Abstract
In this article, the existence of ghost classes for the Shimura varieties associated to algebraic groups of orthogonal similitudes of signature (2, n) is investigated. We make use of the study of the weights in the mixed Hodge structures associated to the corresponding cohomology spaces and results on the Eisenstein cohomology of the algebraic group of orthogonal similitudes of signature (1, n-1). For the values of n = 4, 5 we prove the non-existence of ghost classes for most of the irreducible representations (including most of those with an irregular highest weight). For the rest of the cases, we prove strong restrictions on the possible weights in the space of ghost classes and, in particular, we show that they satisfy the weak middle weight property.
Highlights
Let (G, X ) be a Shimura pair, and let ρ : G → GL(V ) be an irreducible finite dimensional representation
It is interesting to study the nontriviality of the space of ghost classes for a Shimura variety and to give some description of the possible weights in its mixed Hodge structure
We say that the Shimura variety satisfies the weak middle weight property if for every finite dimensional highest weight representation Vλ of G and every nonnegative integer q, the only possible weights in the mixed Hodge structure on the space of q-ghost classes, in H q (∂ S, Vλ), are the middle weight and the middle weight plus one
Summary
Let (G, X ) be a Shimura pair, and let ρ : G → GL(V ) be an irreducible finite dimensional representation (not necessarily defined over Q). It is interesting to study the nontriviality of the space of ghost classes for a Shimura variety and to give some description of the possible weights in its mixed Hodge structure. We say that the Shimura variety satisfies the weak middle weight property if for every finite dimensional highest weight representation Vλ of G and every nonnegative integer q, the only possible weights in the mixed Hodge structure on the space of q-ghost classes, in H q (∂ S, Vλ), are the middle weight and the middle weight plus one. In the remaining cases we restrict the list of degrees in cohomology in which there could exist ghost classes and prove that there is, in each degree, only one possible weight in their corresponding mixed Hodge structure which is in all cases the middle weight or the middle weight plus one
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