Whittaker Measure: Integrable Probability

• Whittaker Measure is a family of probability measures defined via Whittaker functions and Haar integration that generalizes the classical Schur measure to continuous and q-deformed settings.
• It underpins integrable models such as directed polymers, q-TASEP, and random matrix ensembles with explicit formulations using Mellin–Barnes integrals and Fredholm determinants.
• Its representation-theoretic and geometric extensions connect symmetric functions, geometric crystals, and Lie groups, unifying scaling limits and asymptotic analyses in integrable probability.

The Whittaker measure refers to a class of probability measures intimately connected to symmetric functions, integrable probability, representation theory, and random polymer models. It arises as the non-discrete analogue of the Schur measure in various contexts: as a probability distribution on partitions or arrays derived from Whittaker/Givental-type functions, as a canonical measure on geometric crystals for complex reductive groups, and, in its $q$-deformations, as the underpinning measure for the integrable structure of random growth processes and interacting particle systems.

1. Scalar and $q$-Whittaker Measures: Symmetric Function and Probability Theory

The scalar Whittaker measure, for the group $\mathrm{GL}_N(\mathbb{R})$, is defined via the product of two class-one Whittaker functions—$\psi_\theta(y)$ and $\psi_\sigma(y)$—and the Haar measure in logarithmic coordinates, incorporating spectral parameters $\theta$ and $\sigma$. Explicitly, for $y=(y_1,\ldots,y_N)\in(0,\infty)^N$,

$$\mathbf{WM}_N^{\mathrm{Whittaker}}(dy;\theta,\sigma) = \frac{1}{Z_N(\theta,\sigma)}\; \psi_\theta(y)\,\psi_\sigma(y)\;\prod_{i=1}^N\frac{d y_i}{y_i},$$

where the normalization constant $Z_N(\theta,\sigma)$ is given by the Bump–Stade integral: $q$0 Here, $q$1 is most concretely realized by integrals over triangular (Gelfand–Tsetlin) arrays, as in the Givental formula, and the measure generalizes the classical Schur measure to the continuous, “positive-temperature” setting. The $q$2-Whittaker measure is a deformation using $q$3-Whittaker polynomials (Macdonald polynomials at $q$4), taking the form

$q$5

for partitions $q$6, $q$7-Whittaker symmetric polynomials $q$8, $q$9, and $\mathrm{GL}_N(\mathbb{R})$0 the $\mathrm{GL}_N(\mathbb{R})$1-deformed Cauchy-type normalization: $\mathrm{GL}_N(\mathbb{R})$2 The $\mathrm{GL}_N(\mathbb{R})$3-Whittaker measure emerges as a degeneration of Macdonald processes and underlies solvable models such as $\mathrm{GL}_N(\mathbb{R})$4-TASEP, the Higher Spin Six Vertex Model, and $\mathrm{GL}_N(\mathbb{R})$5-Hahn processes (Imamura et al., 2021, Imamura et al., 2019, Corwin et al., 2011).

2. Whittaker Measures and Geometric RSK: Representation-Theoretic and Combinatorial Foundations

A foundational connection arises from the geometric Robinson–Schensted–Knuth (gRSK) correspondence, which maps arrays of positive weights to (tropical) Gelfand–Tsetlin patterns. When the weights are inverse-Gamma distributed, pushing forward the product measure under gRSK yields a probability law on the “shape” vector (i.e., the bottom row) whose density is the product of two Whittaker functions divided by normalization constants (Corwin et al., 2011, O'Connell et al., 2012). The volume-preserving property of the gRSK mapping is critical: it enables rewriting the law of the shapes as

$\mathrm{GL}_N(\mathbb{R})$6

with $\mathrm{GL}_N(\mathbb{R})$7 and the normalization a product of Gamma functions. Analogue identities—Bump–Stade and generalized Cauchy–Littlewood integrals—ensure normalization and furnish Mellin–Barnes-type tools for moments and Laplace transforms (O'Connell et al., 2012).

3. Geometric and Lie-Theoretic Generalizations: Whittaker Measure on Geometric Crystals

In the setting of complex semisimple Lie groups, the Whittaker measure generalizes to geometric crystals. The geometric crystal $\mathrm{GL}_N(\mathbb{R})$8 of highest weight $\mathrm{GL}_N(\mathbb{R})$9 is endowed with a canonical measure, which in Lusztig coordinates $\psi_\theta(y)$0 has “toric” reference form $\psi_\theta(y)$1. This is then twisted by exponential weights involving the crystal weight map $\psi_\theta(y)$2 and the Landau–Ginzburg superpotential $\psi_\theta(y)$3: $\psi_\theta(y)$4 The Laplace transform of the pushforward by the weight map yields the Archimedean Whittaker function,

$\psi_\theta(y)$5

satisfying quantum Toda eigenfunction equations and providing a unifying representation-theoretic, analytic, and probabilistic object (Chhaibi, 2015, Chhaibi, 2013).

