Odd and even major indices and one-dimensional characters for classical Weyl groups

We define and study odd and even analogues of the major index statistics for the classical Weyl groups. More precisely, we show that the generating functions of these statistics, twisted by the one-dimensional characters of the corresponding groups, always factor in an explicit way. In particular, we obtain odd and even analogues of Carlitz's identity, of the Gessel--Simion Theorem, and a parabolic extension, and refinement, of a result of Wachs.