Branching of Representations to Symmetric Subgroups

Abstract

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

28 pages, with a number of LiE programs for branching of representations to subgroups defined by automorphism, while keeping track of the action on both the center and the semisimple part of the subgroup