Restriction De La Cohomologie D’une Variété De Shimura à Ses Sous-variétés

Abstract

Let G be a connected semisimple group over \( \mathbb{Q} \). Given a maximal compact subgroup K ⊂ G(\( \mathbb{R} \)) such that X = G(\( \mathbb{R} \))/K is a Hermitian symmetric domain, and a convenient arithmetic subgroup Γ ⊂ G(\( \mathbb{Q} \)), one constructs a (connected) Shimura variety S = S(Γ) = Γ\X. If H ⊂ G is a connected semisimple subgroup such that H(\( \mathbb{R} \)) / K is maximal compact, then Y = H(\( \mathbb{R} \))/K is a Hermitian symmetric subdomain of X. For each g ∈ G(\( \mathbb{Q} \)) one can construct a connected Shimura variety S(H, g) = (H(\( \mathbb{Q} \)) ∩ g−1Γg)\Y and a natural holomorphic map jg: S(H, g) → S induced by the map H(\( \mathbb{A} \)) → G(\( \mathbb{A} \)), 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 $$ R_{g} :H^{k} {\left( {S,\mathbb{C}} \right)} \to H^{k} {\left( {S{\left( {H,g} \right)},\mathbb{C}} \right)}. $$