Abstract
In this paper we define a two-variable, generic Hecke algebra,H\mathcal H, for each complex reflection groupG(b,1,n)G(b,1,n). The algebraH\mathcal Hspecializes to the group algebra ofG(b,1,n)G(b,1,n)and also to an endomorphism algebra of a representation ofGLn(Fq)\operatorname {GL}_n(\mathbb F_q)induced from a solvable subgroup. We construct Kazhdan-Lusztig “RR-polynomials” forH\mathcal {H}and show that they may be used to define a partial order onG(b,1,n)G(b,1,n). Using a generalization of Deodhar’s notion of distinguished subexpressions we give a closed formula for theRR-polynomials. After passing to a one-variable quotient of the ring of scalars, we construct Kazhdan-Lusztig polynomials forH\mathcal Hthat reduce to the usual Kazhdan-Lusztig polynomials for the symmetric group whenb=1b=1.
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