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Contragredient representations over local fields of positive characteristic. (English) Zbl 1442.11081
In this paper, the author discusses the characterization of the contragredient representation in terms of Langlands correspondence in certain cases. In a broader context, there is a conjecture by Adams, Vogan, Prasad which says: let \(\pi\) be an irreducible representation of a reductive group \(G\) over a local field \(F\). Assume that the Langlands correspondence holds for the group \(G\) and assume that \(\pi\) belongs to the Langlands packet attached to a parameter \(\phi\). Then, \(\check{\pi}\) (the contragredient representation of \(\pi\)) belongs to the \(L\)-parameter \(^{L}\theta \circ \phi,\) where \(^{L}\theta \) denotes the Chevalley involution on \(^{L}G\) (here \(^{L}G\) denotes the \(L\)-group of \(G\), arising in the local Langlands correspondence for \(G\)). There is another layer of this conjecture; it addresses the precise way in which parameter of \(\check{\pi}\) is given inside this \(L\)-packet, in terms of that of \(\pi\).
The author here is concerned only in the first layer of this conjecture, and he proves it in the case of all connected reductive groups \(G\) over local fields of positive characteristic. He uses the local Langlands parametrization of A. Genestier and V. Lafforgue [“Chtoucas restreints pour les groupes réductifs et paramétrisation de Langlands locale”, Preprint, arXiv:1709.00978], but the main ingredient in his proof is a local-global interplay using V. Lafforgue’s global Langlands parametrization of cuspidal automorphic representations over function fields [J. Am. Math. Soc. 31, No. 3, 719–891 (2018; Zbl 1395.14017)]. He uses especially the excursion operators and the proof is geometric and does not rely on any form of Arthur-Selberg trace formula. The Chevalley involution on the dual group \(^{L}G\) turns out to be dual to the duality involution (in the sense of D. Prasad [Trans. Am. Math. Soc. 372, No. 1, 615–633 (2019; Zbl 1458.11083)]) on the original group. This duality involution on the original group coincides with the MVW involution in the case of classical groups.
MSC:
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11R58 Arithmetic theory of algebraic function fields
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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