On the cohomological purity of the affine Springer fibers
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910195844644864 |
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| author | Chen, Zongbin |
| author_facet | Chen, Zongbin |
| contents | We address questions posed by Gérard Laumon and Jean-Loup Waldspurger concerning the cohomological purity of affine Springer fibers. More precisely, we show that an affine Springer fiber is cohomologically pure if and only if its $ξ$-stable quotient is cohomologically pure, and that this is further equivalent to the cohomological purity of a certain sequence of truncated affine Springer fibers. We deduce from this a sheaf-theoretic reformulation of cohomological purity for affine Springer fibers. We then compare this new criterion with a previously known one via a microlocal analysis of the relevant intersection complexes. As a corollary, we show that both the primitive part of the cohomology of an affine Springer fiber and the cohomology of its $ξ$-stable quotient depend only on the root valuation datum of the defining element. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_05596 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the cohomological purity of the affine Springer fibers Chen, Zongbin Algebraic Geometry We address questions posed by Gérard Laumon and Jean-Loup Waldspurger concerning the cohomological purity of affine Springer fibers. More precisely, we show that an affine Springer fiber is cohomologically pure if and only if its $ξ$-stable quotient is cohomologically pure, and that this is further equivalent to the cohomological purity of a certain sequence of truncated affine Springer fibers. We deduce from this a sheaf-theoretic reformulation of cohomological purity for affine Springer fibers. We then compare this new criterion with a previously known one via a microlocal analysis of the relevant intersection complexes. As a corollary, we show that both the primitive part of the cohomology of an affine Springer fiber and the cohomology of its $ξ$-stable quotient depend only on the root valuation datum of the defining element. |
| title | On the cohomological purity of the affine Springer fibers |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2605.05596 |