## 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

Zbl 0558.12007
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