The second paper of Kazhdan and Lusztig established a geometric setting for definition of Kazhdan–Lusztig polynomials, namely, the geometry of singularities of Schubert varieties in the flag variety. Much of the later work of Lusztig explored analogues of Kazhdan–Lusztig polynomials in the context of other natural singular algebraic varieties arising in representation theory, in particular, closures of nilpotent orbits and quiver varieties.
Kazhdan and Lusztig's second paper provided a geometric foundation for Kazhdan–Lusztig polynomials by studying the geometry of Schubert varieties' singularities within flag varieties. Later work by Lusztig explored analogous polynomials in various natural singular algebraic varieties that appear in representation theory, including closures of nilpotent orbits and quiver varieties.