Usual torus action on Kazhdan-Lusztig varieties

Kazhdan–Lusztig varieties, defined as intersections of Schubert varieties with opposite Bruhat cells, are central objects in geometric representation theory, encoding information about singularities and Kazhdan–Lusztig polynomials. We give an insight into how to view them combinatorially and study the natural torus action on these varieties. Its complexity can be read off from associated graphs, providing a combinatorial approach to a geometric invariant. We study how the complexity depends on the permutations defining the Kazhdan-Lusztig variety.