Covexillary Schubert varieties and Kazhdan-Lusztig Polynomials

We establish combinatorial and inductive formulas for Kazhdan-Lusztig polynomials associated to covexillary elements in classical types, extending results of Boe, Lascoux-Sch"{u}tzenberger, Sankaran-Vanchinathan, and Zelevinsky for Grassmannians of classical types. The proof uses intersection cohomology theory and the isomorphism of Kazhdan-Lusztig varieties from Anderson-Ikeda-Jeon-Kawago.