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

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 $\gamma$ of $G(k((t)))$, the affine Springer fiber $F{l}_{\gamma }$ can be presented as a union of closed subvarieties $F{l}_{\gamma }^{\le w}$, defined as the intersection of $F{l}_{\gamma }$ with an affine Schubert variety $F{l}^{\le w}$. The main result of this paper asserts that if elements $w_1,\dots ,w_n$ are sufficiently regular, then the natural map $H_i({\bigcup }_{j=1}^{n}F{l}_{\gamma }^{\le w_j})\to H_i(F{l}_{\gamma })$ 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.

arXiv comment34 pages, revised version

References in corpus (1)

• Affine Springer fibers and depth zero L-packets