Locally Mixed Symmetric Spaces

This chapter introduces the notion which is the main interest in this book and is responsible for the title. Locally mixed symmetric spaces are a very natural construction which enrich the notion of locally symmetric spaces, studied in Chap. 2. While for locally symmetric spaces there is a pair of data entering, \((G_{\mathbb Q}, {\varGamma })\), where \(G_{\mathbb Q}\) is a semisimple \({\mathbb Q}\)-group such that \(X=G_{\mathbb R}/K\) is a symmetric space of non-compact type for a maximal compact subgroup \(K\subset G_{\mathbb R}\) and \( {\varGamma }\subset G_{\mathbb Q}\) is an arithmetic group, there is now a triple defining the situation: \((G_{\mathbb Q}, {\varGamma },{\boldsymbol{\rho }})\), where \({\boldsymbol{\rho }}:G_{\mathbb Q}\longrightarrow GL(V)\) is a faithful rational representation (not necessarily defined over \({\mathbb Q}\)).