Semi-Simple Algebraic Groups Defined Over a Real Closed Field

Abstract

Introduction. In this paper, we extend some of the classical theory of semi-simple algebraic groups and Lie algebras over the real numbers to an arbitrary real closed field. The existence of a Cartan decomposition for a semi-simple Lie algebra over a real closed field k is shown in ? 2. Such a decomposition is unique up to an inner automorphism of the Lie algebra. In ? 3, the theory of k-compact algebraic groups is developed. For such groups, any two maximal tori defined over k are conjugate by a k-rational element in the group . It is this fact that is utilized in ? 4 to show that the classification theory of connected semi-simple algebraic groups over k is the same as over the field of real numbers. A consequence of the existence of a Cartan decomposition for semi-simple Lie algebras over k is the existence of an Iwasawa decomposition. The corresponding decomposition for connected semi-simple algebraic groups defined over k also exists. These proofs are given in ? 5 and several application in ? 6. 1. Cartan decomposition for elements. In this section, we shall let k be a real closed field and C kV- 1. The non-trivial automorphism of C over k will be denoted by z zo. Also, G shall denote an algebraic group