Quantum Whittaker Coefficients

• Quantum Whittaker coefficients are explicit coefficient systems derived by q-deformation and quantum-group quantization of Whittaker objects, encompassing arithmetic and combinatorial models.
• They are constructed using methods such as metaplectic ice partition functions, twisted quantum Fock spaces, and deformed Macdonald theory to capture complex spectral data.
• The framework unifies diverse techniques—from Drinfeld twists and Gauss sums to categorical coefficient functors in quantum geometric Langlands—yielding actionable insights across several fields.

Searching arXiv for recent and foundational work on quantum Whittaker coefficients. Quantum Whittaker coefficients are coefficient data attached to Whittaker-type objects after $q$-deformation, metaplectic covering, quantum-group quantization, or categorical quantization. In recent literature they appear as explicit coefficients in the $q$-Whittaker basis of symmetric functions, finite ice partition functions equal to metaplectic Whittaker functions, matrix elements of half-vertex or Baxter-type operators, and functors such as $\coeff_D$ or $\coeff^{\loc}$ that extract Whittaker data from twisted sheaves and $D$-modules (Bergeron, 2020, Brubaker et al., 2018, Bogdanova, 26 Aug 2025, Bogdanova, 28 Jun 2026). The unifying feature is that a Whittaker object—function, polynomial, module, or sheaf—is paired with an operator, character, or geometric correspondence so that its Whittaker content becomes an explicit coefficient system.

1. Metaplectic arithmetic, solvable lattice models, and quantum Fock space

A central arithmetic realization occurs for spherical Whittaker functions on an $n$-fold metaplectic cover

$$1\longrightarrow \mu_{2n}\longrightarrow \widetilde{GL}_r(F)\longrightarrow GL_r(F)\longrightarrow 1$$

of $GL_r(F)$, where $F$ is nonarchimedean. For a principal series representation $\pi_{\mathbf z}$ with Langlands parameters $q$0, there are $q$1 spherical Whittaker functions $q$2, indexed by $q$3. Their values at

$q$4

are encoded by finite metaplectic ice partition functions. For Delta ice one has

$q$5

and the same statement holds for Gamma ice. In this setting the local field determines Gauss-sum data

$q$6

for $q$7; these are the arithmetic inputs that later become twisted wedge coefficients (Brubaker et al., 2018).

The representation-theoretic mechanism is a Drinfeld-twisted quantum Fock space $q$8 for $q$9. Its basis consists of semi-infinite monomials

$\coeff_D$0

and the relevant $\coeff_D$1-wedge is defined by quotienting tensor powers by the $\coeff_D$2-antisymmetrizer $\coeff_D$3. The key comparison theorem identifies the affine Hecke action coming from metaplectic Whittaker coinvariants with the GRV action from quantum affine Schur–Weyl duality. After a Drinfeld twist

$\coeff_D$4

the coefficients are chosen so that

$\coeff_D$5

thereby inserting Gauss sums directly into the twisted wedge relations. This is the precise point at which metaplectic arithmetic enters the quantum Fock-space construction (Brubaker et al., 2018).

The main operator identity is

$\coeff_D$6

with

$\coeff_D$7

Thus the row transfer matrices of infinite Gamma and Delta ice are half-vertex operators on $\coeff_D$8. Their matrix elements define the metaplectic symmetric functions

$\coeff_D$9

These are symmetric in the $\coeff^{\loc}$0, vanish unless $\coeff^{\loc}$1 has the appropriate $\coeff^{\loc}$2-core behavior, and satisfy

$\coeff^{\loc}$3

so they are specializations of Lam’s supersymmetric LLT polynomials. The operator formalism also yields the Cauchy identity

$\coeff^{\loc}$4

where

$\coeff^{\loc}$5

Finally, finite metaplectic Whittaker functions are reconstructed from these symmetric functions by

$\coeff^{\loc}$6

In this framework, metaplectic Whittaker coefficients are simultaneously finite ice partition functions, linear combinations of metaplectic symmetric functions, and matrix elements of Drinfeld-twisted half-vertex operators (Brubaker et al., 2018).

