Odd symplectic flag manifolds (Q2463785)

This paper studies the odd symplectic group introduced by \textit{R. A.~Proctor} [Invent. Math. 92, 307--332 (1998; Zbl 0621.22009)] from a geometric point of view. This group \(\text{Sp}_{2n+1}\) (where \(n\) is a positive integer) is defined as the of group linear transformations of a complex vector space \(E\) of dimension \(2n+1\) that preserve a skew form \(\omega \in \bigwedge^2 E^*\) of rank \(2n\). It is a \textit{non}-reductive connected linear algebraic group. Mihai introduces odd symplectic [generalized] flag varieties as the varieties of flags of isotropic subspaces of \((E, \omega)\): \[ \mathbb{F}_{\omega}(d_1,\dots,d_r,E) := \{(V_{d_1} \subset \cdots \subset V_{d_r} \subset E)\mid \dim V_{d_i}= d_i, V_{d_i} \text{ isotropic }\forall 1\leq i \leq r\} \] where \(1\leq d_1 < \cdots < d_r \leq n+1\) are integers. He shows that they are smooth projective varieties which are quasi-homogeneous (i.e. have finitely many orbits) under the natural action by \( \text{Sp}_{2n+1}\). He does not consider the homogeneous spaces \(\text{Sp}_{2n+1}/P\) (where \(P\) is a parabolic subgroup) because they are isomorphic to the flag manifolds of the symplectic group \(\text{Sp}_{2n}\) and ``therefore do not constitute representative examples for the odd symplectic sytuation.'' The author then studies the odd symplectic Grassmannian \(G_{\omega}(k,E) := \mathbb{F}_{\omega}(k,E)\) and the odd symplectic flag variety \({F}_{\omega}(E) := {F}_{\omega}(1,\dots,n+1,E)\) in more detail. In the first main result of the paper he determines the automorphism group of the variety \(G_{\omega}(k,E)\). Combining this with the corresponding result for the (even) symplectic result he has thus obtained the following theorem: for integers \(N\) and \(k\) such that \(2 \leq k \leq [N/2]\), the automorphism group of the variety \(G_{\omega}(k, \mathbb{C}^N)\) is \(\text{Sp}_{N}/\{\pm 1\}\). The second main result is an analogue of the Borel--Weil theorem. Because the odd symplectic group is not reductive this requires some care: which representations should replace the simple modules in the Borel-Weil theorem? The author considers the \(\text{Sp}_{2n+1}\)-modules introduced by Proctor [loc. cit.] which ``are defined by carrying over to the odd symplectic setting Weyl's construction of the simple \(\text{Sp}_{2n}\)-modules.'' Here too he obtains a formulation for all integers \(N\) which reduces to the usual Borel--Weil theorem when \(N\) is even. As the author says, these two results ``allow us to picture the symplectic and odd symplectic flag manifolds as a `series'.''