On Maximal Subgroups of Real Lie Groups

Abstract

1.1. Let V denote a finite dimensional vector space over a field F. If x is an endomorphism of V and a e F, we denote by Va(x) the subset of V annihilated by some power of x a. A semi-simple endomorphism x (i.e., x has simple elementary divisors) is called F-diagonal if its eigenvalues are in F. A set S of endomorphisms is called F-diagonal if there is a base B in V such that the matrix of each element of S with respect to B is diagonal. If G is a real Lie algebra and h e G, we denote by Ga(h), G+(h), G_(h) and G*(h), respectively, the subspaces Ga(ad h), Ia>O Ga(h), Ia<oGa and Go(h) + G+(h). The main purpose of this paper is the proof of Theorem 3.1: Let G be a semi-simple real analytic group of linear transformations and let M be a maximal proper analytic subgroup. If M is unimodular, then its radical is compact. If M is not unimodular, then G/M is compact and the Lie algebra of M has the form G* (h) for some real-diagonal element ad h. The line of reasoning employed allows one to prove the conjugacy of maximal triangular subgroups (cf., ? 3) and to describe the maximal solvable analytic subgroups (? 4) of a Lie group. The conjugacy of the maximal triangular subgroups of an algebraic linear group over a perfect field of arbitrary characteristic has been proved by A. Borel and R. Godement, (cf., [2]) and our result on triangular subgroups is a straightforward consequence of theirs. We nevertheless present here our independent proof based on Lie algebras for the sake of methodological completeness.