Macdonald's symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces

Quantum analogues of the homogeneous spaces \GL(n)/\SO(n) and \GL(2n)/\Sp(2n) are introduced. The zonal spherical functions on these quantum homogeneous spaces are represented by Macdonald's symmetric polynomials P_{\ld}=P_{\ld}(x_1,\cdots,x_n;q,t) with $t=q^{1 \over 2}$ or $t=q^2$

Birational Weyl group action arising from a nilpotent Poisson algebra

We propose a general method to realize an arbitrary Weyl group of Kac-Moody type as a group of birational canonical transformations, by means of a nilpotent Poisson algebra. We also give a Lie theoretic interpretation of this realization in terms of Kac-Moody Lie algebras and Kac-Moody groups.Comment: 31 pages, LaTe

Symmetries in the fourth Painleve equation and Okamoto polynomials

We propose a new representation of the fourth Painlev\'e equation in which the $A^{(1)}_2$-symmetries become clearly visible. By means of this representation, we clarify the internal relation between the fourth Painlev\'e equation and the modified KP hierarchy. We obtain in particular a complete description of the rational solutions of the fourth Painlev\'e equation in terms of Schur functions. This implies that the so-called Okamoto polynomials, which arise from the $\tau$-functions for rational solutions, are in fact expressible by the 3-reduced Schur functions.Comment: 25 pages, amslate