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On the affine analogue of Jack and Macdonald polynomials. (English) Zbl 0873.33011

The authors study the Jack symmetric polynomials and their generalizations for affine Lie algebras and quantum groups. They first prove that the Jack symmetric polynomials for the (generalized) root system \(A_{n-1}\) with root multiplicity \(k\) can be defined as trace of an intertwining operator. More exactly, let \(\lambda\) be a dominant weight and let \(J_\lambda\) be the corresponding Jack symmetric polynomials. Consider an intertwining operator \(\Phi\) from the representation with highest weight \(\lambda + (k-1)\rho\) to its tensor product with the symmetric power \(S^{(k-1)n}\mathbb{C}^n\); here \(\rho\) is as usual the half sum of positive roots. It is proved that up to certain multiplier the trace of \(\Phi\) is the Jack symmetric polynomial \(J_\lambda\). They further define Jack symmetric polynomials for the affine Lie algebras and affine quantum groups as eigenfunctions of the Calogero-Sutherland operator, and establish similar results using the trace of intertwining operators. The authors also present certain conjectures on the unitarity of modular action of \(SL_2(\mathbb{Z})\) on the space of (affine) Jack symmetric polynomials.
Reviewer: G.Zhang (Karlstad)

MSC:

33C80 Connections of hypergeometric functions with groups and algebras, and related topics
17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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