Relations between Characters of Lie Algebras and Symmetric Spaces

Abstract

Let Φ be an irreducible root system. The Classification Theorem, ([Hum72, Section 11.4]), then states that its Dynkin diagram must be one of A[subscript n], B[subscript n], C[subscript n], D[subscript n], E₆, E₇, E₈, F₄, or G₂. This is fundamental to the study of finite-dimensional semisimple Lie algebras over algebraically closed fields. In [Helm88] A. G. Helminck established an analogous result for local symmetric spaces where he identified twenty-four graphical structures called involution or θ-diagrams. Implicit in each of these diagrams are two root systems Φ(a) and Φ(b) with a a maximal torus in a local symmetric space p and t ⊃ a a maximal torus in the corresponding semisimple Lie algebra g which contains a. In Chapter 2 we describe Φ(a) as the image of Φ(t) under a projection π derived from an involution θ on φ (t). The weight lattices associated with φ(t) and φ(a) are denoted by Λ[subscript t] and Λ[subscript a], respectively. We consider a linear extension of π from φ(t) to the lattice Λ[subscript t]. It was shown, again in [Helm88], that π(Λ[subscript t]) ⊆ Λ[subscript a] for cases where φ(a) is not of type BC[subscript n]. In this thesis we prove the converse of this result. For cases where φ(a) is of type BC[subscript n] it was shown in this same paper that &#960(Λ[subscript t]) = Λ[subscript a] = R[subscript a]. For these cases we offer a direct proof and for both cases provide explicit formulas for the characters of each in terms of the other.