On @Gjergji Zaimi's request, I turn my comment into an answer (and extend it a bit):

There were many (as far as I know unsuccessful) attempts to understand the major index in a way that makes it possible to uniformly generalize it to other finite Weyl/Coxeter groups, see e.g. http://arxiv.org/abs/math/0002245 and http://www.emis.de/journals/SLC/wpapers/s61Abiazen.pdf and the references therein.

Observe that any geometric approach that does generalize in this sense would provide such a generalized definition. I would therefore rather hope for a geometric interpretation that is strongly attached to the combinatorics/geometry of the symmetric group.

Finally: when not looking at the complete symmetric group $\mathcal{S}_n$ but only on **noncrossing permutations**, or, equivalently, on **Dyck paths**, the major index is related to **MacMahon's q-Catalan numbers**, see http://arxiv.org/abs/0808.2822.

There is a uniform generalization thereof as a bigraded Hilbert series of certain graded modules over the **rational Cherednik algebra** for any **well-generated complex reflection group**. Those can be found in http://arxiv.org/abs/math/0208138 and in http://www.maths.ed.ac.uk/~igordon/pubs/diag3.pdf.

But it is not at all clear if

- these notions can be used to define a statistic on noncrossing partitions (as a subset of all elements in a finite Weyl group),
- can be extended to notions of statistics on the complete finite Weyl group, or
- have anything to do with the weak length function on finite Weyl groups.