A vanishing theorem for modular symbols on locally symmetric spaces

Abstract

A modular symbol is the fundamental class of a totally geodesic submanifold ${\Gamma }^{\prime }\backslash G^{\prime }/K^{\prime }$ embedded in a locally Riemannian symmetric space $\Gamma \backslash G/K$ , which is defined by a subsymmetric space $G^{\prime }/K^{\prime }\hookrightarrow G/K$ . 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 $(\pi \in \hat{G})$ in the Matsushima-Murakami formula based on the discretely decomposable theorem of the restriction $\pi {|}_{G^{\prime }}$ . In particular, we determine explicitly the middle Hodge components of certain totally real modular symbols on the locally Hermitian symmetric spaces of type IV.