2011/04/20 by Cherednik, Ivan, Orr, Daniel
#33C15 #33C52 #33C67 #33D67 #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1104.3918
This work, to be published in Transformation Groups in two parts, is devoted to the theory of nil-DAHA for the root system A1 and its applications to symmetric and nonsymmetric (spinor) global q-Whittaker functions. These functions integrate the q-Toda eigenvalue problem and its Dunkl-type nonsymmetric version. The global symmetric function can be interpreted as the generating function of the Demazure characters for dominant weights, which describe the algebraic-geometric properties of the corresponding affine Schubert varieties. Its Harish-Chandra-type asymptotic expansion appeared directly related to the solution of the q-Toda eigenvalue problem obtained by Givental and Lee in the quantum K-theory of flag varieties. It provides an exact mathematical relation between the corresponding physics A-type and B-type models. The spinor global functions extend the symmetric ones to the case of all Demazure characters (not only those for the dominant weights); the corresponding Gromov-Witten theory is not known. The main result of the paper is a complete algebraic theory of these functions in terms of the induced modules of the core subalgebra of nil-DAHA. It is the first instance of the DAHA theory of canonical-crystal bases, quite non-trivial even for A1.