Abstract: Whenever a subgroup of $GL_n$ acts on the flag variety $GL_n/B$ with finitely many orbits, it is a general fact that a polynomial representative of the cohomology class of a given orbit closure $Y$ can be found by first giving a representative of a closed orbit $Q$ below $Y$ in weak order, then applying to that representative some sequence of divided difference operators. However, there are choices of that initial representative to be made, and some choices may be more geometrically natural than others. In the case of the $B$-action on $GL_n/B$, the orbit closures are the Schubert varieties, and it is more or less a settled matter that the Schubert polynomials give the most natural choices of representatives of the classes, both from a combinatorial standpoint and from a geometric one, the latter having been elucidated by A. Knutson and E. Miller via Gröbner degenerations of matrix Schubert varieties. In this talk, I will consider the $K$-action on $GL_n/B$, where $K$ is the symmetric subgroup $GL_p \times GL_q$. Here, the question of which representatives are most natural is not as well-studied. I will describe a particular choice of representatives which seems particularly natural, and detail some of their known properties, including their self-consistency, their monomial positivity, and their known interpretation as multidegrees of Gröbner degenerations in certain cases, making contact with the aforementioned work of Knutson-Miller, and also related work of Knutson-Miller-Yong. I will also describe a number of conjectures regarding the cases which are not currently as well-understood from the standpoint of Gröbner geometry. Finally, I will indicate how the methods used to investigate these cohomological questions can be adapted to the problem of studying the local structure of $K$-orbit closures on $GL_n/B$. This is joint work with Alexander Yong.