Semi-infinite orbits in affine flag varieties and homology of affine Springer fibers
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866929719129145344 |
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| author | Bezrukavnikov, Roman Varshavsky, Yakov |
| author_facet | Bezrukavnikov, Roman Varshavsky, Yakov |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_13213 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Semi-infinite orbits in affine flag varieties and homology of affine Springer fibers Bezrukavnikov, Roman Varshavsky, Yakov Algebraic Geometry Representation Theory 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. |
| title | Semi-infinite orbits in affine flag varieties and homology of affine Springer fibers |
| topic | Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2104.13213 |