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Abstract We geometrize the modpSatake isomorphism of Herzig and Henniart–Vignéras using Witt vector affine flag varieties for reductive groups in mixed characteristic. We deduce this as a special case of a formula, stated in terms of the geometry of generalized Mirković–Vilonen cycles, for the Satake transform of an arbitrary parahoric modpHecke algebra with respect to an arbitrary Levi subgroup. Moreover, we prove an explicit formula for the convolution product in an arbitrary parahoric modpHecke algebra. Our methods involve the constant term functors inspired from the geometric Langlands program, and we also treat the case of reductive groups in equal characteristic. We expect this to be a first step toward a geometrization of a modpLocal Langlands Correspondence.
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This content will become publicly available on May 16, 2026
Cellular pavings of fibers of convolution morphisms
This article proves, in the case of split groups over arbitrary fields, that all fibers of convolution morphisms attached to parahoric affine flag varieties are paved by products of affine lines and affine lines minus a point. This applies in particular to the affine Grassmannian and to the convolution morphisms in the context of the geometric Satake correspondence. The second part of the article extends these results over $$\mathbb Z$$. Those in turn relate to the recent work of Cass-van den Hove-Scholbach on the geometric Satake equivalence for integral motives, and provide some alternative proofs for some of their results. Comment: 24 pages. Minor error corrected with the addition of Lemma 7.2. Lemma 7.3 added. Material on triviality of morphisms added to section 5. Minor changes in notation. Published version
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- Award ID(s):
- 2200873
- PAR ID:
- 10635657
- Publisher / Repository:
- Episciences
- Date Published:
- Journal Name:
- Épijournal de Géométrie Algébrique
- Volume:
- Volume 9
- ISSN:
- 2491-6765
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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