Simple supercuspidals and the Langlands correspondence
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)].
scientific article; zbMATH DE number 7599278
scientific article