Simple subquotients of big parabolically induced representations of \(p\)-adic groups. (English) Zbl 1402.22018
Summary: This note is motivated by the problem of “uniqueness of supercuspidal support” in the modular representation theory of \(p\)-adic groups. We show that any counterexample to the same property for a finite reductive group lifts to a counterexample for the corresponding unramified \(p\)-adic group. To this end, we need to prove the following natural property: any simple subquotient of a parabolically induced representation is isomorphic to a subquotient of the parabolic induction of some simple subquotient of the original representation. The point is that we put no finiteness assumption on the original representation.
MSC:
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
Keywords:
representation theory; \(p\)-adic groups; modular representations; parabolic induction; supercuspidal representationsReferences:
| [1] | Bernstein, I. N.; Zelevinski, A. V., Induced representations on reductive p-adic groups, Ann. Sci. Éc. Norm. Supér., 10, 441-472, (1977) · Zbl 0412.22015 |
| [2] | Dat, J.-F., Finitude pour LES représentations lisses de groupes p-adiques, J. Inst. Math. Jussieu, 8, 2, 261-333, (2009) · Zbl 1158.22020 |
| [3] | Emerton, M.; Helm, D., The local Langlands correspondence for \(\operatorname{GL}_n\) in families, Ann. Sci. Éc. Norm. Supér. (4), 47, 4, 655-722, (2014) · Zbl 1321.11055 |
| [4] | Helm, D., The Bernstein center of the category of smooth \(W(k) [\operatorname{GL}_n(F)]\)-modules, Forum Math. Sigma, 4, 11, 98, (2016) · Zbl 1364.11097 |
| [5] | Mínguez, A.; Sécherre, V., Représentations lisses modulo ℓ de \(\operatorname{GL}_m(D)\), Duke Math. J., 163, 4, 795-887, (2014) · Zbl 1293.22005 |
| [6] | Morris, L., Tamely ramified supercuspidal representations, Ann. Sci. Éc. Norm. Supér., 29, 5, 639-667, (1996) · Zbl 0870.22012 |
| [7] | Vignéras, M.-F., Induced R-representations of p-adic reductive groups, Selecta Math. (N.S.), 4, 4, 549-623, (1998) · Zbl 0943.22017 |
| [8] | Vignéras, M.-F., Correspondance de Langlands semi-simple pour \(G L(n, F)\) modulo \(\ell \ne p\), Invent. Math., 144, 177-223, (2001) · Zbl 1031.11068 |
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