The actual question I want to ask is whether there is a geometric proof of this famous identity $$\sum_{\sigma \in S_n} q^{\operatorname{inv} \sigma}=\sum_{\sigma\in S_n}q^{\operatorname{maj}\sigma},$$ along the lines of interpreting both sides as the Poincare polynomial of some nice variety computed in two different ways.

Here $\operatorname{inv}$ and $\operatorname{maj}$ stand for number of inversions and Major index, respectively.

One possible candidate variety would be the full flag variety, where LHS can be easily interpreted as the generating function for the dimensions of the Schubert cells. Is there a way to see the major index in that context?