[Paper Review] Propriétés de Lefschetz automorphes pour les groupes unitaires et orthogonaux

Nantel Bergeron|ArXiv.org|Mar 3, 2005

Finite Group Theory Research4 citations

TL;DR

This paper establishes automorphic Lefschetz properties for arithmetic manifolds associated with unitary and orthogonal groups $U(p,q)$ and $O(p,q)$, proving injectivity criteria for restriction maps on cohomology and dual cup-product maps using representation-theoretic techniques. It extends classical Lefschetz theorems to the context of real arithmetic manifolds, providing explicit conditions for the non-vanishing of cohomology classes.

ABSTRACT

Let $G$ be a connected semisimple group over ${\Bbb Q}$. Given a maximal compact subgroup and a convenient arithmetic subgroup $Γ\subset G({\Bbb Q})$, one constructs an arithmetic manifold $S=S(Γ)=Γ\backslash X$. If $H\subset G$ is a connected, $θ$-stable, semisimple subgroup, for each $g\in G({\Bbb Q})$ one can construct an immersion between arithmetic manifolds $j_g \colon S(H,g) o S$ induced by the map $H({\Bbb A}) o G({\Bbb A}), h\mapsto gh$. 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 $$R_g \colon H^{k}(S, {\Bbb C}) o H^{k}(S(H,g), {\Bbb C}).$$ 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 $$H^{k}(S(H), {\Bbb C}) o H^{k+{ m dim} S - { m dim} S(H)} (S, {\Bbb C})$$ 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.

Motivation & Objective

• To establish an automorphic analogue of the classical Lefschetz theorems in the context of real arithmetic manifolds associated with unitary and orthogonal groups.
• To derive explicit criteria for the injectivity of the product of restriction maps $R_g$ on the strongly primitive cohomology part of $H^k(S,\mathbb{C})$.
• To determine conditions under which the dual map $H^k(S(H),\mathbb{C}) \to H^{k+\dim S - \dim S(H)}(S,\mathbb{C})$ is injective.
• To reduce global cohomological problems to local representation-theoretic questions using Burger–Sarnak theorems and isolation properties of cohomological representations.
• To apply these results to prove the non-vanishing of certain cohomology classes in arithmetic manifolds.

Proposed method

• Uses the framework of arithmetic quotients $S = \Gamma \backslash X$ for $X = G(\mathbb{R})/K$, where $G$ is anisotropic and $\Gamma$ is an arithmetic subgroup.
• Constructs restriction maps $R_g: H^k(S,\mathbb{C}) \to H^k(S(H,g),\mathbb{C})$ via natural immersions $j_g: S(H,g) \to S$ induced by $h \mapsto gh$.
• Applies theorems of Burger and Sarnak to reduce global cohomological questions to local analysis of cohomological representations.
• Employs isolation techniques in the automorphic dual to separate cohomological representations from other automorphic forms.
• Utilizes representation-theoretic tools including $K$-type decompositions, discrete series, and tensor product decompositions.
• Applies results on $L^2$-cohomology and harmonic forms to analyze the cohomology of symmetric spaces and their subspaces.

Experimental results

Research questions

• RQ1 Under what conditions is the product of restriction maps $R_g$ injective on the strongly primitive part of $H^k(S,\mathbb{C})$ for $S$ associated with $U(p,q)$ or $O(p,q)$?
• RQ2 When is the dual map $H^k(S(H),\mathbb{C}) \to H^{k+\dim S - \dim S(H)}(S,\mathbb{C})$ injective?
• RQ3 How can global cohomological problems for arithmetic manifolds be reduced to local representation-theoretic problems?
• RQ4 What are the conditions under which certain cohomology classes in $S$ do not vanish?
• RQ5 To what extent does the classical Lefschetz theorem analogy hold in the automorphic setting for real arithmetic manifolds?

Key findings

• The paper provides explicit criteria for the injectivity of the product of restriction maps $R_g$ on the strongly primitive cohomology part of $H^k(S,\mathbb{C})$ for $G = U(p,q)$ and $O(p,q)$.
• It establishes injectivity of the dual cup-product map $H^k(S(H),\mathbb{C}) \to H^{k+\dim S - \dim S(H)}(S,\mathbb{C})$ under specific representation-theoretic conditions.
• The results are derived by reducing global problems to local ones using Burger–Sarnak theorems and isolation of cohomological representations in the automorphic dual.
• The cohomology classes in $S$ are shown to be non-vanishing under the derived criteria, particularly for discrete series and cohomological representations.
• The paper confirms that the classical Lefschetz analogy persists in the automorphic setting for real arithmetic manifolds, despite the absence of algebraic geometry.
• An explicit construction of harmonic forms via $L^2$-cohomology techniques is used to verify injectivity in key cases, particularly in the context of $O(p,q)$ and $U(p,q)$.