×

Hereditary orders, Gauss sums and supercuspidal representations of \(GL_ N\). (English) Zbl 0601.12025

F is a p-adic local field, \(N\geq 2\), \(G=GL_ N(F).\) This paper is concerned with the behaviour of irreducible admissible representations \(\pi\) of G on certain open compact or compact-mod-centre subgroups K of G. The aim is to show that, when K is suitably chosen, the restriction \(\pi| K\) contains a representation of K of a very special kind. The first result (conjectured and proved independently by A. Moy) concerns an arbitrary \(\pi\), in the case where K is a parahoric subgroup of G \((= unit\) group of a hereditary order in \(M_ N(F))\). The other concerns the restriction of a supercuspidal \(\pi\) to a maximal compact mod-centre subgroup K. It shows that K may be chosen so that \(\pi| K\) contains a ”nondegenerate” representation in the sense of the reviewer and A. Fröhlich [Proc. Lond. Math. Soc., III. Ser. 50, 207-264 (1985; Zbl 0558.12007)]. As a consequence, one obtains a formula for the Godement- Jacquet local constant in terms of the Gauss sums of [loc. cit.], and a local proof of Carayol’s exhaustion theorem for supercuspidals in the case where N is prime.

MSC:

11S45 Algebras and orders, and their zeta functions
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
22E50 Representations of Lie and linear algebraic groups over local fields
11S37 Langlands-Weil conjectures, nonabelian class field theory
11L10 Jacobsthal and Brewer sums; other complete character sums

Citations:

Zbl 0558.12007
PDF BibTeX XML Cite
Full Text: DOI Crelle EuDML