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I will describe a new geometric construction, joint with A. Oblomkov, expanding previous work on the diagonal coinvariant algebra $DR_n$. In that work, we identified a monomial basis corresponding to one version of the shuffle theorem, leading to an independent proof based affine Schubert calculus. Our current construction takes the form of a paving of a certain rational affine Springer fiber $Sp_{n,n+1}$ by vector bundles over smooth base spaces, which is closely related to the Hessenberg paving of affine Springer fibers.