Semi-infinite orbits in affine flag varieties and homology of affine Springer fibers

Let G be a connected reductive group over an algebraically closed field k, and let $\operatorname {Fl}$ be the affine flag variety of G. For every regular semisimple element $\gamma $ of $G(k((t)))$ , the affine Springer fiber $\operatorname {Fl}_\gamma $ can be presented as a union of closed subvarieties $\operatorname {Fl}<^>{\leq w}_{\gamma }$ , defined as the intersection of $\operatorname {Fl}_{\gamma }$ with an affine Schubert variety $\operatorname {Fl}<^>{\leq w}$ .The main result of this paper asserts that if elements $w_1,\ldots ,w_n$ are sufficiently regular, then the natural map $H_i(\bigcup _{j=1}<^>n \operatorname {Fl}<^>{\leq w_j}_{\gamma })\to H_i(\operatorname {Fl}_{\gamma })$ is injective for every $i\in \mathbb Z$ . It plays an important role in our work [BV], where our result is used to construct good filtrations of $H_i(\operatorname {Fl}_{\gamma })$ . Along the way, we also show that every affine Schubert variety can be written as an intersection of closures of semi-infinite orbits.