Relative Root Systems

Abstract

In this chapter, we will consider root systems and Weyl groups associated with a Lie group G. We will assume that G satisfies the assumptions in Hypothesis 31.1 of the last chapter. Thus, G is semisimple and comes with a compact dual G c , In Chapter 19, we associated with G c ,a root system and Weyl group. That root system and Weyl group we will call the absolute root system Φ and Weyl group W. We will introduce another root system Φ rel called the relative or restricted root system, and a Weyl group Wrel called the relative Weyl group. The relation between the two root systems will be discussed. The structures that we will find give Iwasawa and Bruhat decompositions in this context.