2005/03/03 by Nantel Bergeron, N. Bergeron, Bergeron, N. · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Number Theory (math.NT) #math.GR #math.NT
paper · pdf · doi:10.48550/arxiv.math/0503062
106 pages
arxiv created 2005/03/03 · openalex publication_date 2005/03/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a connected semisimple group over \Bbb Q. Given a maximal compact subgroup and a convenient arithmetic subgroup Γ⊂ G(\Bbb Q), one constructs an arithmetic manifold S=S(Γ)=Γ\backslash X. If H⊂ G is a connected, θ-stable, semisimple subgroup, for each g∈ G(\Bbb Q) one can construct an immersion between arithmetic manifolds jg \colon S(H,g)→ S induced by the map H(\Bbb A)→ G(\Bbb A), h↦ 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 jg induces a restriction map Rg \colon Hk(S, \Bbb C)→ Hk(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 Rg (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 Hk(S(H), \Bbb C) → H^k+\rm dim S - \rm dim S(H) (S, \Bbb C) dual to the restriction map Re. 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.