Ghost classes in $${\mathbb {Q}}$$ Q -rank two orthogonal Shimura varieties
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Mathematische Zeitschrift
Ghost classes in Q-rank two orthogonal Shimura varieties Jitendra Bajpai1 · Matias V. Moya Giusti2 Received: 7 May 2018 / Accepted: 31 December 2019 © The Author(s) 2020
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. Keywords Shimura varieties · Ghost classes · Mixed Hodge structures Mathematics Subject Classification Primary: 14G35 · 14D07; Secondary: 14M27
1 Introduction Let (G, X ) be a Shimura pair, and let ρ : G → GL(V ) be an irreducible finite dimensional representation (not necessarily defined over Q). For every open compact subgroup K f ⊂ G(A f ) of the group of finite adelic points of G, we consider the level variety S K = G(Q)\X × (G(A f )/K f ) and we denote by S the projective limit, over the directed set of open compact subgroups, of these level varieties (i.e. the space of complex points of the corresponding Shimura variety). on the Shimura variety S associated to One can define in a natural way a local system V (G, X ), underlying a variation of complex Hodge structure of a given weight wt(V ).
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Jitendra Bajpai [email protected] Matias V. Moya Giusti [email protected]
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Mathematisches Institut, Georg-August Universität Göttingen, Bunsenstrasse 3-5, 37073 Göttingen, Germany
2
IHES, 35 Route de Chartres, 91440 Bures-sur-Yvette, France
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J. Bajpai, M.V. Moya Giusti
Let A ⊂ G be a maximal Q-split torus and T ⊂ G a maximal torus defined over Q such that A ⊂ T. We choose systems of positive roots in the corresponding root systems (G, T), (G, A) so that they are compatible, i.e. the restriction to A, of a positive root in (G, T) is either zero or positive in (G, A). Let λ : T(C) −→ C× be the highest weight of V . We will usually denote V by Vλ . The choice of the system of positive roots + (G, A) in (G, A) defines a set of standard proper Q-parabolic subgroups denoted by PQ (G). From now on we will assume that the semisimple Q-rank of G is 2. In this case PQ (G) consists of three elements: two maximal Q-parabolic subgroups denoted by P1 and P2 , and a minimal Q-parabolic subgroup denoted by P0 . We consider the Borel–Serre compactification S of S (see [2]). The inclusion S → S is a λ can be extended naturally to S. The corresponding local system homotopy equivalence and V λ . In fact there is a natural isomorphism H • (S, V λ ) ∼
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