A geometric interpretation of the Delta Conjecture

TL;DR

This paper introduces a geometric object called the affine Δ-Springer fiber, linking its homology to the Delta Conjecture and providing a geometric interpretation for the Rational Shuffle Theorem, thus bridging algebraic combinatorics and geometry.

Contribution

It defines the affine Δ-Springer fiber and establishes its homology as a geometric realization of the Delta Conjecture and Rational Shuffle Theorem.

Findings

Homology of the affine Δ-Springer fiber matches the Delta Conjecture symmetric function.

Provides a geometric interpretation for the Rational Shuffle Theorem.

Connects affine Springer fibers with combinatorial symmetric functions.

Abstract

We introduce a variety $Y_{n,k}$, which we call the \textit{affine $\Delta$-Springer fiber}, generalizing the affine Springer fiber studied by Hikita, whose Borel-Moore homology has an $S_n$ action and a bigrading that corresponds to the Delta Conjecture symmetric function $\mathrm{rev}_q\,\omega \Delta'_{e_{k-1}}e_n$ under the Frobenius character map. We similarly provide a geometric interpretation for the Rational Shuffle Theorem in the integer slope case $(km,k)$. The variety $Y_{n,k}$ has a map to the affine Grassmannian whose fibers are the $\Delta$-Springer fibers introduced by Levinson, Woo, and the third author. Part of our proof of our geometric realization relies on our previous work on a Schur skewing operator formula relating the Rational Shuffle Theorem to the Delta Conjecture.