A subspace H of V is isotropic if the restriction of the form to H is identically zero, <H,H> = 0. The dimension of an isotropic subspace is at most n. The orthogonal Grassmannian OG(n) is the set of isotropic subspaces of V with this maximal dimension. This is an algebraic manifold of dimension N := n(n+1)/2.
A complete flag F. in V is isotropic if Fn is isotropic and <Fi,F2n+1-i> = 0 for all i=1, 2, ..., 2n-1. Given an isotropic flag F., the orthogonal Grassmannian has Schubert varieties Wf indexed by decreasing sequences f : n+1 > f1 > f2 > ... > fm > 0 of positive integers, called strict partitions. (Here m can be any integer between 0 and n). The codimension of the Schubert variety Wf is |f| := f1 + f2 + ... + fm. The orthogonal Schubert calculus asks for the number of points in a transverse zero-dimensional intersection of Schubert varieties.
The simple Schubert variety W1F. consists of those maximal isotropic subspaces meeting the subspace Fn+1 non-trivially. The simple orthogonal Schubert calculus is fully real.
It remains an open problem whether the general Schubert calculus for the orthogonal Grassmannian is fully real.