Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 116, 55 pages      arXiv:2002.00243      https://doi.org/10.3842/SIGMA.2020.116
Contribution to the Special Issue on Elliptic Integrable Systems, Special Functions and Quantum Field Theory

Non-Stationary Ruijsenaars Functions for $\kappa=t^{-1/N}$ and Intertwining Operators of Ding-Iohara-Miki Algebra

Masayuki Fukuda a, Yusuke Ohkubo b and Jun'ichi Shiraishi b
a) Department of Physics, Faculty of Science, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033 Japan
b) Graduate School of Mathematical Sciences, The University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153-8914 Japan

Received April 23, 2020, in final form November 01, 2020; Published online November 18, 2020

Abstract
We construct the non-stationary Ruijsenaars functions (affine analogue of the Macdonald functions) in the special case $\kappa=t^{-1/N}$, using the intertwining operators of the Ding-Iohara-Miki algebra (DIM algebra) associated with $N$-fold Fock tensor spaces. By the $S$-duality of the intertwiners, another expression is obtained for the non-stationary Ruijsenaars functions with $\kappa=t^{-1/N}$, which can be regarded as a natural elliptic lift of the asymptotic Macdonald functions to the multivariate elliptic hypergeometric series. We also investigate some properties of the vertex operator of the DIM algebra appearing in the present algebraic framework; an integral operator which commutes with the elliptic Ruijsenaars operator, and the degeneration of the vertex operators to the Virasoro primary fields in the conformal limit $q \rightarrow 1$.

Key words: Macdonald function; Rujisenaars function; Ding-Iohara-Miki algebra.

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