Characterization of Involutions of SP(2n,k)

Abstract

In this thesis, we discuss the relationship between involutions of the two matrix groups SL(2n,k) and SP(2n,k). Involutions determine symmetric spaces hence a complete classification of involutions of both SL(n,k) and SP(2n,k) will in turn classify the symmetric spaces coming from these involutions. We begin by giving a complete classification of involutions of the group SL(n,k) over the algebraically closed fields, the real numbers, the rational numbers, and the finite fields. As a method of classifying a particular type of involution of SL(n,k) we focus on how they may be obtained from a non-degenerate symmetric or skew-symmetric bilinear form. With the classification of involutions of SL(n,k) in hand we focus our attention on the subgroup SP(2n,k) of SL(2n,k). We first show that all involutions of SP(2n,k) are the restriction of an involution of SL(2n,k) to SP(2n,k). We determine that an automorphism theta=Inn_A leaves SP(2n,k) invariant if and only if A=pM for some p in k bar and M in SP(2n,k). Next we give specific criteria to characterize which involutions of SL(2n,k) remain involutions when restricted to SP(2n,k). Lastly, we determine that if two involutions of SP(2n,k) are isomorphic under SP(2n,k) then they are isomorphic under SL(2n,k).