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Integrable connections related to zonal spherical functions. (English) Zbl 0801.35131

Summary: We define a system of differential equations of first order for a function valued in the group algebra of the Weyl group associated with an arbitrary root system. This is equivalent to the system of differential equations given by Heckman and Opdam which is a deformation of the system satisfied by the zonal spherical function of the Riemannian symmetric space \(G/K\) of non-compact type. When the root system is \(A_ n\)-type, our equation is related to the Knizhnik-Zamolodchikov equation in conformal field theory.

MSC:

35Q58 Other completely integrable PDE (MSC2000)
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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