Motivic homotopy theory of the classifying stack of finite groups of Lie type
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866929437574955008 |
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| author | Yaylali, Can |
| author_facet | Yaylali, Can |
| contents | Let $G$ be a reductive group over $\mathbb{F}_{p}$ with associated finite group of Lie type $G^{F}$. Let $T$ be a maximal torus contained inside a Borel $B$ of $G$. We relate the (rational) Tate motives of $\text{B}G^{F}$ with the $T$-equivariant Tate motives of the flag variety $G/B$. On the way, we show that for a reductive group $G$ over a field $k$, with maximal Torus $T$ and absolute Weyl group $W$, acting on a smooth finite type $k$-scheme $X$, we have an isomorphism $A^{n}_{G}(X,m)_{\mathbb{Q}}\cong A^{n}_{T}(X,m)_{\mathbb{Q}}^{W}$ extending the classical result of Edidin-Graham to higher equivariant Chow groups in the non-split case. We also extend our main result to reductive group schemes over a regular base that admit maximal tori. Further, we apply our methods to more general quotient stacks. In this way, we are able to compute the motive of the stack of $G$-zips introduced by Pink-Wedhorn-Ziegler for reductive groups over fields of positive characteristic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_09808 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Motivic homotopy theory of the classifying stack of finite groups of Lie type Yaylali, Can Algebraic Geometry K-Theory and Homology Representation Theory 14C15 (Primary) 20C33, 14A20 (Secondary) Let $G$ be a reductive group over $\mathbb{F}_{p}$ with associated finite group of Lie type $G^{F}$. Let $T$ be a maximal torus contained inside a Borel $B$ of $G$. We relate the (rational) Tate motives of $\text{B}G^{F}$ with the $T$-equivariant Tate motives of the flag variety $G/B$. On the way, we show that for a reductive group $G$ over a field $k$, with maximal Torus $T$ and absolute Weyl group $W$, acting on a smooth finite type $k$-scheme $X$, we have an isomorphism $A^{n}_{G}(X,m)_{\mathbb{Q}}\cong A^{n}_{T}(X,m)_{\mathbb{Q}}^{W}$ extending the classical result of Edidin-Graham to higher equivariant Chow groups in the non-split case. We also extend our main result to reductive group schemes over a regular base that admit maximal tori. Further, we apply our methods to more general quotient stacks. In this way, we are able to compute the motive of the stack of $G$-zips introduced by Pink-Wedhorn-Ziegler for reductive groups over fields of positive characteristic. |
| title | Motivic homotopy theory of the classifying stack of finite groups of Lie type |
| topic | Algebraic Geometry K-Theory and Homology Representation Theory 14C15 (Primary) 20C33, 14A20 (Secondary) |
| url | https://arxiv.org/abs/2306.09808 |