Double Whittaker Polynomials

• Double Whittaker polynomials are multivariable special functions defined through two sets of spectral parameters, unifying classical, q-, spin, and factorial deformations.
• They use solvable lattice models and Yang–Baxter integrability to establish recursion relations and connect combinatorial models with quantum group symmetries.
• They underpin applications in nonarchimedean harmonic analysis, representation theory, and geometry by encoding stable envelopes, motivic Chern classes, and matrix coefficients.

Double Whittaker polynomials are multivariable special functions exhibiting deep combinatorial, representation-theoretic, and integrable-system structures, simultaneously generalizing the classical Whittaker polynomials, $q$-Whittaker polynomials, and their “spin” and “factorial” deformations. Most distinctively, double Whittaker polynomials depend on two sets of spectral variables and encode two interacting gradings or deformation parameters, typically arising in context of nonarchimedean harmonic analysis, solvable lattice models, and quantum group symmetries. They are closely linked to the representation theory of $\mathrm{GL}(n,F)$ over nonarchimedean local fields, motivic Chern classes, matrix coefficients of intertwining operators, and geometric objects such as motivic stable envelopes and double Grothendieck/Schubert polynomials. For the current state of the art, see (Feigin et al., 2016, Rahman et al., 2017, Bhattacharya et al., 28 Nov 2024, Mucciconi, 1 Feb 2025, Brubaker et al., 22 Sep 2025).

1. Definition and Characterization

Double Whittaker polynomials, as formalized in (Brubaker et al., 22 Sep 2025), are families of functions depending on two sets of variables, say $x = (x_1, \dots, x_n)$ and $y = (y_1, y_2, \dots)$, and parameterized by suitable combinatorial or representation-theoretic data (such as Weyl group elements, partitions, or signatures). The canonical realization uses the partition functions of exactly solvable vertex models, with boundary conditions and Boltzmann weights carefully chosen so that the resulting polynomial encodes combinatorial and geometric data for double (bispectral) objects:

$$\mathrm{DW}_\lambda(x,y;\mathbf{a},\mathbf{b}; \kappa) = \sum_{T\in\mathcal{T}_\lambda} W(T;x,y;\mathbf{a},\mathbf{b};\kappa)$$

where $\mathcal{T}_\lambda$ indexes admissible states (for example, Gelfand–Tsetlin patterns, column strict fillings, or colored lattice path states) and $W(T; \ldots)$ is the explicit monomial or partition function weight, depending sensitively on two sets of “spectral” variables $x, y$, and typically on deformation or “spin” variables $\mathbf{a}, \mathbf{b}$, as well as a parameter $\kappa$.

In the boundary specialization $y_i\to 0$, double Whittaker polynomials reduce to colored lattice model partition functions representing Iwahori Whittaker functions for $\mathrm{GL}(n,F)$ (Brubaker et al., 22 Sep 2025). With boundary conditions chosen for motivic Chern classes, they encode stable envelopes and Kazhdan–Lusztig polynomial deformations.

2. Lattice Model Formulation and Structural Properties

The central construction uses solvable (integrable) lattice models whose states are colored lattice paths or decorated vertex configurations, with partition functions of the form

$$Z^\Box_{x,y} = \sum_{\text{states}} \left[ \prod_{\text{vertices}} \text{Boltzmann weight}(x, y, \mathbf{a}, \mathbf{b}) \right]$$

The partition function $Z^\Box$ depends on row variables $x_i$ and column variables $y_j$, with the specific arrangements of colored paths and vertex weights designed to realize Yang–Baxter integrability and Demazure-Lusztig operator recursions.

Key combinatorial operations (such as color merging, pipe dream enumeration, and decorated pattern overlays) govern the transition between double Whittaker, double Grothendieck/Schubert, and factorial Schur polynomial limits (Brubaker et al., 22 Sep 2025). Notably, the recursion relations for corresponding $r_{u,v}$-polynomials (Kazhdan–Lusztig deformations) emerge as direct reflections of the lattice model’s integrability structure.

3. Connections to Representation Theory and Harmonic Analysis

Double Whittaker polynomials generalize the graded character picture of generalized global Weyl modules for current/affine Lie algebras (Feigin et al., 2016), capturing a two-parameter refinement (energy grading and $t$-grading) and thus interpolating between nonsymmetric Macdonald polynomials and their Whittaker specialized limits.