2. $\coeff^{\loc}$7-Whittaker coefficients in symmetric-function theory

In symmetric-function theory, $\coeff^{\loc}$8-Whittaker polynomials are the $\coeff^{\loc}$9 specialization of modified Macdonald theory: $D$0 They also admit the Schur expansion

$D$1

and the specializations

$D$2

A dual basis is obtained from

$D$3

and the associated Cauchy kernel is

$D$4

Here the coefficients are basis coefficients in a $D$5-deformed Hall pairing and structure constants in Whittaker-basis expansions (Bergeron, 2020).

A more explicit coefficient problem is the expansion of the power sum $D$6 in the $D$7-Whittaker basis. For a simple operator $D$8 on $D$9, the number

$n$0

is equal to the coefficient problem for $n$1 in $n$2. The paper gives an explicit formula for $n$3, proves it by a double-counting argument using partial maps and defect dimensions, and shows that the coefficient of $n$4 in $n$5 is the same finite-field quantity up to the sign $n$6 and the standard quadratic exponent in $n$7. It also identifies Niederreiter’s splitting-subspace enumeration as the rectangular-profile case $n$8 (Ram, 2024).

The monomial coefficients of $n$9-Whittaker polynomials admit three combinatorially equivalent descriptions. For $$1\longrightarrow \mu_{2n}\longrightarrow \widetilde{GL}_r(F)\longrightarrow GL_r(F)\longrightarrow 1$$0, the fermionic formula uses Gelfand–Tsetlin patterns or partition overlaid patterns, the $$1\longrightarrow \mu_{2n}\longrightarrow \widetilde{GL}_r(F)\longrightarrow GL_r(F)\longrightarrow 1$$1 model uses column strict fillings with a reflected inversion statistic, and the $$1\longrightarrow \mu_{2n}\longrightarrow \widetilde{GL}_r(F)\longrightarrow GL_r(F)\longrightarrow 1$$2 model uses column strict fillings with rightward triple statistics. The bijections

$$1\longrightarrow \mu_{2n}\longrightarrow \widetilde{GL}_r(F)\longrightarrow GL_r(F)\longrightarrow 1$$3

preserve both the monomial weight and the $$1\longrightarrow \mu_{2n}\longrightarrow \widetilde{GL}_r(F)\longrightarrow GL_r(F)\longrightarrow 1$$4-weight: $$1\longrightarrow \mu_{2n}\longrightarrow \widetilde{GL}_r(F)\longrightarrow GL_r(F)\longrightarrow 1$$5 They are compatible with row-sorting, projection to Gelfand–Tsetlin patterns, and branching. Moreover,

$$1\longrightarrow \mu_{2n}\longrightarrow \widetilde{GL}_r(F)\longrightarrow GL_r(F)\longrightarrow 1$$6

so the two statistics are complementary on each row-sorting fiber (Bhattacharya et al., 2023).

Taken together, these results show that $$1\longrightarrow \mu_{2n}\longrightarrow \widetilde{GL}_r(F)\longrightarrow GL_r(F)\longrightarrow 1$$7-Whittaker coefficients are not confined to one basis expansion. They can be Schur coefficients, power-sum coefficients, monomial coefficients, or local $$1\longrightarrow \mu_{2n}\longrightarrow \widetilde{GL}_r(F)\longrightarrow GL_r(F)\longrightarrow 1$$8-binomial factors, and the same coefficients may simultaneously encode finite-field subspace counts, local Weyl module characters, and explicit filling statistics (Bergeron, 2020, Ram, 2024, Bhattacharya et al., 2023).

3. Coefficient data in quantum algebras and Whittaker modules

For quantum groups, Whittaker coefficients often appear as the scalar data controlling induced modules, PBW monomials, and Whittaker vectors. In a path-model realization for $$1\longrightarrow \mu_{2n}\longrightarrow \widetilde{GL}_r(F)\longrightarrow GL_r(F)\longrightarrow 1$$9, the Whittaker vector is expanded as

$GL_r(F)$0

where the coefficients are path partition functions over the positive root lattice. In the quantum case the local weights are edge weights rather than vertex weights: $GL_r(F)$1 The associated $GL_r(F)$2-Whittaker function has the series form

$GL_r(F)$3

and the partition functions satisfy a recursion that yields the $GL_r(F)$4-difference Toda equation. Here the coefficients $GL_r(F)$5 are the explicit Whittaker coefficients of the path model (Francesco et al., 2014).

