Paquets d’Arthur Spéciaux Unipotents aux Places Archimédiennes et Correspondance de Howe

Abstract

Let G be a classical quasi-split group defined over a number fields, F. Arthur has proved that the square integrable irreducible automorphic representations of the adeles points of G satisfy a relatively strong form of the strong multiplicity-one theorem true for general linear groups. More precisely, let π be such a representation and fix S a finite number of places in the number field such that for all place v not in S, the situation is unramified. Denote by n∗ the dimension of the natural representation of the L-group of G and for all v not in S, denote by πvGL the unramified representation of GL(n∗, Fv) corresponding to the local component πv under the unramified Langlands correspondence. Using the stabilization of the untwisted and twisted trace formula (and a lot of other ideas), Arthur has proved that there exists a unique irreducible automorphic representation πGL of \(\mathrm{GL}(n^{{\ast}}, \mathbb{A}_{F})\) such that the local component of πGL at each place v ∉ S is precisely πvGL. Moreover, at any place v of F, the local component πvGL determines a semi-simple representation of G(Fv) of finite length such that πv is an irreducible component of this representation. One would like to determine explicitly this semi-simple representation and in particular the multiplicity appearing in it. This is done if v is p-adic but not known if v is Archimedean. In this chapter one studies the case where πvGL is induced from a quadratic character of a Levi subgroup of GL(n∗, Fv). We impose some parity condition explained in the text. This is the special unipotent case of Barbasch and Vogan. In that case, using the theta correspondence, we obtain a precise description.