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 W _J , 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 W _J -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).