Generic Hecke algebras for monomial groups

In this paper we define a two-variable, generic Hecke algebra, $\mathcal H$, for each complex reflection group $G(b,1,n)$. The algebra $\mathcal H$ specializes to the group algebra of $G(b,1,n)$ and also to an endomorphism algebra of a representation of $\operatorname{GL}_n(\mathbb{F}_q)$ induced from a solvable subgroup. We construct Kazhdan-Lusztig ``$R$-polynomials'' for $\mathcal{H}$ and show that they may be used to define a partial order on $G(b,1,n)$. Using a generalization of Deodhar's notion of distinguished subexpressions we give a closed formula for the $R$-polynomials. After passing to a one-variable quotient of the ring of scalars, we construct Kazhdan-Lusztig polynomials for $\mathcal H$ that reduce to the usual Kazhdan-Lusztig polynomials for the symmetric group when $b=1$.