# Commentarii Mathematici Helvetici

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**Volume 73, Issue 1, 1998, pp. 45–70**

**DOI: 10.1007/s000140050045**

A vanishing theorem for modular symbols on locally symmetric spaces

Toshiyuki Kobayashi^{[1]}and Tadao Oda

^{[2]}(1) Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, 153-8914, Tokyo, Japan

(2) Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, 153-8914, Tokyo, Japan

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.

*Keywords: *Modular symbols, semisimple Lie group, Zuckerman-Vogan module, Matsushima-Murakami formula, modular varieties, discrete decomposable restriction, bounded symmetric domain, discontinuous group, symmetric space

Kobayashi Toshiyuki, Oda Tadao: A vanishing theorem for modular symbols on locally symmetric spaces. *Comment. Math. Helv.* 73 (1998), 45-70. doi: 10.1007/s000140050045