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 28.1 of the last chapter. Thus, G is semisimple and comes with a compact dual G c . In Chap. 18, 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 W rel 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. This chapter may be skipped with no loss of continuity.