Propriétés de Lefschetz automorphes pour les groupes unitaires et orthogonaux
[Automorphic Lefschetz properties for unitary and orthogonal groups]
Mémoires de la Société Mathématique de France, no. 106 (2006) , 131 p.

Let G be a connected semisimple group over . Given a maximal compact subgroup KG() – such that X=G()/K is a Riemannian symmetric space – and a convenient arithmetic subgroup ΓG(), one constructs an arithmetic manifold S=S(Γ)=ΓX. If HG is a connected semisimple subgroup such that H()K is maximal compact, then Y=H()/H()K is a symmetric subspace of X. For each gG() one can construct an arithmetic manifold S(H,g)=(H()g -1 Γg)Y and a natural immersion j g :S(H,g)S induced by the map H(𝔸)G(𝔸),hgh. Let us assume that G is anisotropic, which implies that S and S(H,g) are compact. Then, for each positive integer k, the map j g induces a restriction map R g : H k ( S , ) H k ( S ( H , g ) , ) . In this paper we focus on symmetric spaces associated to the unitary and orthogonal groups, namely O(p,q) and U(p,q), and give explicit criterions for the injectivity of the product of the maps R g (for g running through G()) when restricted to the strongly primitive (in the sense of Vogan and Zuckerman) part of the cohomology. We also give explicit criterions for the injectivity of the map H k ( S ( H ) , ) H k + dim S - dim S ( H ) ( S , ) dual to the restriction map dual to the restriction map R e . The results we obtain fit into a larger conjectural picture that we describe and which bare a strong analogy with the classical Lefschetz Theorems. This may sound quite surprising that such an analogy still exists in the case of the real arithmetic manifolds. We reduce the global problems mentioned above to local ones by using Theorems of Burger and Sarnak and isolations properties of cohomological representations in the automorphic dual. The methods used then are mainly representation-theoretic. We finally derive some applications concerning the non vanishing of some cohomology classes in arithmetic manifolds.

Soit S=S(Γ)=ΓX une variété arithmétique obtenue comme quotient d’un espace symétrique X=G()/K – G étant un groupe semi-simple connexe sur , KG() un sous-groupe compact maximal – par un sous-groupe arithmétique Γ de G(). Si HG est un sous-groupe semi-simple connexe tel que H()K soit un sous-groupe compact maximal, alors Y=H()/H()K est un sous-espace symétrique de X. Pour tout gG() on peut former la variété arithmétique S(H,g)=(H()g -1 Γg)Y et considérer l’immersion naturelle j g :S(H,g)S induite par l’application H(𝔸)G(𝔸), hgh. Supposons G anisotrope ce qui implique que S et S(H,g) sont compactes. Alors, pour tout entier positif k, l’application j g induit l’application de restriction R g : H k ( S , ) H k ( S ( H , g ) , ) . Dans ce Mémoire nous nous concentrons sur le cas des espaces symétriques associés aux groupes orthogonaux et unitaires, Dans ce Mémoire nous nous concentrons sur le cas des espaces symétriques associés aux groupes orthogonaux et unitaires, O(p,q) et U(p,q) ; nous démontrons des critères explicites d’injectivité du produit (sur les gG()) des applications R g en restriction à la partie fortement primitive (au sens de Vogan et Zuckerman) de la cohomologie. Nous démontrons également des critères explicites d’injectivité de l’application H k ( S ( H ) , ) H k + dim S - dim S ( H ) ( S , ) duale à l’application de restriction R e . Les résultats obtenus s’inscrivent naturellement dans une programme conjectural plus large que nous décrivons et auquel on peut penser comme à un analogue automorphe des Théorèmes de Lefschetz classiques sur les variétés projectives. Il est peut-être un peu surprenant qu’une telle analogie subsiste dans le cas de variétés arithmétiques réelles. La démonstration consiste à réduire les problèmes globaux mentionnés ci-dessus à leurs analogues locaux à l’aide de Théorèmes de Burger et Sarnak et de propriétés d’isolation des représentations cohomologiques dans le dual automorphe. Les méthodes utilisées sont alors essentiellement issues de la théorie des représentations. Finalement, nous déduisons de ces résultats des applications à la construction de classes de cohomologie non nulles dans certaines variétés arithmétiques.

DOI: 10.24033/msmf.418
Classification: 11F75,  22E47,  22E55,  11G18,  14G35,  58J50
Keywords: Arithmetic manifolds, cohomology of locally symmetric spaces, automorphic spectrum, cohomological representations
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Bergeron, Nicolas. Propriétés de Lefschetz automorphes pour les groupes unitaires et orthogonaux. Mémoires de la Société Mathématique de France, Serie 2, , no. 106 (2006), 131 p. doi : 10.24033/msmf.418. http://numdam.org/item/MSMF_2006_2_106__1_0/

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