Relations between characters of Lie algebras and symmetric spaces

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_n, B_n, C_n, D_n, E ₆, E₇, E₈, F₄, or G₂. This is fundamental to the study of finite-dimensional semisimple Lie algebras over algebraically closed fields. In [Hel88] A. G. Helminck established an analogous result for local symmetric spaces where he identified twenty-four graphical structures called involution or &thetas;-diagrams. Implicit in each of these diagrams are two root systems Φ($a$) and Φ($t$) with $a$ a maximal torus in a local symmetric space $p$ and $t\supset a$ a maximal torus in the corresponding semisimple Lie algebra $g$. In Chapter 2 we describe Φ($a$) as the image of Φ($t$) under a projection π derived from an involution &thetas; on Φ($t$). The weight lattices associated with Φ($t$) and Φ($a$) are denoted by $Lt$ and $La$, respectively. We consider a linear extension of π from Φ($t$) to the lattice $Lt$. It was shown, again in [Hel88], that π for cases where Φ($a$) is not of type BC_n. In this thesis we prove the converse of this result. For cases where Φ($a$) is of type BC_n it was shown in this same paper that π. 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.