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Parabolic induction in characteristic p

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Let \(\mathrm{F}\) (resp. \(\mathbb F\)) be a nonarchimedean locally compact field with residue characteristic p (resp. a finite field with characteristic p). For \(k=\mathrm{F}\) or \(k=\mathbb F\), let \(\mathbf {G}\) be a connected reductive group over k and R be a commutative ring. We denote by \(\mathrm{Rep}( \mathbf G(k)) \) the category of smooth R-representations of \( \mathbf G(k) \). To a parabolic k-subgroup \({\mathbf P}=\mathbf {MN}\) of \(\mathbf G\) corresponds the parabolic induction functor \(\mathrm{Ind}_{\mathbf P(k)}^{\mathbf G(k)}:\mathrm{Rep}( \mathbf M(k)) \rightarrow \mathrm{Rep}( \mathbf G(k))\). This functor has a left and a right adjoint. Let \({{\mathcal {U}}}\) (resp. \({\mathbb {U}}\)) be a pro-p Iwahori (resp. a p-Sylow) subgroup of \( \mathbf G(k) \) compatible with \({\mathbf P}(k)\) when \(k=\mathrm{F}\) (resp. \(\mathbb F\)). Let \({H_{ \mathbf G(k)}}\) denote the pro-p Iwahori (resp. unipotent) Hecke algebra of \( \mathbf G(k) \) over R and \(\mathrm{Mod}({H_{ \mathbf G(k)}})\) the category of right modules over \({H_{ \mathbf G(k)}}\). There is a functor \(\mathrm{Ind}_{{H_{ \mathbf M(k)}}}^{{H_{ \mathbf G(k)}}}: \mathrm{Mod}({H_{ \mathbf M(k)}}) \rightarrow \mathrm{Mod}({H_{ \mathbf G(k) }})\) called parabolic induction for Hecke modules; it has a left and a right adjoint. We prove that the pro-p Iwahori (resp. unipotent) invariant functors commute with the parabolic induction functors, namely that \(\mathrm{Ind}_{\mathbf P(k)}^{\mathbf G(k)}\) and \(\mathrm{Ind}_{{H_{ \mathbf M(k)}}}^{{H_{ \mathbf G(k)}}}\) form a commutative diagram with the \({{\mathcal {U}}}\) and \({{\mathcal {U}}}\cap \mathbf M(\mathrm{F})\) (resp. \({\mathbb {U}}\) and \({\mathbb {U}}\cap \mathbf M(\mathbb F) \)) invariant functors. We prove that the pro-p Iwahori (resp. unipotent) invariant functors also commute with the right adjoints of the parabolic induction functors. However, they do not commute with the left adjoints of the parabolic induction functors in general; they do if p is invertible in R. When R is an algebraically closed field of characteristic p, we show that an irreducible admissible R-representation of \( \mathbf G(\mathrm{F}) \) is supercuspidal (or equivalently supersingular) if and only if the \({H_{ \mathbf G(\mathrm{F})}}\)-module \({\mathfrak {m}}\) of its \({{\mathcal {U}}}\)-invariants admits a supersingular subquotient, if and only if \({\mathfrak {m}}\) is supersingular.

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Acknowledgements

We thank Noriyuki Abe for suggesting the counter example of Prop. 4.12 and for generously sharing his recent results with us. We are also thankful to Guy Henniart for his continuous interest and helpful remarks. Our work was carried out at the Institut de Mathematiques de Jussieu – Paris 7, the University of British Columbia and the Mathematical Sciences Research Institute. We would like to acknowledge the support of these institutions. The first author is partially funded by NSERC Discovery Grant.

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  1. University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada

    Rachel Ollivier

  2. Institut de Mathématiques de Jussieu, UMR 7586, 175 Rue du Chevaleret, 75013, Paris, France

    Marie-France Vignéras

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Ollivier, R., Vignéras, MF. Parabolic induction in characteristic p. Sel. Math. New Ser. 24, 3973–4039 (2018). https://doi.org/10.1007/s00029-018-0440-0

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