q-Whittaker Functions: Theory & Applications

• q-Whittaker functions are defined as the t=0 specialization of Macdonald polynomials, serving as q-deformations of Schur functions with key applications in representation theory and combinatorics.
• They exhibit orthogonality under the Hall scalar product and admit tableau expansions and Cauchy identities that elucidate finite field flag enumerations.
• Their algebraic structure connects quantum q-Toda systems, graded Demazure modules, and nonsymmetric generalizations, offering rich combinatorial and geometric insights.

The $q$-Whittaker functions form a cornerstone in the interplay between the theory of symmetric functions, quantum integrable systems, and the combinatorial geometry of finite fields and flag varieties. Defined as the $t=0$ specialization of Macdonald polynomials, $q$-Whittaker functions $W_\lambda(\mathbf{x};q)$ serve as $q$-deformations of Schur functions, interpolating between complex and $p$-adic Whittaker functions, and providing an organizing basis for explicit symmetric function expansions connected to subspace profiles, operator enumeration, and representation theory.

1. Definitions, Normalizations, and Fundamental Properties

Let $\Lambda_{(q)}$ denote the ring of symmetric functions in variables $\mathbf{x}=(x_1, x_2, \ldots)$ with coefficients in $\mathbb{Q}(q)$. The $q$-Whittaker functions are defined as

$t=0$0

where $t=0$1 is the (integral form of the) Macdonald polynomial indexed by the partition $t=0$2.

Orthogonality and Normalization: In the Hall scalar product $t=0$3 (with orthogonal power sums),

$t=0$4

for $t=0$5, the $t=0$6-Whittaker functions are unitriangular in the monomial basis and satisfy

$t=0$7

The dual $t=0$8-Whittaker basis $t=0$9 is characterized by

$q$0

with explicit normalization

$q$1

where $q$2, $q$3.

2. Combinatorial Models and Flag Expansions

The $q$4-Whittaker functions admit a tableau expansion

$q$5

where the $q$6-weight is

$q$7

Finite Field Interpretation: For a finite field $q$8 of size $q$9 and a nilpotent endomorphism $W_\lambda(\mathbf{x};q)$0 of $W_\lambda(\mathbf{x};q)$1 with Jordan type $W_\lambda(\mathbf{x};q)$2, the coefficient of $W_\lambda(\mathbf{x};q)$3 in $W_\lambda(\mathbf{x};q)$4 counts flags $W_\lambda(\mathbf{x};q)$5 strictly compatible with $W_\lambda(\mathbf{x};q)$6: $W_\lambda(\mathbf{x};q)$7 This underlies the geometric connection between $W_\lambda(\mathbf{x};q)$8-Whittaker functions and Springer fibers, flag varieties, and quiver loci (Karp et al., 2022).

3. Algebraic and Representation-Theoretic Constructions

$W_\lambda(\mathbf{x};q)$9-Whittaker functions are characterized as unique $q$0-invariant eigenfunctions of the quantum $q$1-Toda system for a (semi-)simple group $q$2: $q$3

where $q$4 is a commutative family of $q$5-difference operators (Braverman et al., 2012, Braverman et al., 2014). In type $q$6, explicit Gelfand–Tsetlin pattern sum formulas relate $q$7-Whittaker functions to graded Demazure module characters and affine Schubert geometry.

Weyl Module Realization: The dual Weyl module $q$8 for the current algebra $q$9 has character

$p$0

providing a geometric model for $p$1-Whittaker functions through (equivariant) $p$2-theory and quasimaps to flag varieties (Braverman et al., 2012, Braverman et al., 2014).

4. Cauchy Identities, $p$3-Burge Correspondence, and Combinatorics

The $p$4-Whittaker Cauchy identity takes the form

$p$5

where $p$6 is the dual $p$7-Whittaker function. Karp–Thomas construct a probabilistic bijection ("$p$8-Burge correspondence") between nonnegative integer matrices and pairs of semistandard tableaux, proving the Cauchy identity via explicit randomization over nilpotents compatible with two flags (Karp et al., 2022). As $p$9, this specialized bijection recovers the classical (column) RSK/Burge correspondence.

5. Explicit Expansions and Subspace Enumeration

The $\Lambda_{(q)}$0-Whittaker basis governs explicit symmetric function expansions and enumeration of subspaces with prescribed operator profile. Central results include:

• Power-sum expansion (Ram, 2024, Ram, 2023):

$\Lambda_{(q)}$1

• Profile enumeration: For an operator $\Lambda_{(q)}$2 on $\Lambda_{(q)}$3, the number of subspaces of $\Lambda_{(q)}$4-profile $\Lambda_{(q)}$5 is expressible as a Hall scalar product involving dual $\Lambda_{(q)}$6-Whittaker functions and an explicit flag-generating function $\Lambda_{(q)}$7:

$\Lambda_{(q)}$8

These formulas encode and solve deep counting problems in finite field geometry and operator theory (Ram, 2024, Ram, 2023).

