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Opdam, E. M.

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

Compos. Math. 67, No. 1, 21-49 (1988).
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.
Reviewer: G.Tu

Cited in 5 Reviews
Cited in 44 Documents

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 \(G_ 2\)

Citations:

Zbl 0669.33008; Zbl 0656.17006; