Key and Lascoux polynomials for symmetric orbit closures

We introduce shifted analogues of key polynomials related to symplectic and orthogonal orbit closures in the complete flag variety. Our definitions are given by applying isobaric divided difference operators to the analogues of Schubert polynomials for orbit closures that correspond to dominant involutions. We show that our shifted key polynomials are linear combinations of key polynomials with nonnegative integer coefficients. We also prove that they are partial versions of the classical Schur $P$- and $Q$-polynomials. Finally, we examine $K$-theoretic generalizations of these functions, which give shifted forms of Lascoux polynomials. In the symplectic case, these generalizations are partial versions of the $GP$-polynomials introduced by Ikeda and Naruse. Besides developing basic properties, we identify a number of conjectures and open problems.