Regular functions on spherical nilpotent orbits in complex symmetric pairs: classical non-Hermitian cases (Q1689609)

Let \(G\) be a connected semisimple complex algebraic group and \(\theta:G\to G\) an algebraic involution. Then the set of fixed points \(K=G^\sigma\) is a reductive group, closed in \(G\). In this situation, the homogeneous space \(G/K\) is said to be \textit{symmetric}. It is well known that the Lie algebra \(\mathfrak g\) of \(G\) splits into a direct sum \(\mathfrak g=\mathfrak k \oplus \mathfrak p\), where \(\mathfrak k\) (the Lie algebra of \(K\)) is the eigenspace of eigenvalue \(1\) for the induced action of \(\theta\), and \(\mathfrak p\) is the eigenspace of eigenvalue \(-1\)). Moreover, \(\mathfrak k\) and \(\mathfrak p\) are stable for the adjoint action of \(G\) on \(\mathfrak g\), once restricted to \(K\). The \textit{isotropy representation} \(\mathfrak p\) of \(K\) presents interesting properties: for example, the \textit{nilpotent cone} \(\mathcal N_{\mathfrak p}\subset \mathfrak p\), defined as the set of elements whose \(K\)-orbit closure contains the origin, is the set of nilpotent elements of \(\mathfrak g\) belonging to \(\mathfrak p\), and by a result of \textit{B. Kostant} and \textit{S. Rallis} [Bull. Am. Math. Soc. 75, 879--883 (1969; Zbl 0223.53049)], \(\mathcal N_{\mathfrak p}\) is a finite union of \(K\)-orbits. In this paper -- presented as the first of a series -- the authors begin the systematic study of the closures of nilpotent \(K\)-orbits when \(K\) is connected and semisimple, or equivalently, when \(\mathfrak p\) is a simple \(K\)-module. In this case, the nilpotent \(K\)-orbits are \textit{spherical} -- that is, a Borel subgroup of \(K\) has an open orbit. The nilpotent orbits where classified by King (as spherical varieties) in [\textit{D. King}, J. Lie Theory 14, No. 2, 339--370 (2004; Zbl 1057.22020)]. The authors provide a full description of the normality of the spherical nilpotent closures; this description is obtained by giving a representative \(e\in {\mathfrak p}\), of each spherical nilpotent orbit, and describing Luna's spherical system associated to \(N_K(K_e)\), the normaliser of the isotropy group of \(e\). This (new) data is also used to compute the ring of regular functions of the closure as a (multiplicity free since \(K/K_e\) is spherical) \(K\)-module. These results are summarized in useful tables in an Appendix to the main text.