The $\gamma$-positivity of basic Eulerian polynomials via group actions

We provide combinatorial interpretation for the $\gamma$-coefficients of the basic Eulerian polynomials that enumerate permutations by the excedance statistic and the major index as well as the corresponding $\gamma$-coefficients for derangements. Our results refine the classical $\gamma$-positivity results for the Eulerian polynomials and the derangement polynomials. The main tools are Br"and'en’s modified Foata–Strehl action on permutations and the recent triple statistic (des, rix,aid) equidistibuted with (exc, fix, maj).