Singularities in the nilpotent scheme of a classical group

Abstract:

If $(X,x)$ is a pointed scheme over a ring k, we introduce a (generalized) partition ${\text {ord}}(x,X/k)$. If G is a reductive group scheme over k, the existence of a nilpotent subscheme $N(G)$ of ${\text {Lie}}(G)$ is discussed. We prove that ${\text {ord}}(x,N(G)/k)$ characterizes the orbits in $N(G)$, their codimension and their adjacency structure, provided that G is $G{l_n}$, or $S{p_n}$ and $1/2 \in k$. For $S{O_n}$ only partial results are obtained. We give presentations of some singularities of $N(G)$. Tables for its orbit structure are added.