A second realization uses the quantum Heisenberg algebra inside $GL_r(F)$6. The basic data are a Whittaker character

$GL_r(F)$7

a central parameter $GL_r(F)$8, a derivation parameter $GL_r(F)$9, and a weight $F$0. The coefficient-extraction identity is

$F$1

This shows that $F$2, $F$3, $F$4, and $F$5 act as coefficient data attached to PBW monomials. The paper proves that $F$6 is irreducible as an $F$7-module when $F$8, and constructs induced imaginary Whittaker modules for quantum affine algebras, with a full proof in type $F$9 for the free-derivation case (Futorny et al., 3 Jun 2026).

For the deformed Virasoro algebra $\pi_{\mathbf z}$0, the Whittaker vector in Fock space is expanded in the Macdonald basis: $\pi_{\mathbf z}$1 The coefficients are completely factorized: $\pi_{\mathbf z}$2 These are explicitly described as the quantum Whittaker coefficients associated with the deformed Virasoro algebra, and their proof uses the Ding–Iohara–Miki currents in the Fock representation (Yanagida, 2014).

The $\pi_{\mathbf z}$3 case shows that coefficient data can also govern submodule structure. Because the quantum Serre relations force $\pi_{\mathbf z}$4, there are no non-singular Whittaker functions for $\pi_{\mathbf z}$5, so the theory uses the singular choice $\pi_{\mathbf z}$6, $\pi_{\mathbf z}$7. The universal Whittaker module $\pi_{\mathbf z}$8 has Whittaker vectors exactly

$\pi_{\mathbf z}$9

and irreducibility of quotients is controlled by the critical polynomials

$q$00

Thus the coefficient algebra $q$01 and the roots of $q$02 replace the polynomial-center picture familiar from $q$03 (Guo et al., 14 Apr 2025).

4. Quantum geometric Langlands and categorical Whittaker coefficient functors

In quantum geometric Langlands, Whittaker coefficients are not scalar coefficients but functors. For a reductive group $q$04 of adjoint type over $q$05, a nondegenerate level $q$06, and a $q$07-valued divisor $q$08 on $q$09, the Whittaker coefficient of $q$10 is

$q$11

The paper proves a non-vanishing theorem: if $q$12 is adjoint, $q$13 is rational, and

$q$14

is cuspidal, then there exists a $q$15-valued divisor $q$16 such that $q$17. It also proves a microlocal theorem: when the relevant singular support meets the $q$18-Kostant slice transversely at $q$19, the coefficient functor becomes a twisted microstalk at $q$20, hence is $q$21-exact and commutes with Verdier duality. In the constructible case its Euler characteristic equals the corresponding characteristic-cycle multiplicity (Bogdanova, 26 Aug 2025).

The Betti construction replaces scalar Fourier coefficients by a local Whittaker coefficient functor

$q$22

Here $q$23 is the quantum Betti Whittaker category, defined via a Kirillov-model quotient of twisted sheaves on the Beilinson–Drinfeld Grassmannian. The local and global Whittaker categories are equivalent, and $q$24 has adjoint Poincaré functors

$q$25

with $q$26 an adjoint pair. The decisive theorem is that

$q$27

factors through the factorization homology category

$q$28

thereby producing the quantum Betti geometric Langlands functor. The whole construction is then upgraded to a sheaf-of-categories statement compatible with the 2-Fourier–Mukai transform between gerbe stacks controlled by $q$29 and $q$30 (Bogdanova, 28 Jun 2026).

These papers give two complementary meanings of quantum geometric Whittaker coefficients. In the $q$31-module setting they are de Rham Fourier–Whittaker coefficient functors detected by microstalks; in the Betti setting they are categorical coefficient functors whose composition with quantum FLE lands in factorization homology. In both cases, non-vanishing and conservativity replace explicit scalar evaluation as the fundamental structural statement (Bogdanova, 26 Aug 2025, Bogdanova, 28 Jun 2026).

