Singularities in the nilpotent scheme of a classical group

Abstract

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