Branching of Representations to Symmetric Subgroups

Abstract

Let $\gg$ be the Lie algebra of a compact Lie group and let $\theta$ be any automorphism of $\gg$. Let $\gk$ denote the fixed point subalgebra $\gg^\theta$. In this paper we present LiE programs that, for any finite dimensional complex representation $\pi$ of $\gg$, give the explicit branching $\pi|_\gk$ of $\pi$ on $\gk$. Cases of special interest include the cases where $\theta$ has order 2 (corresponding to compact riemannian symmetric spaces $G/K$), where $\theta$ has order 3 (corresponding to compact nearly--kaehler homogeneous spaces $G/K$), where $\theta$ has order 5 (which include the fascinating 5--symmetric space $E_8/A_4A_4$), and the cases where $\gk$ is the centralizer of a toral subalgebra of $\gg$.