An inversion formula for induced Kazhdan–Lusztig polynomials and duality ofW-graphs

Abstract

Let \(\fancyscript{H}\) be the Hecke algebra associated with a Coxeter group W. Many interesting \(\fancyscript{H}\)-modules can be described using the concept of a W-graph, as introduced in the influential paper [Invent. Math. 53: 165–184,1979] of Kazhdan and Lusztig. In particular, Kazhdan and Lusztig showed that the regular representation of \(\fancyscript{H}\) has an associated W-graph. Let \(\fancyscript{H}_{J}\) be the Hecke algebra associated with WJ, a parabolic subgroup of W. In [Math Z 244:415–431, 2003], an algorithm was described for the construction of a so-called induced W-graph for an induced module \(\fancyscript{H} \bigotimes_{\fancyscript{H}_{J}}\)V, where V is an \(\fancyscript{H}_{J}\)-module derived from a WJ-graph. In this note, we continue to analyse the induced W-graphs and prove the following results: the induced Kazhdan–Lusztig polynomials for a pair of dual, induced W-graphs are related by an inversion formula. This result generalizes a result of Kazhdan and Lusztig [Invent Math 53:165–184, 1979. Theorem 3.1] that has already been generalized independently by Deodhar J Algebra 190:214–225,1997 and Matthew [Comm Algebra 18:371–387, 1990] the dual of an induced W-graph and the W-graph induced from the dual are associated to isomorphic \(\fancyscript{H}\)-modules(or simply, duality commutes with induction).