Abstract.
A modular symbol is the fundamental class of a totally geodesic submanifold \( \Gamma'\backslash G'/K' \) embedded in a locally Riemannian symmetric space \( \Gamma \backslash G / K \), 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 \( \pi \)-component \( (\pi \in \widehat {G}) \) in the Matsushima-Murakami formula based on the discretely decomposable theorem of the restriction \( \pi |_{G'} \). 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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Received: December 8, 1996
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Kobayashi, T., Oda, T. A vanishing theorem for modular symbols on locally symmetric spaces. Comment. Math. Helv. 73, 45–70 (1998). https://doi.org/10.1007/s000140050045
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DOI: https://doi.org/10.1007/s000140050045