On adjacencies of orbits of a symplectic group action on the manifold of complete flags in a symplectic space (Q1356269)

Adjacencies and local singularities of the stratification of Schubert cells on the manifold of complete flags are investigated in [\textit{M. Eh. Kazaryan}, Russ. Math. Surv. 46, No. 5, 91-136 (1991); translation from Usp. Mat. Nauk 46, No. 5(281), 79-119 (1991; Zbl 0783.32015)]. Adjacencies of orbits of the symplectic group \(\text{Sp}(2n,\mathbb{R})\) on the manifold of complete flags \(\mathbb{F}_{2n}\) in the linear symplectic space \((\mathbb{R}^{2n},\omega)\) are studied in this article. The orbit \({\mathbf v}_1\) adjoins the orbit \({\mathbf v}_2\) if \({\mathbf v}_1\subset\overline {\mathbf v}_2\). Let us fix the Darboux coordinates \((p, q)\), \(\omega = dp \wedge dq\). It is easy to prove that some coordinate flag consisting of coordinate planes is contained in each orbit. A simple computation shows that the number of these orbits is finite and this number is equal to \((2n-1)!!=1\cdot 3 \dots(2n-1)\). The orbits of the action of \(\text{Sp} (2n,\mathbb{R})\) on \(\mathbb{F}_{2n}\) are characterized by degenerations of restrictions of the symplectic form \(\omega\) on the subspaces of the flags of these orbits. For low dimensions, the adjacency diagrams have the following form. For \(n = 2\), we have three orbits and the adjacency diagram \({\mathbf M}_1\leftarrow {\mathbf M}_2\leftarrow {\mathbf M}_3\), where \({\mathbf M}_1\) consists of flags \({\mathbf V}^1\subset {\mathbf V}^2\subset {\mathbf V}^3\subset\mathbb{R}^4\) with symplectic subspace \({\mathbf V}^2\), \({\mathbf M}_2\) consists of flags with Lagrange subspaces \({\mathbf V}^2\) and \({\mathbf V}^1\) different from \(\ker\omega|_{{\mathbf V}^3}\), and \({\mathbf M}_3\) consists of flags with Lagrange subspaces \({\mathbf V}^2\) and \({\mathbf V}^1\) coinciding with \(\ker\omega |_{{\mathbf V}^3}\). When \(n = 3\), we have fifteen orbits and a certain adjacency diagram shown in the article. This diagram is symmetrical with respect to the plane of the vertices 1, 2, 3, 4, 5 if it is considered to be a three-dimensional diagram. A geometrical sense of this symmetry will be explained below.