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arXiv · 2211.16226

Geometrization of the Satake transform for mod $p$ Hecke algebras

Robert Cass·Yujie Xu
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Abstract

We geometrize the mod $p$ Satake isomorphism of Herzig and Henniart-Vign\'eras 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 Mirkovi\'c-Vilonen cycles, for the Satake transform of an arbitrary pararhoric mod $p$ Hecke algebra with respect to an arbitrary Levi subgroup. Moreover, we prove an explicit formula for the convolution product in an arbitrary parahoric mod $p$ Hecke 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 towards a geometrization of a mod $p$ Local Langlands Correspondence.

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Robert Cass, Yujie Xu. 2022-11-29. Geometrization of the Satake transform for mod $p$ Hecke algebras. https://doi.org/10.1017/fms.2024.130

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