5. Integrable models and deformed special functions

In analytic representation theory, $q$32-Whittaker functions are eigenfunctions of the modular $q$33-deformed $q$34 open Toda system with deformation parameter

$q$35

They define a unitary transform

$q$36

intertwining the Toda Hamiltonians with multiplication by elementary symmetric polynomials in $q$37. The same paper identifies the $q$38-Whittaker functions as the quantum Whittaker coefficients governing decomposition of tensor products of positive representations of $q$39. It derives a modular $q$40-analogue of Givental’s integral formula, a Mellin–Barnes recursion, a Baxter-operator description via quantum cluster transformations, a Dehn twist eigenvalue equation, and $q$41-deformations of continuous Cauchy–Littlewood and Stade-type identities (Schrader et al., 2018).

A closely related lattice-model realization appears in free-fermionic six-vertex models with reflecting boundaries. The wavefunctions are matrix elements of double-row creation operators $q$42 and $q$43, and the type II case is expressed by generalized Bump–Friedberg–Hoffstein Whittaker functions. The Izergin–Korepin characterization determines the full wavefunctions and dual wavefunctions by symmetry, inversion invariance, and recursion, while the factorized domain wall partition functions produce dual Cauchy identities for the generalized Whittaker functions. In this setting the wavefunction coefficients of the quantum inverse scattering model are realized exactly as Whittaker-type symmetric functions (Motegi et al., 2018).

A different special-function deformation uses the $q$44-confluent hypergeometric function. The $q$45-Whittaker function is

$q$46

and its coefficient expansion is

$q$47

The generalized coefficients are therefore

$q$48

which reduce to the usual hypergeometric coefficients at $q$49. The paper also gives integral representations, a $q$50 transformation law, and derivative identities shifting $q$51 (Rahman et al., 2017).

These constructions are analytically different, but they share the same structural pattern: a Whittaker object is produced as an eigenfunction or matrix element, and its coefficients are then realized by explicit kernels, wavefunctions, or deformed hypergeometric series (Schrader et al., 2018, Motegi et al., 2018, Rahman et al., 2017).

6. Conceptual themes and common misconceptions

A recurrent misconception is that a Whittaker coefficient must be a scalar Fourier coefficient. Recent work shows several nonequivalent meanings. In the categorical quantum geometric Langlands setting it is a functor $q$52 or $q$53, not a number; in quantum affine Whittaker-module theory it is the scalar system $q$54, $q$55, $q$56, and $q$57 that controls PBW monomials; in deformed Virasoro theory it is a partition-indexed product formula $q$58; and in lattice-model realizations it is a partition function or wavefunction matrix element (Bogdanova, 26 Aug 2025, Futorny et al., 3 Jun 2026, Yanagida, 2014, Brubaker et al., 2018).

A second misconception is that metaplectic Whittaker coefficients should behave like the normalized $q$59 Whittaker coefficient. For $q$60, the geometric Whittaker coefficient of an Eisenstein series is not normalized to $q$61. The most non-degenerate Fourier coefficient on $q$62 is the central value complex

$q$63

up to a canonical one-dimensional twist, and the conjectural Whittaker sheaf $q$64 satisfies

$q$65

Thus the metaplectic coefficient is governed by a square root of a quadratic $q$66-value complex rather than by a trivial normalization (Lysenko, 2012).

A third theme is that Whittaker functions behave like geometric or quantum characters. For a real semisimple group, one has the geometric-crystal integral

$q$67

where $q$68 is a highest-weight decorated geometric crystal, $q$69 is the decoration or superpotential, and $q$70 is the canonical volume form. This integral is a Whittaker function, solves the entire quantum Toda lattice, and is interpreted as a geometric analogue of an irreducible character. A plausible implication is that many later “quantum Whittaker coefficient” constructions can be read as different quantizations of the same character-theoretic paradigm: sums or integrals over a geometric, combinatorial, or categorical model in which Whittaker data become explicit coefficients (Lam, 2013).

Across arithmetic, symmetric-function theory, quantum groups, integrable systems, and quantum geometric Langlands, quantum Whittaker coefficients therefore form a family of related constructions rather than a single invariant. What remains stable is the role they play: they convert Whittaker conditions into explicit coefficient data, and they do so in forms adapted to the ambient theory—Gauss sums in metaplectic wedges, $q$71-binomial and Kostka–Foulkes data in symmetric functions, PBW extractors in quantum modules, and microstalk or Kirillov-model functors in geometric Langlands.