Construction of tame supercuspidal representations (Q2719032)

Let \(F\) be a non-archimedean local field. The author constructs a list of supercuspidal representations for any connected reductive linear algebraic group \(G\) that is defined over \(F\) and splits over the maximal tamely ramified extension field of \(F\). The supercuspidal representations on the list are called tame, because the list is expected to be complete if the group is tame, whereas in general the list is incomplete. For \(G=Gl_n\) the tameness of the group means that \((n,p)=1\), where \(p\) is the characteristic of the residual field of \(F\). In general, the author expects that \(G\) is tame (i.e. the list is complete), if \(p\) is large enough.NEWLINENEWLINENEWLINEThe construction depends on a set of data \((\vec G,\pi_0, \vec\varphi)\), where the components \(\vec G=(G^0, \dots,G^d)\) and \(\vec\varphi =(\varphi_0, \dots, \varphi_d)\) are generalizations of the tower of generic characters as first defined in the work [J. Algebra 131, 388-424 (1990; Zbl 0715.22019)] of \textit{R. Howe} and \textit{A. Moy} for tame \(Gl_n\). The datum \(\pi_0\) is a supercuspidal representation of \(G_0\) of depth 0 and \(G_0\) is an algebraic subgroup of \(G\). All irreducible supercuspidal representations of depth 0 are induced from a compact mod center subgroup [\textit{L. Morris}, Compos. Math. 118, 135-157 (1999; Zbl 0937.22011); \textit{A. Moy} and \textit{G. Prasad}, Comment. Math. Helv. 71, 98-121 (1996; Zbl 0860.22006)]. The author constructs from such an inducing representation and the other data an inducing representation of the associated tame supercuspidal irreducible representation \(\pi=\pi(\vec G,\pi_0,\vec \varphi)\). Finally, the author weakens the assumptions on the data and proves a generalization of a Hecke algebra isomorphism of \textit{R. Howe} and \textit{A. Moy} [op. cit.]. This can be considered also in the context of the covering type due to \textit{C. J. Bushnell} and \textit{Ph. C. Kutzko} [Proc. Lond. Math. Soc. (3) 77, 582-634 (1998; Zbl 0911.22014)].