6. Monomial Expansions and Rook/Set Partition Combinatorics

The expansion of $\Lambda_{(q)}$9-Whittaker functions in the monomial basis is governed by explicit positive polynomials $\mathbf{x}=(x_1, x_2, \ldots)$0: $\mathbf{x}=(x_1, x_2, \ldots)$1 The $\mathbf{x}=(x_1, x_2, \ldots)$2 admit both tableau and set partition/Mahonian statistic formulas, linking $\mathbf{x}=(x_1, x_2, \ldots)$3-Whittaker expansions with rook theory, $\mathbf{x}=(x_1, x_2, \ldots)$4-Stirling numbers, and Touchard–Riordan generating functions (Ram et al., 2023).

7. Generalizations, Spin and Nonsymmetric $\mathbf{x}=(x_1, x_2, \ldots)$5-Whittaker Theory

• Spin and inhomogeneous variants: Spin $\mathbf{x}=(x_1, x_2, \ldots)$6-Whittaker polynomials introduce a 1-parameter deformation related to integrable vertex models, stochastic processes, and deformed quantum Toda chains, with established Pieri/Cauchy rules and orthogonality in a Sklyanin-type torus measure (Borodin et al., 2017, Mucciconi et al., 2020, Mucciconi, 1 Feb 2025).
• Nonsymmetric $\mathbf{x}=(x_1, x_2, \ldots)$7-Whittaker functions: These arise as specializations of nonsymmetric Macdonald polynomials and serve as generating functions for graded Weyl module characters, satisfying explicit eigenvalue problems for $\mathbf{x}=(x_1, x_2, \ldots)$8-Toda Dunkl operators (Feigin et al., 2016, Cherednik et al., 2011, Cherednik et al., 2013).
• Classical limits: In the limit $\mathbf{x}=(x_1, x_2, \ldots)$9, $\mathbb{Q}(q)$0-Whittaker sums over Gelfand–Tsetlin patterns degenerate into Givental-type integral formulas for $\mathbb{Q}(q)$1 Whittaker functions; as $\mathbb{Q}(q)$2, the functions reduce to $\mathbb{Q}(q)$3-adic Whittaker–Shintani–Casselman–Shalika formulas (Gerasimov et al., 2011).

8. Summary Table of Core Formulas

Representation	Formula/expression	Context
$\mathbb{Q}(q)$4-Whittaker via tableau sum	$\mathbb{Q}(q)$5	Semistandard Young tableaux
Monomial expansion	$\mathbb{Q}(q)$6	Combinatorics, set partitions
Cauchy identity	$\mathbb{Q}(q)$7	Symmetric functions, RSK/Rand. matrix models
Subspace profile counting	$\mathbb{Q}(q)$8	Finite fields, operator enumeration
$\mathbb{Q}(q)$9 expansion	$q$0	Symmetric function theory
Macdonald specialization	$q$1	Macdonald/Weyl modules

$q$2 and their variants provide an algebraic-combinatorial and representation-theoretic framework unifying topics in symmetric functions, finite field flags, $q$3-deformed integrable systems, and subspace enumeration (Karp et al., 2022, Ram et al., 2023, Ram, 2023, Ram, 2024, Braverman et al., 2012, Feigin et al., 2016).

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References:

• (Karp et al., 2022) "q-Whittaker functions, finite fields, and Jordan forms"
• (Ram et al., 2023) "Diagonal operators, $q$4-Whittaker functions and rook theory"
• (Ram, 2023) "Subspace Profiles over Finite Fields and $q$5-Whittaker Expansions of Symmetric Functions"
• (Ram, 2024) "Simple operators and $q$6-Whittaker coefficients of power sum symmetric functions"
• (Braverman et al., 2012) "Weyl modules and q-Whittaker functions"
• (Feigin et al., 2016) "Generalized Weyl modules and nonsymmetric $q$7-Whittaker functions"
• (Borodin et al., 2017) "Spin $q$8-Whittaker polynomials"
• (Mucciconi et al., 2020) "Spin q-Whittaker polynomials and deformed quantum Toda"
• (Mucciconi, 1 Feb 2025) "Orthogonality of spin $q$9-Whittaker polynomials"
• (Braverman et al., 2014) "Twisted zastava and $t=0$00-Whittaker functions"
• (Gerasimov et al., 2011) "On a classical limit of q-deformed Whittaker functions"
• (Cherednik et al., 2011) "One-dimensional nil-DAHA and Whittaker functions"
• (Cherednik et al., 2013) "Nonsymmetric difference Whittaker functions"
• (Bhattacharya, 2024) "Equating Inv-Quinv formulas for the $t=0$01-Whittaker and modified Hall-Littlewood functions"