Let \(V\) be a \(2d\)-dimensional vector space with a non-degenerate quadratic form. The subset of \(\mathrm{Gr}(d,2d)\) consisting of \(d\)-dimensional isotropic subspaces is a subvariety with two isomorphic connected components, which are two orbits for the action of \(\mathrm{SO}(V)\). Every such a orbit is called the Orthogonal Grassmannian. On fix a maximal torus \(T\subset \mathrm{SO}(V)\) and a Borel subgroup \(T\subset B\). The closure of \(B\)-orbits (resp. of \(B^-\)-orbits) are called Schubert varieties (resp. opposite Schubert varieties). The intersection of a Schubert variety with an opposite Schubert variety is called a Richardson variety; in particuliar, Schubert varieties are Richardson varieties.
K. N. Raghavan and S. Upadhyay [J. Comb. Theory, Ser. A 116, No. 3, 663–683 (2009; Zbl 1168.14033)] gave an explicit description of the initial ideal (with respect to certain conveniently chosen term orders) for the ideal of the tangent cone at any \(T\)-fixed point of a Schubert variety in the Orthogonal Grassmannian. In this paper, this result is generalized to the case of Richardson varieties in the Orthogonal Grassmannian. One can remark that the local properties of the Schubert varieties at \(T\)-fixed points determine their local properties at all other points, because of the \(B\)-action; but this does not extend to the Richardson varieties, since Richardson varieties only have a \(T\)-action.
The proof is based on a generalization of the Robinson-Schensted-Knuth correspondence.