A geometric interpretation of the Delta Conjecture

Abstract

We introduce a variety $Y_{n,k}$, which we call the \textit{affine $\hat{I}”$-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 ${\mathrm{rev}}_q\ddot{I}‰\hat{I}{”}_{e_{k-1}}^{\prime }{e}_n$ 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 $\hat{I}”$-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.

39 pages