## Journal of the Mathematical Society of Japan

### Multiplicity-free branching rules for outer automorphisms of simple Lie algebras

Hidehisa ALIKAWA

#### Abstract

We find explicit multiplicity-free branching rules of some series of irreducible finite dimensional representations of simple Lie algebras $\mathfrak g$ to the fixed point subalgebras $\mathfrak g^{\sigma}$ of outer automorphisms $\sigma$. 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 $(\mathfrak g, \mathfrak g^{\sigma})$ includes an exceptional symmetric pair $(E_6, F_4)$ and also a non-symmetric pair $(D_4, G_2)$ 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 $\mathfrak g$ in terms of the subalgebras $\mathfrak g^{\sigma}$ on one hand, and the length function on the other hand.

#### Article information

Source
J. Math. Soc. Japan, Volume 59, Number 1 (2007), 151-177.

Dates
First available in Project Euclid: 25 May 2007

https://projecteuclid.org/euclid.jmsj/1180135505

Digital Object Identifier
doi:10.2969/jmsj/1180135505

Mathematical Reviews number (MathSciNet)
MR2302667

Zentralblatt MATH identifier
1136.17005

#### Citation

ALIKAWA, Hidehisa. Multiplicity-free branching rules for outer automorphisms of simple Lie algebras. J. Math. Soc. Japan 59 (2007), no. 1, 151--177. doi:10.2969/jmsj/1180135505. https://projecteuclid.org/euclid.jmsj/1180135505

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