Nonsymmetric difference Whittaker functions

TL;DR

This paper develops a comprehensive theory of nonsymmetric global q-Whittaker functions for arbitrary root systems, introducing spinor q-Toda-Dunkl operators and their algebraic properties, advancing the understanding of DAHA-related eigenvalue problems.

Contribution

It defines and computes spinor global q-Whittaker functions for all reduced root systems, introducing new spinor Dunkl operators and algebraic techniques in DAHA theory.

Findings

Constructed spinor q-Whittaker functions as reproducing kernels.

Introduced spinor q-Toda-Dunkl operators as limits of Dunkl operators.

Developed algebraic theory for these operators across all reduced root systems.

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.