On a specialization of Toda eigenfunctions

TL;DR

This paper explores specialized Toda eigenfunctions related to quantum groups, providing explicit formulas for their denominators and numerators, and proposing a geometric interpretation in type A.

Contribution

It introduces a new class of rational functions linked to Toda systems, with explicit combinatorial formulas and conjectural geometric realizations in type A.

Findings

Explicit combinatorial formula for the numerator in type A.

Analysis of the denominator of the rational functions.

Conjectural geometric interpretation as Poincaré polynomial in type A.

Abstract

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.