Memoirs of the American Mathematical Society 2014; 109 pp; softcover Volume: 228 ISBN10: 0821890956 ISBN13: 9780821890950 List Price: US$76 Individual Members: US$45.60 Institutional Members: US$60.80 Order Code: MEMO/228/1072
 For a finite real reflection group \(W\) and a \(W\)orbit \(\mathcal{O}\) of flats in its reflection arrangementor equivalently a conjugacy class of its parabolic subgroupsthe 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 rightmultiplication within the group algebra \(\mathbb{C} W\) by the element that has \(\operatorname{noninv}_\mathcal{O}(w)\) as its coefficient on \(w\). Table of Contents  Introduction
 Defining the operators
 The case where \(\mathcal O\) contains only hyperplanes
 Equivariant theory of \(\mathsf{BHR}\) random walks
 The family \(\nu_{(2^k,1^{n2k})}\)
 The original family \(\nu_{(k,1^{nk})}\)
 Acknowledgements
 Appendix A. \(\mathfrak{G}_n\)module decomposition of \(\nu_{(k,1^{nk})}\)
 Bibliography
 List of Symbols
 Index
