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 F1 be the affine flag variety of G. For every regular semisimple element γ of G(k((t))), the affine Springer fiber Fl_γ can be presented as a union of closed subvarieties Fl_γ^≤w, defined as the intersection of Fl_γ with an affine Schubert variety Fl^≤w. The main result of this paper asserts that if elements w₁, … w_n are sufficiently regular, then the natural map (Figure presented) is injective for every i ∈ Z. It plays an important role in our work [BV], where our result is used to construct good filtrations of H_i(Fl_γ). Along the way, we also show that every affine Schubert variety can be written as an intersection of closures of semi-infinite orbits.