Signed Euler–Mahonian identities

A relationship between signed Eulerian polynomials and the classical Eulerian polynomials on S_n was given by Désarménien and Foata in 1992, and a refined version, called signed Euler–Mahonian identity, together with a bijective proof was proposed by Wachs in the same year. By generalizing this bijection, in this paper we extend the above results to the Coxeter groups of types B_n, D_n, and the complex reflection group G(r,1,n), where the ‘sign’ is taken to be any one-dimensional character. Some obtained identities can be further restricted on some particular set of permutations. We also derive some new interesting sign-balance polynomials for types B_n and D_n.