cohomology ring $H^*(Y_{n,\lambda,s})$ and show that there is a symmetric group action on this ring generalizing the Springer action on the cohomology of a Springer fiber. In particular, the top cohomology groups are induction products of Specht modules with trivial modules. The $\lambda=(1^k)$ case of this construction gives a compact geometric realization for the expression in the Delta Conjecture at $t=0$. Finally, we generalize results of De Concini and Procesi on the scheme of diagonal nilpotent matrices by constructing an ind-variety $Y_{n,\lambda}$ whose cohomology ring is isomorphic to the coordinate ring of the scheme-theoretic intersection of an Eisenbud--Saltman rank variety and diagonal matrices.