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arXiv · 2204.11581

The Left Adjoint of Derived Parabolic Induction

Claudius Heyer
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Abstract

We prove that the derived parabolic induction functor, defined on the unbounded derived category of smooth mod $p$ representations of a $p$-adic reductive group, admits a left adjoint $\mathrm{L}(U,-)$. We study the cohomology functors $\mathrm{H}^i\circ \mathrm{L}(U,-)$ in some detail and deduce that $\mathrm{L}(U,-)$ preserves bounded complexes and global admissibility in the sense of Schneider--Sorensen. Using $\mathrm{L}(U,-)$ we define a derived Satake homomorphism und prove that it encodes the mod $p$ Satake homomorphisms defined explicitly by Herzig.

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Claudius Heyer. 2022-04-25. The Left Adjoint of Derived Parabolic Induction. https://doi.org/10.1007/s00209-023-03385-5

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