Affine Schubert calculus and double coinvariants

We first define an action of the double coinvariant algebra DR_n on the homology of the affine flag variety Fl_n in type A, and use affine Schubert calculus to prove that it preserves the image of the homology of the rational (n,m)-affine Springer fiber H_*(S̃_n,m)⊂ H_*(Fl_n) under the pushforward of the inclusion map. In our main result, we define a filtration by ℚ[𝐱]-submodules of DR_n≅ H_*(S̃_n,n+1) indexed by compositions, whose leading terms are the Garsia-Stanton "descent monomials" in the y-variables. We find an explicit presentation of the subquotients as submodules of the single-variable coinvariant algebra R_n(x)≅ H_*(Fl_n), by identifying the leading torus fixed points with a subset ℋ⊂ S_n of the torus fixed points of the regular nilpotent Hessenberg variety, and comparing them to a cell decomposition of S̃_n,n+1 due to Goresky, Kottwitz, and MacPherson. We also discover an explicit monomial basis of DR_n, and in particular an independent proof of the Haglund-Loehr formula.