Propri\'et\'es de Lefschetz automorphes pour les groupes unitaires et orthogonaux

Abstract

Let $G$ be a connected semisimple group over ${\Bbb Q}$. Given a maximal compact subgroup and a convenient arithmetic subgroup $\Gamma\subset G({\Bbb Q})$, one constructs an arithmetic manifold $S=S(\Gamma)=\Gamma\backslash X$. If $H\subset G$ is a connected, $\theta$-stable, semisimple subgroup, for each $g\in G({\Bbb Q})$ one can construct an immersion between arithmetic manifolds $j_g \colon S(H,g)\to S$ induced by the map $H({\Bbb A})\to G({\Bbb A}), h\mapsto gh$. 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 $$R_g \colon H^{k}(S, {\Bbb C})\to H^{k}(S(H,g), {\Bbb C}).$$ 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 $G({\Bbb Q})$) 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 $$H^{k}(S(H), {\Bbb C}) \to H^{k+{\rm dim} S - {\rm dim} S(H)} (S, {\Bbb C})$$ 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 finally derive some applications concerning the non vanishing of some cohomology classes in arithmetic manifolds.