WEYL GROUPS ARE FINITE - AND OTHER FINITENESS PROPERTIES OF CARTAN SUBALGEBRAS

For each pair (g, a) consisting of a real Lie algebra g and a subalgeb ra a of some Gal-tan subalgebra h of g such that [a, h] subset of or e qual to [a, a] we define a Weyl group W(g, a) and show that it is fini te. In particular, W(g, h) is finite for any Cartan subalgebra h. The proof involves the embedding of g into the Lie algebra of a complex al gebraic linear Lie group to which the structure theory of Lie algebras and algebraic groups is applied. If G is a real connected Lie group w ith Lie algebra g, the normalizer N(h, G) acts on the finite set Lambd a of roots of the complexification g(c) with respect to h(c), giving a representation pi :N(h, G) --> S(Lambda) into the symmetric group on the set Lambda. We call the kernel of this map the Gal-tan subgroup C( h) of G with respect to h; the image is isomorphic to W(g, h), and C(h ) = {g is an element of G : Ad(g)(h) - h is an element of [h, h] for a ll h is an element of h}. All concepts introduced and discussed reduce in special situations to the familiar ones. The information on the fi niteness of the Weyl groups is applied to show that under very general circumstance, for b subset of or equal to h the set h boolean AND phi (b) remains finite as phi ranges through the full group of inner autom orphisms of g.