  We first define an action of the double coinvariant algebra $DR_n$ on the
homology of the affine flag variety $\widetilde{Fl}_n$ 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 $H_*(\tilde{S}_{n,m})\subset
H_*(\widetilde{Fl}_n)$ under the pushforward of the inclusion map. In our main
result, we define a filtration by $\mathbb{Q}[\mathbf{x}]$-submodules of
$DR_n\cong H_*(\tilde{S}_{n,n+1})$ 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 $R_n(x)\cong H_*(Fl_n)$, by identifying the leading torus
fixed points with a subset $\mathcal{H}\subset S_n$ of the torus fixed points
of the regular nilpotent Hessenberg variety, and comparing them to a cell
decomposition of $\tilde{S}_{n,n+1}$ 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.
