A geometric interpretation of the Delta Conjecture

We introduce a variety $Y_{n,k}$, which we call the $\mathit{\text{affine}}\mathrm{\Delta }\mathit{\text{-Springer}}\mathit{\text{fiber}}$, 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 ${\mathrm{r}\mathrm{e}\mathrm{v}}_q\omega {\mathrm{\Delta }}_{e_{k-1}}^{\mathrm{\prime }}{e}_n$ 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 $\mathrm{\Delta }$-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.