A vanishing theorem for modular symbols on locally symmetric spaces

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.