Weyl Groups are Finite — and Other Finiteness Properties of Cartan Subalgebras

Abstract

AbstractFor each pair (𝔤,𝔞) consisting of a real Lie algebra 𝔤 and a subalgebra a of some Cartan subalgebra 𝔥 of 𝔤 such that [𝔞, 𝔥]∪ [𝔞, 𝔞] we define a Weyl group W(𝔤, 𝔞) and show that it is finite. In particular, W(𝔤, 𝔥,) is finite for any Cartan subalgebra h. The proof involves the embedding of 0 into the Lie algebra of a complex algebraic linear Lie group to which the structure theory of Lie algebras and algebraic groups is applied. If G is a real connected Lie group with Lie algebra 𝔤, the normalizer N(𝔥, G) acts on the finite set Λ of roots of the complexification 𝔤c with respect to hc, giving a representation π : N(𝔥, G)→ S(Λ) into the symmetric group on the set Λ. We call the kernel of this map the Cartan subgroup C(𝔥) of G with respect to h; the image is isomorphic to W(𝔤, 𝔥), and C(𝔥)= {g G : Ad(g)(h)— h ε [h,h] for all h ε h }. All concepts introduced and discussed reduce in special situations to the familiar ones.The information on the finiteness of the Weyl groups is applied to show that under very general circumstance, for b ∪ 𝔥 the set 𝔥⊂ ϕ(b) remains finite as ϕ ranges through the full group of inner automorphisms of 𝔤.