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.

Key Words:

• Coxeter group
• Hecke algebra
• Kazhdan–Lusztig polynomial
• Solomon's ideals

2000 Mathematics Subject Classification:

• Primary and Secondary 20C08

ACKNOWLEDGMENTS

The author would like to thank the referee for many helpful comments.

The project is sponsored by NSFC 10771068 and SRF for ROCS, SEM.

Notes

Communicated by D. K. Nakano.