Regular functions on spherical nilpotent orbits in complex symmetric pairs: exceptional cases (Q2216694)

Let \(G\) be a connected simple complex algebraic group and let \(K\) be the fixed point subgroup of an involution of \(G\). Let \(\mathfrak{p}\) be the \(-1\)-eigenspace for \(\theta\) in the Lie algebra \(\mathfrak{g}\) of \(G\), which is stable under the adjoint action of \(K\). The objects under study are the \(K\)-orbits of nilpotent elements in \(\mathfrak{p}\) which are spherical (classified by \textit{D. R. King} [J. Lie Theory 14, No. 2, 339--370 (2004; Zbl 1057.22020)]), and their closures. In a series of three articles (the article under review, together with [\textit{P. Bravi} and \textit{J. Gandini}, Kyoto J. Math. 60, No. 2, 405--450 (2020; Zbl 1470.14105)] and [\textit{P. Bravi} et al., Kyoto J. Math. 57, No. 4, 717--787 (2017; Zbl 1402.14068)]), with a case by case analysis, the authors determine the spherical systems encoding the orbits as spherical homogeneous spaces, then they use the theory of spherical varieties to determine if multiplication of sections of globally generated line bundles on a wonderful variety associated with such an orbit is surjective, finally, they use this to conclude on the normality of orbit closures. This article is the third and final of the series. It is not to be read without the first two as it makes use of notations and results from those. The first two papers in the series dealt with symmetric pairs under classical groups. The remaining exceptional cases are dealt with in the paper under review. In this situation, there is only one nilpotent orbit closure which is not normal (in the case \(G=G_2\) and \(K\) of type \(A_1\times A_1\)).