Mutual Interlacing and Eulerian-like Polynomials for Weyl Groups

We use the method of mutual interlacing to prove two conjectures on the real-rootedness of Eulerian-like polynomials: Brenti's conjecture on $q$-Eulerian polynomials for Weyl groups of type $D$, and Dilks, Petersen, and Stembridge's conjecture on affine Eulerian polynomials for irreducible finite Weyl groups. For the former, we obtain a refinement of Brenti's $q$-Eulerian polynomials of type $D$, and then show that these refined Eulerian polynomials satisfy certain recurrence relation. By using the Routh--Hurwitz theory and the recurrence relation, we prove that these polynomials form a mutually interlacing sequence for any positive $q$, and hence prove Brenti's conjecture. For $q=1$, our result reduces to the real-rootedness of the Eulerian polynomials of type $D$, which were originally conjectured by Brenti and recently proved by Savage and Visontai. For the latter, we introduce a family of polynomials based on Savage and Visontai's refinement of Eulerian polynomials of type $D$. We show that these new polynomials satisfy the same recurrence relation as Savage and Visontai's refined Eulerian polynomials. As a result, we get the real-rootedness of the affine Eulerian polynomials of type $D$. Combining the previous results for other types, we completely prove Dilks, Petersen, and Stembridge's conjecture, which states that, for every irreducible finite Weyl group, the affine descent polynomial has only real zeros.