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On a specialization of Toda eigenfunctions

Antoine Labelle

TL;DR

This work analyzes the specialization $\mathfrak{J}_\alpha$ of Toda-eigenfunction-type rational functions for $\alpha$ in the positive root lattice, establishing a simple denominator and palindromic numerator; it yields an explicit type $A$ combinatorial formula for the numerator and proposes a conjectural topological interpretation via a smooth projective variety whose Poincaré polynomial matches the numerator. The study unifies geometric, representation-theoretic, and combinatorial perspectives: $\mathfrak{J}_\alpha$ equals the Hilbert series of Zastava spaces and arises from a Shapovalov pairing of Whittaker vectors in quantum-group Verma modules, while the type $A$ formula expresses the numerator as a weighted sum over triangular arrays with Gaussian-binomial weights. In type $A$, the construction recovers Narayana and Catalan-type polynomials in special cases and connects to a conjectured topological interpretation, with partial evidence and clear open problems on positivity, unimodality, and potential Hikita-type correspondences. The work thus links algebraic combinatorics, geometric representation theory, and the topology of moduli spaces, suggesting deep structural ties across these domains.

Abstract

This paper studies rational functions $\mathfrak{J}_α(q)$, which depend on a positive element $α$ of the root lattice of a root system. These functions arise as Shapovalov pairings of Whittaker vectors in Verma modules of highest weight $-ρ$ 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 Poincaré polynomial of a smooth variety in type A.

On a specialization of Toda eigenfunctions

TL;DR

This work analyzes the specialization of Toda-eigenfunction-type rational functions for in the positive root lattice, establishing a simple denominator and palindromic numerator; it yields an explicit type combinatorial formula for the numerator and proposes a conjectural topological interpretation via a smooth projective variety whose Poincaré polynomial matches the numerator. The study unifies geometric, representation-theoretic, and combinatorial perspectives: equals the Hilbert series of Zastava spaces and arises from a Shapovalov pairing of Whittaker vectors in quantum-group Verma modules, while the type formula expresses the numerator as a weighted sum over triangular arrays with Gaussian-binomial weights. In type , the construction recovers Narayana and Catalan-type polynomials in special cases and connects to a conjectured topological interpretation, with partial evidence and clear open problems on positivity, unimodality, and potential Hikita-type correspondences. The work thus links algebraic combinatorics, geometric representation theory, and the topology of moduli spaces, suggesting deep structural ties across these domains.

Abstract

This paper studies rational functions , which depend on a positive element of the root lattice of a root system. These functions arise as Shapovalov pairings of Whittaker vectors in Verma modules of highest weight 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 Poincaré polynomial of a smooth variety in type A.

Paper Structure

This paper contains 8 sections, 13 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Geometric interpretation
  3. Representation-theoretic interpretation
  4. Denominator
  5. Combinatorial formula in type A
  6. A conjectural topological interpretation
  7. Conclusion and open problems
  8. Acknowledgements

Key Result

Proposition 1.1

In type $A_n$, with the standard ordering of the simple roots, and $\alpha=\sum_{i=1}^n a_i \alpha_i$, we have the recursion with the conventions that $a_i=0$ if $i\not\in\{1, \ldots, n\}$ and $\mathfrak{J}_\alpha=0$ if $\alpha \not\in Q^{\ge 0}$.

Theorems & Definitions (43)

  • Definition 1.1
  • Proposition 1.1: Toda recursion
  • Remark 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Remark 1.2
  • Theorem 2.1
  • ...and 33 more