In the context of nonarchimedean local harmonic analysis, the lattice model realization yields Iwahori Whittaker functions for $\mathrm{GL}(n,F)$ (upon $y_i\to 0$), and provides explicit formulas for motivic Chern classes and matrix coefficients of intertwining operators (Brubaker et al., 22 Sep 2025). The denominator conjecture for Kazhdan–Lusztig polynomials is reproved combinatorially by exhibiting the analytic cancellation of apparent poles upon multiplication by suitable Weyl denominator factors.

When indexed appropriately, double Whittaker polynomials specialize to double Schubert or double Grothendieck polynomials and encode pipe dream combinatorics, reflecting the geometric structure of Schubert varieties and K-theoretic stable envelopes (Brubaker et al., 22 Sep 2025).

4. Combinatorial Models, Compression, and Branching

Double Whittaker polynomials admit multiple combinatorial representations using generalized pattern overlays, Gelfand–Tsetlin arrays with two gradings, colored lattice paths, or (p,q)-deformed tableaux (Bhattacharya et al., 28 Nov 2024, Rahman et al., 2017). In particular, compression phenomena arise naturally: summing over “richer” sets of fillings or path configurations can be grouped and compressed to classical formulas analogously to the passage from alcove walk/Ram–Yip formulations to Tokuyama-type formulas for spherical Whittaker polynomials (Lenart et al., 2021).

Branching and projection maps in the relevant combinatorial models mirror the representation-theoretic direct limits, e.g., for chains of local Weyl modules for affine Lie algebras (Bhattacharya et al., 28 Nov 2024). Bijections preserving combinatorial weight and gradings allow double Whittaker polynomials to interpolate between different basis families and connect lattice path intersection statistics to polynomial gradings.

5. Recursion Formulae, Yang–Baxter Equations, and Analytical Structure

Recursion relations such as those for $r_{u,v}$-polynomials (Kazhdan–Lusztig deformations) are derivable from the Yang–Baxter equations governing the underlying lattice models (Brubaker et al., 22 Sep 2025). Demazure-Lusztig type operators acting in the space of spectral variables realize analytic continuation and symmetries, and their preservation of key “GKM-type” subrings guarantees analytic behavior and cancellation of potential poles.

Upon specialization (e.g., $t=0$, $y_i=0$), these recursions reduce to classical Pieri-type rules, Tokuyama identities, or Demazure operator formulas for Whittaker/Grothendieck polynomials.

6. Parameter Specializations and Limiting Cases

Double Whittaker polynomials unify and generalize several function families via parameter specializations:

• Setting $y$ variables to zero yields colored lattice models matching Iwahori Whittaker functions and single-parameter Whittaker polynomials (Brubaker et al., 22 Sep 2025).
• Setting deformation parameters $t=0$, spin parameters $\mathbf{a}, \mathbf{b}=0$, or merging colors recovers double Grothendieck polynomials (and, for appropriate choices, Schur or factorial Schur functions).
• Limiting procedures connect the discrete $q$-Whittaker setting to continuous spin Whittaker functions upon $q\to 1$ or suitable scaling of spectral variables (Mucciconi, 1 Feb 2025).

A plausible implication is that further generalizations, such as double spin Whittaker polynomials or mixed bispectral objects, are achievable via modifications to the lattice model boundary and parameter sets.

7. Applications in Geometry, Probability, and Quantum Integrable Systems

Double Whittaker polynomials, through their lattice model and motivic Chern class connections, have implications for the paper of matrix coefficients in nonarchimedean representation theory, factorization phenomena in Weyl groups, and partition functions in integrable statistical mechanics models.

They encode stable envelopes in K-theory for equivariant cohomology, arise in enumerative geometry of Schubert varieties, and provide state sum and combinatorial proofs for conjectures such as the denominator conjecture for Kazhdan–Lusztig R-polynomials.

In stochastic integrable systems, generalizations of the double Whittaker structure underpin multilevel stochastic vertex models, polymer models, and correspondences with quantum Toda Hamiltonian eigenfunctions (Mucciconi et al., 2020, Bhattacharya et al., 28 Nov 2024).

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In summary, double Whittaker polynomials constitute a foundational family in modern algebraic combinatorics, representation theory, and integrable systems, realized via solvable lattice models depending on two variable sets and encoding two gradings. Their analytic, geometric, and combinatorial properties are governed by integrable recursions, representation-theoretic direct limits, and compression phenomena, with broad applications in the algebraic structures associated to symmetric functions, quantum groups, and nonarchimedean harmonic analysis.