Appendix: non-uniqueness of supercuspidal support for finite reductive groups
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 \({\mathbb G}\) of type \(F_4\). This work appears as an appendix to work of \textit{J.-F. Dat} [J. Algebra 510, 499--507 (2018; Zbl 1402.22018)], in which it is shown (in the initial generality) that if non-conjugate pairs arise over \({\bar G}\), then they also do over \({\mathbb G}(F)\). The counter-examples provided in the work under review are in contrast to the case when \({\mathbb G}\) is a general linear group, where \textit{M.-F. Vignéras} [Sel. Math., New Ser. 4, No. 4, 549--623 (1998; Zbl 0943.22017)] showed that supercuspidal supports are ``unique''.
scientific article; zbMATH DE number 6910269
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