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**Root systems and hypergeometric functions. III.**
*(English)*
Zbl 0669.33007

In parts I, II [ibid. 64, 329-352, 353-373 (1987; Zbl 0656.17006, Zbl 0656.17007)] the author and G. J. Heckman studied a class of differential equations associated with a root system R, the solutions of these equations are then used to introduce multivariable hypergeometric functions. In their construction they made essential use of the hypothesis that the so-called Harish-Chandra homomorphism is an isomorphism onto.

This third part under review gives a proof of that fact when R is a root system of type \(G_ 2\). This result gives a possibility for constructing the hypergeometric function associated with arbitrary root systems.

This third part under review gives a proof of that fact when R is a root system of type \(G_ 2\). This result gives a possibility for constructing the hypergeometric function associated with arbitrary root systems.

Reviewer: G.Tu

### MSC:

33C60 | Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions) |

17B20 | Simple, semisimple, reductive (super)algebras |

### Keywords:

multivariable hypergeometric functions; Harish-Chandra homomorphism; root system of type \(G_ 2\)### References:

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[4] | G.J. Heckman and E.M. Opdam : Root systems and hypergeometric functions I . Comp. Math. 64 (1987) 329-352. · Zbl 0656.17006 |

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[6] | H. V.D.Lek : The homotopy type of complex hyperplane complements . Thesis, Nijmegen (1983). |

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