Affine Schubert calculus and double coinvariants

Abstract

We define an action of the double coinvariant algebra $D{R}_n$ on the equivariant Borel-Moore homology of the affine flag variety ${\widetilde{Fl}}_n$ in type $A$, which has an explicit form in terms of the left and right action of the (extended) affine Weyl group and multiplication by Chern classes. Up to first order in the augmentation ideal, we show that it coincides with the action of the Cherednik algebra on the equivariant homology of the homogeneous affine Springer fiber ${\tilde{S}}_{n,n+1}\subset {\widetilde{Fl}}_n$ due to Yun and the second author, and therefore preserves the non-equivariant Borel-Moore homology groups $H_{\ast }({\tilde{S}}_{n,n+1})\hookrightarrow H_{\ast }({\widetilde{Fl}}_n)$. We then define a geometric filtration $F_a{H}_{\ast }({\tilde{S}}_{n,n+1})=H_{\ast }(\tilde{S}(a))$ by closed subspaces $\tilde{S}(a)\subset {\tilde{S}}_{n,n+1}$, which we prove recovers the Garsia-Stanton descent order on $D{R}_n$. We use this to deduce an explicit monomial basis of $D{R}_n$, as well as an independent proof of the (non-compositional) Shuffle Theorem.

71 pages, correccted typos, improved exposition, explained and used previously-known results by Iarrobino and Goettsche on the geometry of the spaces that we use; some typos are corrected

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Cited by in corpus (3)

• The Hilbert series of the superspace coinvariant ring
• Lefschetz theory for exterior algebras and fermionic diagonal coinvariants
• Affine Springer Fibers, Procesi bundles, and Cherednik algebras