Algorithms for Computing Characters for Symmetric Spaces

Abstract

Symmetric spaces or more general symmetric k-varieties can be defined as the homogeneous spaces G _k /K _k , where G is a reductive algebraic group defined over a field k of characteristic not 2, K the fixed point group of an involution θ of G and G _k resp. K _k the sets k-rational points of G resp. K. These symmetric spaces have a fine structure of root systems, characters, Weyl groups etc., similar to the underlying algebraic group G. The relationship between the fine structure of the symmetric space and the group plays an important role in the study of these symmetric spaces and their applications. To develop a computer algebra package for symmetric spaces one needs explicit formulas expressing the fine structure of the symmetric space and group in terms of each other. In this paper we consider the case that k is algebraically closed and give explicit algorithmic formulas for expressing the characters of the weight lattice $\Lambda_{\mathfrak{a}}$ of the symmetric space in terms of the characters of the weight lattice $\Lambda_{\mathfrak {t}}$ of the group. These algorithms can easily be implemented in a computer algebra package.

The root system of the symmetric space $\Phi(\mathfrak{a})$ can be described as the image of the root system of the group $\Phi(\mathfrak{t})$ under a projection π derived from an involution θ on $\Phi(\mathfrak{t})$ . This implies that $\pi(\Lambda_{\mathfrak{t}})\subseteq \Lambda_{\mathfrak{a}}$ . Using these formulas for the characters of each of these lattices we show that in fact $\pi(\Lambda_{\mathfrak{t}})=\Lambda_{\mathfrak {a}}$ .