A geometric interpretation of the Delta Conjecture

TL;DR

This work constructs affine geometric models—affine ($Δ$-)Springer fibers $X_{n,k}$ and $Y_{n,k}$—to give geometric realizations of the Rectangular Shuffle Theorem and the Delta Conjecture. It proves that $H_*^{BM}(X_{n,k})$ realizes the operator $E_{K,k}\cdot 1$ and that $H_*^{BM}(Y_{n,k})$ realizes $Δ'_{e_{k-1}}e_n$ up to explicit $q,t$-gradings, with a precise Schur-skewing relation between them. The approach weaves together affine Springer theory, Schubert calculus on the affine Grassmannian, and combinatorics of $(K,k)$ parking functions via γ-restricted affine permutations. The results unify algebraic, geometric, and combinatorial perspectives on diagonal coinvariants and related symmetric functions, and point to stable geometric limits and potential extensions to broader Macdonald/Elliptic Hall setups. Overall, the paper provides a robust bridge between geometric representation theory and the combinatorics of shuffle-type symmetric functions, yielding explicit, computable Frobenius characters for key geometric objects.

Abstract

We introduce a variety $Y_{n,k}$, which we call the \textit{affine $Δ$-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 $\mathrm{rev}_q\,ωΔ'_{e_{k-1}}e_n$ 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 $Δ$-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.

Figures (7)

• Figure 1: Examples of $(3,3)$ and $(6,3)$ parking functions
• Figure 2: Ranks for $(K,k)=(12,4)$
• Figure 3: At left, computing pathdinv of the red path $D$ as the number of boxes marked with $\ast$, whose arm and leg satisfy the conditions of Lemma \ref{['lem:alt-pathdinv']}. We therefore have $\mathrm{pathdinv}(D)=9$. The remaining figures show the three types of complementary boxes as used in the proof of Lemma \ref{['lem: codinv of a path']}.
• Figure 4: The map $F$ that shrinks a $(K,k)$-parking function to a stacked parking function by removing the big labels. Here $k=3$ and $n=6$, so $K=9$.
• Figure 5: The labeled rational Dyck path $\pi$ corresponding to $\omega_\pi = [1,5,11,16,6,9,20,10,12,14,15,19]$. The ranks are in bold to the left of the Dyck path, and the parking function is to the right of the path.

Theorems & Definitions (130)

• Remark 1.1
• Definition 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5: GGG1
• Theorem 1.6
• Remark 1.7
• Definition 1.8
• Theorem 1.9: GG
• Theorem 1.10