An Inversion Formula for -Relative Kazhdan–Lusztig Polynomials

Abstract

The article proves a relative version of one of the results from the influential article [4] of Kazhdan and Lusztig which introduced the Kazhdan–Lusztig polynomials. Given a Coxeter group W and a set S of simple reflections, let ℋ denote the corresponding Hecke algebra, it has a “standard” basis T w and another basis C w with many remarkable properties. The Kazhdan–Lusztig polynomials p x, w give the transition matrix between these bases. One of their results proved by Kazhdan and Lusztig is an inversion formula, which states that if W is finite with longest element w 0, then for all x ≤ w in W. The main result of this article generalizes this result to the following setting: for any subset J of S, we define elements η J , and , and consider the two “dual” ideals and , where their standard basis and a Kazhdan–Lusztig basis are, respectively, indexed by subsets E J and of W.