Higher mod p Satake transforms: explicit formulas and images for GL2
Abstract
Heyer's left adjoint to derived parabolic induction gives rise to higher mod p Satake transforms 𝒮Vi: ℋG(V) ⟶ ℋM(Hi(KU,V)), whose extreme degrees recover Herzig's invariant and coinvariant transforms. For G=G(F), where G is a connected reductive group defined over a finite extension F/ℚp, we describe these maps explicitly using Mackey decompositions and corestriction in group cohomology. For G=GLn(F), we evaluate them on the standard Hecke generators, showing that their support lies in a single Weyl orbit and that their values on the individual support cosets are concrete corestriction operators. For G=GL2(F), we determine the image in every cohomological degree for each non-Steinberg Serre weight. The endpoint images are the classical polynomial Satake images, while every interior image is of the form k[x,y±1]/(xr) for some positive integer r. Thus, every interior image is a finite nilpotent thickening of the Laurent polynomial algebra k[y±1]. We give explicit bounds for its thickness, prove that they are sharp, and show that in the unramified case every interior image is either reduced or a square-zero thickening.
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BibTeX
@misc{bhatia2026higher,
title = {{Higher mod $p$ Satake transforms: explicit formulas and images for $\mathrmGL_2$}},
author = {Hargun Bhatia and Yashi Jain and Casey Perdue and Ben Savoie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13203},
eprint = {2610.13203},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13203},
}
RIS
TY - UNPB
TI - Higher mod p Satake transforms: explicit formulas and images for GL_2
AU - Hargun Bhatia
AU - Yashi Jain
AU - Casey Perdue
AU - Ben Savoie
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13203
UR - https://arxiv.org/abs/2610.13203
AB - Heyer's left adjoint to derived parabolic induction gives rise to higher mod p Satake transforms 𝒮_V^i: ℋ_G(V) ⟶ ℋ_M(H^i(K_U,V)), whose extreme degrees recover Herzig's invariant and coinvariant transforms. For G=G(F), where G is a connected reductive group defined over a finite extension F/ℚ_p, we describe these maps explicitly using Mackey decompositions and corestriction in group cohomology. For G=GL_n(F), we evaluate them on the standard Hecke generators, showing that their support lies in a single Weyl orbit and that their values on the individual support cosets are concrete corestriction operators. For G=GL_2(F), we determine the image in every cohomological degree for each non-Steinberg Serre weight. The endpoint images are the classical polynomial Satake images, while every interior image is of the form k[x,y^±1]/(x^r) for some positive integer r. Thus, every interior image is a finite nilpotent thickening of the Laurent polynomial algebra k[y^±1]. We give explicit bounds for its thickness, prove that they are sharp, and show that in the unramified case every interior image is either reduced or a square-zero thickening.
ER -