Abstract
The article proves a relative version of one of the results from the influential article [Citation4] 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
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.
2000 Mathematics Subject Classification:
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.