On compactifications of the Steinberg zero-fiber

Let G be a connected semisimple linear algebraic group over an algebraically closed field k of positive characteristic and let X denote an equivariant embedding of G. We define a distinguished Steinberg fiber N in G, called the zero-fiber, and prove that the closure of N within X is normal and Cohen–Macaulay. Furthermore, when X is smooth we prove that the closure of N is a local complete intersection.