Homotopy type of the nilpotent orbits in classical Lie algebras (Q6185319)

Let \(\mathfrak{g}\) be a real simple Lie algebra, and let \(G\) be the associated adjoint Lie group. An element \(X \in\mathfrak{g}\) is called \textit{nilpotent} if \(\mathrm{ad}(X) : \mathfrak{g}\to \mathfrak{g}\) is a nilpotent operator. For any nilpotent \(X\), let \(\mathcal{O}_X := \{\mathrm{Ad}(g)X\mid g\in G\}\) be the corresponding nilpotent orbit under the adjoint action of \(G\) on \(\mathfrak{g}\). Such nilpotent orbits form a rich class of homogeneous spaces. In fact, they lie on the interface of several disciplines in mathematics such as Lie theory, symplectic geometry, representation theory, and algebraic geometry (see e.g., [\textit{D. H. Collingwood} and \textit{W. M. McGovern}, Nilpotent orbits in semisimple Lie algebras. New York, NY: Van Nostrand Reinhold Company (1993; Zbl 0972.17008)]. Nevertheless, surprisingly, for a very long period there seems to have been hardly any literature on the topological invariants of these orbits, other than the description of the fundamental group in the case of simple Lie algebras. In recent work by the present authors [\textit{I. Biswas} et al., Kyoto J. Math. 60, No. 2, 717--799 (2020; Zbl 1519.57028)], homotopy types of nilpotent orbits have been explicitly described in the case of real simple classical Lie algebras for which any maximal compact subgroup in the associated adjoint group is not semisimple. The objective of this article is to complete and extend the above description of homotopy types of nilpotent orbits to the remaining cases of real simpleclassical Lie algebras for which any maximal compact subgroup in the associated adjoint group is semisimple.