4. Matrix Whittaker Measures and Noncommutative Extensions

The matrix Whittaker measure provides a noncommutative extension: it is defined on tuples of real, symmetric positive-definite $\psi_\theta(y)$6 matrices $\psi_\theta(y)$7, using integrals over arrays of matrices,

$\psi_\theta(y)$8

where $\psi_\theta(y)$9 is the multivariate Gamma function, and $\psi_\sigma(y)$0, $\psi_\sigma(y)$1 are matrix-valued Whittaker functions obtained from recursive integrals over triangular matrix arrays. For $\psi_\sigma(y)$2, this construction recovers the classical scalar Whittaker measure, establishing continuity between scalar and matrix-valued frameworks (Arista et al., 2022). The measure arises as the fixed-time law for the bottom edge in certain random matrix-valued Markovian processes driven by inverse Wishart random variables, and under Laplace approximations concentrates on energy-minimizing configurations solvable by saddle point equations.

5. Integrable Probabilistic Models, Scaling Limits, and Asymptotics

Whittaker measures occupy a central role in integrable probability, as they encode the law of partition functions in log-gamma and directed polymer models, stochastic vertex models, and their $\psi_\sigma(y)$3-deformations. Their Laplace transforms and one-point distributions admit explicit contour or Fredholm determinant formulas, facilitating rigorous analysis of fluctuations and Tracy–Widom-type asymptotics. A central result is that, under suitable scaling, the one-point fluctuations of the height function in models such as the Higher Spin Six Vertex Model or $\psi_\sigma(y)$4-TASEP converge to Baik–Rains distributions with $\psi_\sigma(y)$5 exponent, as established in (Imamura et al., 2019). The Whittaker measure forms the scaling limit (as $\psi_\sigma(y)$6) of $\psi_\sigma(y)$7-Whittaker measures and lies at the core of asymptotic analysis in random interface growth, non-intersecting polymers, and last passage percolation (Johnston et al., 2019).

6. Degenerations, Limiting Regimes, and Representation-Theoretic Connections

Degenerations of the Whittaker measure relate it to uniform measures on polytopes (e.g., string polytopes under tropicalization), and to classical objects in representation theory such as Harish-Chandra characters. For example, as $\psi_\sigma(y)$8, $\psi_\sigma(y)$9-Whittaker polynomials become rational Whittaker functions, and the $\theta$0-Whittaker measure converges to the classical Whittaker measure. Within the geometric crystal framework, the canonical measure deforms continuously to the uniform measure on the string polytope—reflecting a crystal-to-tropical transition mirroring the degeneration of Markov processes to Brownian motion conditioned to remain in a Weyl chamber. Representation-theoretically, the Whittaker measure encapsulates the spectral theory of the quantum Toda lattice and encodes tensor product decompositions via crystal combinatorics (Chhaibi, 2015, Chhaibi, 2013).

7. Summary Table: Key Variants of the Whittaker Measure

Setting	Measure Definition	Probabilistic Model
Scalar (GL$\theta$1)	Product of Whittaker functions times Haar measure; Bump–Stade norm.	Directed polymer with inverse-Gamma weights; geometric RSK.
$\theta$2-Whittaker (Macdonald)	Product of $\theta$3-Whittaker polynomials, $\theta$4-Pochhammer normalization.	$\theta$5-TASEP, Higher Spin Six Vertex, $\theta$6-Hahn processes, partitions.
Geometric crystal / Lie type	Integration on geometric crystal with toric measure, spectral twisting, superpotential.	Brownian motion in Cartan with Doob transform; geometric Pitman map; string polytopes.
Matrix-valued	Higher-dimensional integration over positive-definite matrices; matrix Whittaker functions.	Matrix log-gamma polymer, inverse Wishart arrays.

The Whittaker measure thus provides a central, unifying structure at the interface of integrable systems, symmetric functions, stochastic processes, and algebraic geometry, encoding in its variants a broad class of exactly solvable models, representation-theoretic transformations, and scaling limits underpinning modern integrable probability (Corwin et al., 2011, O'Connell et al., 2012, Chhaibi, 2015, Imamura et al., 2021, Chhaibi, 2013, Arista et al., 2022, Imamura et al., 2019, Johnston et al., 2019).