Relations between Characters of Lie Algebras and Symmetric Spaces

by Gagliardi, Daniel James

Gagliardi, Daniel James Gagliardi, Daniel James. Relations between Characters of Lie Algebras and Symmetric Spaces. (Under the direction of Dr. Aloysius Helminck.)

Let F be an irreducible root system. The Classification Theorem, ([Hum72, Section 11.4]), then states that its Dynkin diagram must be one of An, Bn, Cn, Dn, 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 q-diagrams. Implicit in each of these diagrams are two root systems F(a) and F(t) with a a maximal torus in a local symmetric space p and t a maximal torus in the corresponding semisimple Lie algebra g which contains a. In Chapter 2 we desribe F(a) as the image of F(t) under a projection p derived from an involution q on F(t). The weight lattices associated with F(t) and F(a) are denoted by Lt and La, respectively. We consider a linear extension of p from F(t) to the lattice Lt. It was shown, again in [Helm88], that p(Lt) is contained in La for cases where F(a) is not of type BCn. In this thesis we prove the converse of this result. For cases where F(a) is of type BCn it was shown in this same paper that p(Lt)=La=Ra. 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.