Bicolored Eulerian quasisymmetric functions, affine lengths, and Theta operators

We introduce a $q$-analogue of the bicolored Eulerian quasisymmetric functions by incorporating an additional grading arising from affine length, obtained by interpreting bicolored permutations as affine permutations. We also provide an intrinsic description of this refinement in terms of bicolored permutations by introducing notions of alignments and crossings, together with an explicit bijection identifying these statistics with the alignments and crossings in the existing literature. Finally, we prove that our refinement is symmetric by expressing it in terms of the Theta operators appearing in the Theta coinvariant Conjecture and the modified Macdonald symmetric functions. Assuming the Theta coinvariant conjecture implies the Schur-positivity of our refinement.