T. Kobayashi and T. Oda,
A vanishing theorem for modular symbols on locally symmetric spaces,
Comment. Math. Helv. 73 (1998), 45-70..
A modular symbol is the fundamental class of a totally geodesic submanifold Γ'\G'/K' embedded in a locally Riemannian symmetric space Γ\G/K, which is defined by a subsymmetric space G'/K' \hookrightarrow G/K. In this paper, we consider the modular symbol defined by a semisimple symmetric pair (G,G'), and prove a vanishing theorem with respect to the π-component (π∈\hat{G}) in the Matsushima-Murakami formula based on the discretely decomposable theorem of the restriction π|G'. In particular, we determine explicitly the middle Hodge components of certain totally real modular symbols on the locally Hermitian symmetric spaces of type IV.

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