Let G be a reductive linear algebraic group defined over an algebraically closed field k whose characteristic is good for G. Let theta be an involution defined on G, and let K be the subgroup of G consisting of elements fixed by theta. The differential of theta, also denoted theta, is an involution of the Lie algebra g = Lie (G), and it decomposes g into +1- and -1-eigenspaces, k and p , respectively. The space p identifies with the tangent space at the identity of the symmetric space G/K. In this dissertation, we are interested in the adjoint action of K on p , or more specifically, on the nullcone Np , which consists of the nilpotent elements of p . The main result is a new classification of the K-orbits on Np .
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