Descents and one-dimensional characters for classical Weyl groups

Abstract

This paper examine all sums of the form σ πϵW χ(π)t d(π) where W is a classical Weyl group, X is a one-dimensional character of W, and d( π) is the descent statistic. This completes a picture which is known when W is the symmetric group S n (the Weyl group A n−1 ). Surprisingly, the answers turn out to be simpler and generalize further for the other classical Weyl groups B n(≊C n) and D n . The B n, case uses sign-reversing involutions, while the D n case follows from a result of independent interest relating statistics for all three groups.