Descents and one-dimensional characters for classical Weyl groups

This paper examine all sums of the form σ πε{lunate}Wχ(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.