Semi-infinite orbits in affine flag varieties and homology of affine Springer fibers
Roman Bezrukavnikov, Yakov Varshavsky
TL;DR
This work analyzes the homology of affine Springer fibers $Fl_{\gamma}$ for regular semisimple $\gamma$ by developing a detailed combinatorial and geometric framework for semi-infinite orbits in the affine flag variety. The authors introduce admissible tuples and semi-infinite stratifications to express affine Schubert varieties as intersections of semi-infinite orbit closures, and they prove an injectivity result: for sufficiently regular $m$-regular elements $w$ in the affine Weyl group, the natural map $H_i(\bigcup_{j=1}^n Fl^{\leq w_j}_{\gamma})\to H_i(Fl_{\gamma})$ is injective for all $i$. This yields a filtration on $H_*(Fl_{\gamma})$ compatible with the affine Springer action and related to weighted orbital integrals, providing a categorification of those integrals. The approach hinges on a combination of localization in equivariant cohomology, stratifications by semi-infinite orbits, and an inductive argument on the semisimple rank, with finite-type reductions and affine-bundle geometry playing key technical roles.
Abstract
Let $G$ be a connected reductive group over an algebraically closed field $k$, and let $Fl$ 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 $Fl^{\leq w}_γ$, defined as the intersection of $Fl_γ$ with an affine Schubert variety $Fl^{\leq w}$. The main result of this paper asserts that if elements $w_1,\ldots,w_n$ are sufficiently regular, then the natural map $H_i(\bigcup_{j=1}^n Fl^{\leq w_j}_γ)\to H_i(Fl_γ)$ is injective for every $i\in{\mathbb Z}$. It plays an important role in our work [BV]. One can view this statement as providing a categorification of the notion of a weighted orbital integral. Along the way we also show that every affine Schubert variety can be written as an intersection of closures of semi-infinite orbits.