On the Isomorphy Classes of Involutions over SO(2n, k)

Abstract

ABSHER, JOHN M. On the Isomorphy Classes of Involutions over SO(2n, k). (Under the direction of Dr. Aloysius Helminck). The study of symmetric spaces involves group theory, ï¬ eld theory, linear algebra, and Lie algebras, as well as involving the related disciplines of topology, manifold theory, and analysis. The notion of symmetric space was generalized in the 1980’s to groups deï¬ ned over arbitrary base ï¬ elds. In particular, if G is an algebraic group deï¬ ned over a ï¬ eld k of characteristic not 2, θ is an automorphism of order 2 of G, and H is the ï¬ xed point group of θ, then the homogeneous space G/H is called a symmetric space. It can be identiï¬ ed with the subvariety Q = {gθ(g)^{−1} | g ∈ G} of G. These generalized symmetric varieties are especially of interest in representation theory, especially when the base ï¬ eld k is the p-adic numbers, a ï¬ nite ï¬ eld or a number ï¬ eld. A full classiï¬ cation of these symmetric spaces for arbitrary ï¬ elds is still an open problem. The main focus for my thesis concerns a classiï¬ cation of these symmetric spaces for G the special orthogonal group deï¬ ned over an arbitrary ï¬ eld.