Algorithms for Computations in Local Symmetric Spaces

In the last two decades much of the algebraic/combinatorial structure of Lie groups, Lie algebras, and their representations has been implemented in several excellent computer algebra packages, including LiE, GAP4, Chevie, Magma and Maple. The structure of reductive symmetric spaces or more generally symmetric k-varieties is very similar to that of the underlying Lie group, with a few additional complications. A computer algebra package enabling one to do computations related to these symmetric spaces would be an important tool for researchers in many areas of mathematics, including representation theory, Harish Chandra modules, singularity theory, differential and algebraic geometry, mathematical physics, character sheaves, Lie theory, etc. However, until a few years ago only very few algorithms existed for computations in these symmetric spaces. This is in part due to the fact that the algebraic/combinatorial structure of these symmetric spaces is much more complicated and also much of this structure was traditionally proved using analytic and geometric methods, which do not lend themselves to designing algorithms. About 15 years ago the concept of real symmetric spaces was generalized to symmetric spaces over general fields of characteristic not 2 and it was shown that these generalized symmetric spaces (called symmetric k-varieties) have many algebraic/combinatorial properties similar to those of the classical real symmetric spaces. Most of this work mainly used algebraic and combinatorial methods, which lend themselves excellently to designing algorithms; and based on these results some of the first algorithms for symmetric spaces were developed a few years ago (see [Hel96, Hel00]). In this paper we lay the groundwork for computing the fine structure of symmetric spaces over the real numbers and other base fields, give a complete set of algorithms for computing the fine structure of symmetric varieties and use this to compute nice bases for the local symmetric varieties.