Nestings of rational homogeneous varieties (Q2120894)

\textit{C. De Concini} and \textit{Z. Reichstein} [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 15, No. 2, 109--118 (2004; Zbl 1219.14052)] classified the possible morphisms \(f: \mathbb G(k,n) \to \mathbb G(r,n)\) where \(k<r\) and \(\mathbb G(k,n)\) denotes the Grassmanian of linear subspaces of dimension \(k\) in \(\mathbb P^n\). They called such morphisms \emph{nestings} and showed that nestings only exist when \(n\) is odd, \(k=0\) and \(r=n-1\). In the paper under review, the authors study a generalisation of such problem in the following way. Let \(G\) be a semisimple algebraic group over \(\mathbb C\) and \(P \subset P' \subset G\) two parabolic subgroups. A \emph{nesting} for \(P \subset P' \subset G\) is a section of the natural projection of rational homogeneous varieties \(G/P \to G/P'\). The main result is a complete classification of nestings when \(G\) is simple of classical type (i.e.\ the Dynkin diagram of \(G\) is \(A_n, B_n, C_n\) or \(D_n\)). More precisely, they prove that besides the ones obtained by De Concini and Reichstein, there exist only another infinite family and a sporadic case (both explicitly described in the paper). As a consequence, they obtain necessary and sufficient conditions for the existence of subbundles of the universal quotient bundle on rational homogeneous spaces of classical type and Picard number 1.