Signed Mahonian polynomials for major and sorting indices

We derive some new signed Mahonian polynomials over the complex reflection group $G(r,1,n)=C_r\wr {\mathfrak{S}}_n$, where the “sign” is taken to be any of the $2r$ $1$-dim characters and the “Mahonian” statistics are the $\mathsf{l}\mathsf{m}\mathsf{a}\mathsf{j}$ defined by Bagno and the $\mathsf{s}\mathsf{o}\mathsf{r}$ defined by Eu et al. Various new signed Mahonian polynomials over Coxeter groups of types $B_n$ and $D_n$ are derived as well. We also investigate the signed counting polynomials on $G(r,1,n)$ for those statistics with the distribution $[r]_q[2r]_q\cdots [nr]_q$.