The Local $h$-Polynomials of Cluster Subdivisions Have Only Real Zeros

Athanasiadis raised the question whether the local $h$-polynomials of type $A$ cluster subdivisions have only real zeros. In this paper, we confirm this conjecture and prove the real-rootedness of local $h$-polynomials for all the other Cartan–Killing types. Our proofs mainly involve multiplier sequences and Chebyshev polynomials of the second kind.