Supercuspidal representations in non-defining characteristics (Q6568815)

Let \(G\) denote a connected reductive group over a non-Archimedean local field \(F\) of residue characteristic \(p\). Let \(R\) denote an algebraically closed field of characteristic \(\ell\neq p\).\N\NWe say that an irreducible \(R\)-representation \(\pi\) of \(G(F)\) is \textit{cuspidal} if it can not be realized as a quotient of any representation parabolically induced from a proper Levi subgroup. Equivalently, all of its Jacquet modules are trivial. We say that \(\pi\) is \textit{supercuspidal} if it cannot be realized as a \textit{sub}quotient of any properly parabolically induced representation. Analogous notions exist for representations of reductive groups over finite fields. It is clear from the definitions that a supercuspidal representation must be cuspidal. The converse is true when \(\ell = 0\), but not in general.\N\NThe author has previously [Mich. Math. J. 72, 331--342 (2022; Zbl 1523.22019)] adapted \textit{J.-K. Yu}'s construction of (complex) (super)cuspidal representations [J. Am. Math. Soc. 14, No. 3, 579--622 (2001; Zbl 0971.22012)] to obtain a construction of cuspidal \(R\)-representations of \(G(F)\). Moreover, she has shown that this construction is exhaustive provided that the center of \(G\) splits over a tamely ramified extension of \(F\), \(p\) is odd, and \(p\) does not divide the order of the absolute Weyl group of \(G\). Thus, if one wants to classify all irreducible supercuspidal \(R\)-representations under these conditions, it only remains to determine which cuspidal representations arising from Yu's construction [loc. cit.] are supercuspidal. The present paper accomplishes this task.\N\NBriefly, the Yu construction builds a cuspidal representation \(\pi\) of \(G(F)\) using the ingredients for building a ``depth-zero'' cuspidal representation \(\pi^0\) of a subgroup \(G^0(F)\), together with some additional data. The key ingredient in the construction of \(\pi^0\) is a cuspidal representation \(\rho^0\) of the reductive quotient of a maximal parahoric subgroup of \(G^0(F)\). (Since this quotient is isomorphic to the group of rational points of a connected reductive group defined over the residue field of \(F\), the notion of cuspidality for \(\rho^0\) makes sense.)\N\NThe main result of this paper (Theorem 1 combined with Corollary 2) is that the cuspidal representation \(\pi\) of \(G(F)\) is supercuspidal if and only if the associated cuspidal representation \(\rho^0\) is supercuspidal.\N\N\textit{G. Henniart} and \textit{M.-F. Vignéras} [Tunis. J. Math. 4, No. 2, 249--305 (2022; Zbl 1535.11074)] have also obtained this result under the assumption that Bernstein's second adjointness holds for \(R\)-representations. Although second adjointness is now known due to work of \textit{J.-F. Dat} et al. [J. Am. Math. Soc. 37, No. 3, 929--949 (2024; Zbl 1544.22019)], their proof relies on a lot of machinery. The present paper gives a very quick route to its main result.