Spectra of Symmetrized Shuffling Operators

For a finite real reflection group $W$ and a $W$-orbit $\mathcal{O}$ of flats in its reflection arrangement—or equivalently a conjugacy class of its parabolic subgroups—the authors introduce a statistic $\operatorname{noninv}_\mathcal{O}(w)$ on $w$ in $W$ that counts the number of “$\mathcal{O}$-noninversions” of $w$. This generalizes the classical (non-)inversion statistic for permutations $w$ in the symmetric group $\mathfrak{S}_n$. The authors then study the operator $\nu_\mathcal{O}$ of right-multiplication within the group algebra $\mathbb{C} W$ by the element that has $\operatorname{noninv}_\mathcal{O}(w)$ as its coefficient on $w$.