On a specialization of Toda eigenfunctions

This paper studies rational functions $\mathfrak{J}_\alpha(q)$, which depend on a positive element $\alpha$ of the root lattice of a root system. These functions arise as Shapovalov pairings of Whittaker vectors in Verma modules of highest weight $-\rho$ for quantum groups and as Hilbert series of Zastava spaces, and are related to the Toda system. They are specializations of multivariate functions more commonly studied in the literature. We investigate the denominator of these rational functions and give an explicit combinatorial formula for the numerator in type A. We also propose a conjectural realization of the numerator as the Poincar\'e polynomial of a smooth variety in type A.