Fundamental Involutory Root Systems and a Branching Theorem for Symmetric Pairs

Abstract

Let 𝔤 be a complex semisimple Lie algebra with an involutory automorphism θ. Let 𝔨 be the corresponding symmetric subalgebra and 𝔥 be a fundamental Cartan subalgebra of 𝔤 containing a Cartan subalgebra 𝔱 of 𝔨. Let Δ(𝔤, 𝔥) and Δ(𝔤, 𝔱) be the respective root systems of 𝔤 with respect to 𝔥 and 𝔱. The pair (Δ(𝔤, 𝔥), θ) is a fundamental involutory root system. We prove that there exists natural 1-1 correspondences between θ-stable Weyl chambers of Δ(𝔤, 𝔥) and Weyl chambers of Δ(𝔤, 𝔱) and that the Weyl group of Δ(𝔤, 𝔱) acts simply transitively on the above two sets of Weyl chambers. Regard a finite dimensional complex irreducible 𝔤-module V as a 𝔨-module, based on the above results we show that some certain irreducible modules of 𝔨 will occur in V.

Key Words:

• Branching
• Involutory root system
• Symmetric subalgebra

2000 Mathematics Subject Classification:

• Primary 17B20, 17B10

ACKNOWLEDGMENTS

I would like to take this opportunity to thank the referee for his suggestions to revise this article. I would also like to thank my colleague Yi-fang Kang for his computation of the cardinality of W ¹ for all the simple symmetric pairs (𝔤, 𝔨) with 𝔤 a simple Lie algebra, which provided the examples of the main branching theorem.

This research was supported by NSFC Grant 10726044 and NSFC Grant 10801116.

Notes

Communicated by A. Elduque.