Closures of conjugacy classes in \(G_2\) (Q584366)

Let \(\mathfrak g\) be the simple Lie algebra of type \(G_2\) and \(C_i\) be the nilpotent conjugacy class in \(\mathfrak g\) of dimension \(i = 6, 8, 10\) and \(12\). In this paper the author shows the following: (a) every conjugacy class of \(\mathfrak g\) except \(C_8\) has a normal closure with rational singularities, (b) [\textit{T. Levasseur} and \textit{S. P. Smith}, J. Algebra 114, 81--105 (1988; Zbl 0644.17005)] \(\bar C_8\) is not normal in \(\bar C_6=\bar C_8\setminus C_8\). The normalization \(\eta_8: \tilde C_8\to \bar C_8\) is bijective and \(\tilde C_8\) has an isolated rational singularity in \(\eta_8^{-1}(0)\), (c) \(\bar C_{12}\) has a singularity of type \(D_4\) in \(C_{10}\) and \(\bar C_{10}\) a singularity of type \(A_1\) in \(C_8\). The proof of these results is based on the same construction as in Levasseur-Smith (loc. cit.). Namely, he embeds \(\mathfrak g\) into \(\mathfrak{so}_7\) by the 7-dimensional standard representation and studies the \(\mathfrak g\)-equivariant projection \(p: \mathfrak{so}_7\to \mathfrak g\). At the end, the author gives a brief summary of what is known about normality of closures of conjugacy classes in reductive groups.