On compactifications of the Steinberg zero-fiber (Q855962)

Let \(G\) be a connected semisimple linear algebraic group over an algebraically closed field of positive characteristic. The set of elements in \(G\) with semisimple part within a fixed \(G\)-conjugacy class is called a Steinberg fiber. The authors consider a certain distinguished Steinberg fiber \(N\) and prove that the closure \(\overline N\) of \(N\) in any smooth equivariant embedding \(X\) of \(G\) is normal and Cohen-Macaulay. Moreover, if \(X\) is smooth, then \(\overline N\) is a local complete intersection. As a biproduct it is deduced that \(\overline N\) has a canonical Frobenius splitting.