
SIGMA 5 (2009), 010, 12 pages arXiv:0810.2581
http://dx.doi.org/10.3842/SIGMA.2009.010
Contribution to the Special Issue on Dunkl Operators and Related Topics
The Rational qKZ Equation and Shifted NonSymmetric Jack Polynomials
Saburo Kakei ^{a}, Michitomo Nishizawa ^{b}, Yoshihisa Saito ^{c} and Yoshihiro Takeyama ^{d}
^{a)} Department of Mathematics, College of Science, Rikkyo University,
NishiIkebukuro, Toshimaku, Tokyo 1718501, Japan
^{b)} Department of Mathematics,
Faculty of Education, Hirosaki University, 1 Bunkyocho, Hirosaki, Aomori 0368560, Japan
^{c)} Graduate School of Mathematical Sciences, University of Tokyo, Tokyo 1538914, Japan
^{d)} Department of Mathematics, Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 3058571, 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 KnizhnikZamolodchikov equation
associated with the Lie algebra gl_{N}.
The main ingredient is a special class of the shifted nonsymmetric 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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