Simple supercuspidals and the Langlands correspondence
type: Item
instance of: scholarly article
title: Simple supercuspidals and the Langlands correspondence
published in: Michigan Mathematical Journal
publication date: 2022-10-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/2005.09078
full work available at URL: 20207202.full
review text: The author explores simple supercuspidal representations, a certain class of representations with a particularly simple formal degree. The formal degree conjecture predicts that when \(\pi\) is an irreducible representation of \(G(k)\) with \(G\) reductive and \(k\) a \(p\)-adic field, which corresponds to the \(L\)-parameter \(\phi\colon W(k)\times\mathrm{SL}_2(\mathbb C)\to\widehat G(\mathbb C)\), then \(\deg(\pi)\) is given in terms of special values of the \(L\)-function \(L(\phi,\mathrm{ad},s):=\det(1-\mathrm{Fr}.q^{-s}|\widehat{\mathfrak g}^{I(k)})^{-1}\). Thus, the formula is particularly simple when the invariants under the inertia \(\widehat{\mathfrak g}^{I(k)}=\widehat{\mathfrak t}\) is the maximal torus. The author gives a construction of such a representation, \emph{simple supercuspidals}, for split, simply connected, almost simple groups \(G\) by inducing characters from the \(p\)-Sylow normal subgroup of the Iwahori subgroup of \(G\). The author describes the \(L\)-parameter, and further explores when they arise as local factors of global automorphic representations. Moreover, the paper explores the function field analogue, and compares the construction of their system of \(\ell\)-adic representations to that of Heinloth, Ngo, and Yun [\textit{J. Heinloth} et al., Ann. Math. (2) 177, No. 1, 241--310 (2013; Zbl 1272.14012)].
reviewed by: Kenta Suzuki
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 7599278
zbMATH Keywords: simple supercuspidal representations
zbMATH Keywords: Langlands correspondence
zbMATH Keywords: formal degree
author: Benedict H. Gross
MaRDI profile type: Publication
OpenAlex ID: W3026482126
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DOI: 10.1307/MMJ/20207202
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description: scientific article; zbMATH DE number 7599278