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

TL;DR

This paper studies the homology of affine Springer fibers in affine flag varieties, proving that unions of certain regular affine Schubert subvarieties inject into the homology of the entire fiber, with implications for orbital integrals.

Contribution

It establishes injectivity of homology maps from unions of regular affine Schubert varieties into affine Springer fibers, and shows affine Schubert varieties as intersections of semi-infinite orbit closures.

Findings

Injectivity of homology maps for unions of regular affine Schubert varieties.

Affine Schubert varieties can be expressed as intersections of semi-infinite orbit closures.

Provides a categorification of weighted orbital integrals.

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 $Fl_{\gamma}$ can be presented as a union of closed subvarieties $Fl^{\leq w}_{\gamma}$, defined as the intersection of $Fl_{\gamma}$ 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 $H_i(\bigcup_{j=1}^n Fl^{\leq w_j}_{\gamma})\to H_i(Fl_{\gamma})$ 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.