A formula for computing Kazhdan Lusztig characters in Hecke algebras of Coxeter groups is given. This formula holds for products of Coxeter elements of parabolic subgroups in an arbitrary Coxeter group, where the parabolic subgroups are of type A and the Kazhdan Lusztig representations are of finite dimension. The formula yields new combinatorial interpretations (of Murnaghan Nakayama type) to these characters and to other related characters. 1997 Academic Press 1. INTRODUCTION 1.1. The Murnaghan Nakayama rule is a most useful recursive rule for computing characters of the symmetric groups. For various applications of this rule cf. [FL, FOW, JK2.7, Ro1, St7.4, and SW]. The classical proofs of this rule use the Littlewood Richardson rule [J, Chap. 21], the Jacobi Trudi formula [JK, 2.4], and Schur polynomials [Mac, Sect. I, 7, Ex. 5].
A new proof of this rule, using the Young orthogonal form, appears in [Gr]. Recent papers generalize these proofs to Hecke algebras of type A, B, and D. See [AK, CK, HR, KW, P, Ra1, and vdJ]. Unfortunately, the machinery used in these proofs is not applicable for general Coxeter groups. In this paper we suggest a new approach to the Murnaghan Nakayama rule. We use Kazhdan Lusztig theory to prove this rule in a more general setting, i.e., general Coxeter groups and their Hecke algebras. The classical Murnaghan Nakayama rule (for type A) is obtained as a special case.
We prove a recursive rule for computing characters of certain elements in Hecke algebras of arbitrary Coxeter groups (Theorems 1 and 2). If q=1 these elements are conjugate to all products of Coxeter elements of commuting parabolic subgroups, where the parabolic subgroups are of type A. Note that in the symmetric group case, such elements represent the complete list of conjugacy classes.