The real-rootedness and log-concavities of coordinator polynomials of Weyl group lattices

It is well-known that the coordinator polynomials of the classical root lattice of type _An$A_n$ and those of type _Cn$C_n$ are real-rooted. They can be obtained, either by the Aissen–Schoenberg–Whitney theorem, or from their recurrence relations. In this paper, we develop a trigonometric substitution approach which can be used to establish the real-rootedness of coordinator polynomials of type _Dn$D_n$. We also find the coordinator polynomials of type _Bn$B_n$ are not real-rooted in general. As a conclusion, we obtain that all coordinator polynomials of Weyl group lattices are log-concave.