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

SIGMA 5 (2009), 010, 12 pages      arXiv:0810.2581      https://doi.org/10.3842/SIGMA.2009.010
Contribution to the Special Issue on Dunkl Operators and Related Topics

### The Rational qKZ Equation and Shifted Non-Symmetric Jack Polynomials

Saburo Kakei a, Michitomo Nishizawa b, Yoshihisa Saito c and Yoshihiro Takeyama d
a) Department of Mathematics, College of Science, Rikkyo University, Nishi-Ikebukuro, Toshima-ku, Tokyo 171-8501, Japan
b) Department of Mathematics, Faculty of Education, Hirosaki University, 1 Bunkyo-cho, Hirosaki, Aomori 036-8560, Japan
c) Graduate School of Mathematical Sciences, University of Tokyo, Tokyo 153-8914, Japan
d) Department of Mathematics, Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan

Received October 15, 2008, in final form January 15, 2009; Published online January 27, 2009

Abstract
We construct special solutions to the rational quantum Knizhnik-Zamolodchikov equation associated with the Lie algebra glN. The main ingredient is a special class of the shifted non-symmetric Jack polynomials. It may be regarded as a shifted version of the singular polynomials studied by Dunkl. We prove that our solutions contain those obtained as a scaling limit of matrix elements of the vertex operators of level one.

Key words: qKZ equation; shifted Jack polynomial; degenerate double affine Hecke algebra.

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