On a remarkable class of subvarieties of a symmetric variety
Let G be a semisimple algebraic group, \(\theta\) : \(G\to G\) an involution and H the subgroup of \(\theta\)-fixed elements of G. Then G/H is a symmetric space for the involution \(\theta\). Denote by X the G- equivariant compactification of G/H constructed by De Concini and Procesi and identify the points x of X with their corresponding subalgebras \(\ell_ x\) in the Lie algebra \({\mathfrak g}\) of G. Let \(T_ 1\subset G\) be a maximal anisotropic torus of G with Lie algebra \({\mathfrak t}_ 1\). For \(y\in {\mathfrak t}_ 1\) define \(Z_ y\) to be the subvariety of all points x in X with \(\ell_ x^{\perp}\ni y\), the orthogonal complement being taken with respect to the Killing form. Then the group \(C_ y:=Stab_ G(y)\) acts on \(Z_ y\). The closure \(\bar T_ 1\) in X of the orbit of the point \(H\in G/H\) under the group \(T_ 1\) is contained in \(Z_ y\) for all \(y\in {\mathfrak t}_ 1\). The author proves the following very interesting theorem: Let \({\mathcal O}\) be a G-orbit in X and \(y\in {\mathfrak t}_ 1.\) (i) \(Z_ y\cap {\mathcal O}\) is a union of a finite number of \(C_ y\)- orbits. (ii) Each \(C_ y\)-orbit in \(Z_ y\) intersects \(\bar T_ 1\) in a union of \(T_ 1\)-orbits, which are transitively permuted by the group \(W_{1y}:=Stab_{W_ 1}(y)\), where \(W_ 1\) denotes the restricted Weyl group. (iii) The \(C_ y\)-orbits in \(Z_ y\cap {\mathcal O}\) are closed in X if and only if \({\mathcal O}\) is the closed orbit in X. (iv) \(Z_ y\) is smooth and connected. (v) The closures of the \(C_ y\)-orbits in \(Z_ y\) are smooth and each one is the transverse intersection of the codimension one orbit closures that contain it. The theorem generalizes a result of \textit{C. De Concini}, \textit{M. Goresky}, \textit{R. MacPherson} and \textit{C. Procesi} [``On the geometry of complete quadrics'' (preprint, Inst. Hautes Étud. Sci. 1986); see also Comment. Math. Helv. 63, No.3, 337-413 (1988))] where the case of the symmetric space of quadrics is considered.
scientific article; zbMATH DE number 4123905
scientific article