Singularities in the Nilpotent Scheme of a Classical Group

If is a pointed scheme over a ring k, we introduce a (generalized) partition <!-- MATH ${\text{ord}}(x,X/k)$ --> . If G is a reductive group scheme over k, the existence of a nilpotent subscheme of <!-- MATH ${\text{Lie}}(G)$ --> is discussed. We prove that <!-- MATH ${\text{ord}}(x,N(G)/k)$ --> characterizes the orbits in , their codimension and their adjacency structure, provided that G is , or and . For only partial results are obtained. We give presentations of some singularities of . Tables for its orbit structure are added.