Appendix: non-uniqueness of supercuspidal support for finite reductive groups
type: Item
instance of: scholarly article
title: Appendix: non-uniqueness of supercuspidal support for finite reductive groups
zbMATH Open document ID: 1402.20020
author: Olivier Dudas
published in: Journal of Algebra
publication date: 2018-07-25T00:00:00Z
review text: Let \(F\) be a \(p\)-adic field, \({\mathcal O}_F\) be its ring of integers, and \(k_{F}\) be its residue field. Let \({\mathbb G}\) be a connected reductive group over \({\mathcal O}_F\) with a parabolic subgroup \({\mathbb P}\) and corresponding Levi quotient \({\mathbb M}\). Further, let \({\bar G} := G(k_F)\), \({\bar P} := P(k_F)\), and \({\bar M} := M(k_F)\) denote the associated finite reductive groups and consider modules over the field \({\overline {\mathbb F}}_{\ell}\), where \(\ell \neq p\). Parabolic induction \(i_{{\bar P}}^{{\bar G}}\) and restriction \(r_{{\bar P}}^{{\bar G}}\) are adjoint functors relating \({\overline {\mathbb F}}_{\ell}{\bar M}\)-modules and \({\overline {\mathbb F}}_{\ell}{\bar G}\)-modules. Given a simple \({\overline {\mathbb F}}_{\ell}{\bar G}\)-module \(\pi\), one can ask whether \(\pi\) occurs as a subquotient of \(i_{{\bar P}}^{{\bar G}}(\sigma)\) for a Levi factor \({\bar M}\) and an \({\overline {\mathbb F}}_{\ell}{\bar M}\)-module \(\sigma\). Such a pair \(({\bar M},\sigma)\) for which \({\bar M}\) is minimal and \(\sigma\) is simple is called a supercuspidal support for \(\pi\). The question of interest here is whether a given representation \(\pi\) can admit non-conjugate supercuspidal supports. Consider the case that \({\mathbb G}\) is of type \(C_4\) with \({\bar G} = \mathrm{Sp}_{8}(q)\), for \(q\) a power of \(p\). Assume that \(\ell > 5\) and \(\ell\) divides \(q^2 + 1\). The author identifies a simple \({\overline {\mathbb F}}_{\ell}{\bar G}\)-module \(\pi\) and a pair of non-conjugate supercuspidal supports for \(\pi\). In one case, the Levi quotient of \({\mathbb G}\) is of type \(C_2\), and, in the other, it is a split torus. The verification makes use of a result of \textit{G. Hiss} [J. Algebra 184, No. 3, 839--851 (1996; Zbl 0859.20009)], which allows one to translate the problem to one involving projective covers of simple modules. The author also observes that a similar example can be found for a group \({\…
reviewed by: Christopher P. Bendel
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 6910269
zbMATH Keywords: finite reductive group
zbMATH Keywords: symplectic group
zbMATH Keywords: \(p\)-adic group
zbMATH Keywords: supercuspidal representation
zbMATH Keywords: supercuspidal support
zbMATH Keywords: simple module
zbMATH Keywords: parabolic induction
zbMATH Keywords: modular representation
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2018.05.021
OpenAlex ID: W2809565900
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cites work: Supercuspidal representations of finite reductive groups
cites work: Q3520011
cites work: Decomposition numbers of \(\text{Sp}(4,q)\)
cites work: The Characters of the Finite Symplectic Group Sp(4,q)
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cites work: Brauer trees of Sp(4,q)
Wikidata QID: Q129653410
DOI: 10.1016/J.JALGEBRA.2018.05.021
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description: scientific article; zbMATH DE number 6910269