Starting with nonsymmetric global difference spherical functions, we define and calculate spinor (nonsymmetric) global q-Whittaker functions for arbitrary reduced root systems, which are reproducing kernels of the DAHA-Fourier transforms of Nil-DAHA and solutions of the q-Toda-Dunkl eigenvalue problem. We introduce the spinor q-Toda-Dunkl operators as limits of the difference Dunkl operators in DAHA theory under the spinor variant of the Ruijsenaars procedure. Their general algebraic theory (any reduced root systems) is the key part of this paper, based on the new technique of W-spinors and corresponding developments in combinatorics of affine root systems.

This work records the details of the Ram-Yip formula for nonsymmetric Macdonald-Koornwinder polynomials for the double affine Hecke algebras of not-necessarily-reduced affine root systems. It is shown that the t → 0 equal-parameter specialization of nonsymmetric Macdonald polynomials admits an explicit combinatorial formula in terms of quantum alcove paths, generalizing the formula of Lenart in the untwisted case. In particular our formula yields a definition of quantum Bruhat graph for all affine root systems. For mixed type the proof requires the Ram-Yip formula for the nonsymmetric Koornwinder polynomials. A quantum alcove path formula is also given at t → ∞. As a consequence we establish the positivity of the coefficients of nonsymmetric Macdonald polynomials under this limit, as conjectured by Cherednik and the first author. Finally, an explicit formula is given at q → ∞, which yields the p-adic Iwahori-Whittaker functions of Brubaker, Bump, and Licata.2.2. Root data. Given a Cartan datum (I, A), a root datum is a triple (X,where X is a lattice (free Z-module) containing elements α i (simple roots) for i ∈ I, and α ∨ i (simple coroots) are elements in X * = Hom Z (X, Z) such that α ∨ i , α j = a ij for i, j ∈ I (2.1) 1 In [1], the authors use E λ (X; q −1 ; t −1 ) and correspondingly send q → 0; [16] also uses this convention for q and t.

We present two new connections between the inhomogeneous stochastic higher spin six vertex model in a quadrant and integrable stochastic systems from the Macdonald processes hierarchy. First, we show how Macdonald q-difference operators with t = 0 (an algebraic tool crucial for studying the corresponding Macdonald processes) can be utilized to get q-moments of the height function h in the higher spin six vertex model first computed in [BP16a] using Bethe ansatz. This result in particular implies that for the vertex model with the step Bernoulli boundary condition, the value of h at an arbitrary point (N + 1, T ) ∈ Z ≥2 × Z ≥1 has the same distribution as the last component λN of a random partition under a specific t = 0 Macdonald measure. On the other hand, it is known that xN := λN − N can be identified with the location of the N th particle in a certain discrete time q-TASEP started from the step initial configuration. The second construction we present is a coupling of this q-TASEP and the higher spin six vertex model (with the step Bernoulli boundary condition) along time-like paths providing an independent probabilistic explanation of the equality of h(N + 1, T ) and xN + N in distribution. Combined with the identification of averages of observables between the stochastic higher spin six vertex model and Schur measures (which are t = q Macdonald measures) obtained recently in [Bor16], this produces GUE Tracy-Widom asymptotics for a discrete time q-TASEP with the step initial configuration and special jump parameters.

We introduce generalized global Weyl modules and relate their graded characters to nonsymmetric Macdonald polynomials and nonsymmetric q-Whittaker functions. In particular, we show that the series part of the nonsymmetric q-Whittaker function is a generating function for the graded characters of generalized global Weyl modules.

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