Nonsymmetric difference Whittaker functions

Abstract

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.

Extremal part of the PBW-filtration and nonsymmetric Macdonald polynomials

“…The formulas for extremal q-degrees of E-dag polynomials for all weights are a significant ingredient of the new theory of nonsymmetric q-Whittaker function [8][9][10]. This link is expected to be important to understand their meaning, but we present (and partially justify) the PBW-E † correspondence in an entirely algebraic way in this paper.…”

“…For the classical root systems, the extremal degrees for the fundamental weights can be calculated by a relatively straightforward induction. The q-positivity of complete E-dag polynomials was conjectured in [10], which is a theorem for their extremal parts and for antidominant weights (see Theorem 3.1 below; it was announced in [10], Corollary 2.6).…”

“…This is the extremal part of Conjecture 2.7 from [10] (with participation of E. Feigin) on the coincidence of the E-dag polynomials and the corresponding characters of Demazure level-one modules for the sum of the Kac-Moody grading and that defined via the PBW-filtration. It was/is stated only for (anti)dominant weights, which is significant, and for the ADE root systems; the twisted case is in progress.…”

“…This becomes especially involved in the Kac-Moody case (the setting of Conjecture 2.7 from [10] and Conjecture 7.5 below). The extremal coefficients of the E-dag polynomials for antidominant weights can be calculated for all root systems (using computers for F 4 , E 6,7,8 ), so the problem with their identification is currently due to the lack of methods on the PBW-side of this surprising correspondence.…”

“…Instead of giving in this section the definition of E-dag polynomials and then restricting ourselves to their extremal parts, we will introduce the latter directly using mainly Proposition 2.5(i) from [10]. Section 7 contains a systematic approach via general E-polynomials.…”