Spherical unitary dual of general linear group over non-Archimidean local field

Tadic, Marko

Annales de l'Institut Fourier, Tome 36 (1986), p. 47-55 / Harvested from Numdam

Résumé

Soit $F$ un corps local non-archimédien. Dans cet article, nous commençons par décrire toutes les représentations irréductibles sphériques de $GL(n,F)$. En particulier, nous montrons que de telles représentations sont paraboliquement induites par des caractères non-ramifiés. Le résultat de Bernstein sur l’irréductibilité de la représentation de $GL(n,F)$ paraboliquement induite par une représentation unitaire irréductible, et la construction de Olshansky des séries complémentaires donnent directement le dual unitaire sphérique de $GL(n,F)$.

Abstract

Let $F$ be a local non-archimedean field. The set of all equivalence classes of irreducible spherical representations of $GL(n,F)$ is described in the first part of the paper. In particular, it is shown that each irreducible spherical representation is parabolically induced by an unramified character. Bernstein’s result on the irreducibility of the parabolically induced representations of $GL(n,F)$ by irreducible unitary ones, and Ol’shanskij’s construction of complementary series give directly a description of all equivalence classes of irreducible unitary spherical representations of $GL(n,F)$.

Détail

Publié le : 1986-01-01

MR 87m:22047 | Zbl 0554.20009

DOI : https://doi.org/10.5802/aif.1046

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