We find explicit multiplicity-free branching rules of some series of irreducible finite dimensional representations of simple Lie algebras to the fixed point subalgebras of outer automorphisms . The representations have highest weights which are scalar multiples of fundamental weights or linear combinations of two scalar ones. Our list of pairs of Lie algebras includes an exceptional symmetric pair and also a non-symmetric pair as well as a number of classical symmetric pairs. Some of the branching rules were known and others are new, but all the rules in this paper are proved by a unified method. Our key lemma is a characterization of the ``middle'' cosets of the Weyl group of in terms of the subalgebras on one hand, and the length function on the other hand.
Hidehisa ALIKAWA. "Multiplicity-free branching rules for outer automorphisms of simple Lie algebras." J. Math. Soc. Japan 59 (1) 151 - 177, January, 2007. https://doi.org/10.2969/jmsj/1180135505