Schubert supercells (Q1086315)

Let T be a linear superspace of dimension \(m| n\) over an algebraically closed field of characteristic \(\neq 2\). Let G denote one of the following algebraic supergroups which has T as the space of fundamental representation: \((a)\quad SL;\) \((b)\quad OSp:\) the automorphism group of an even symmetric form \(b: T{\tilde \to}T^*;\) \((c)\quad \Pi Sp:\) The same as in (b) but for an odd skewsymmetric form b (here \(m=n)\); \((d)\quad Q:\) the automorphism group of an odd involution \(p: T{\tilde \to}T,\) \(p^ 2=id\) \((m=n).\) The authors investigate here the homogeneous spaces \({}^ GF\) of complete flags in T which are invariant with respect to b or p for \(G\neq SL\). The decomposition of \(F\times F\) into G-orbits (in the sense of the theory of schemes) is obtained; the intersections of such orbits with the fibers of the projection onto F are classically called the Schubert cells. In a contrast with the classical case, F is, in general, reducible and G-orbits in \(F\times F\) are supervarieties. Formulas for the dimension of these orbits are given.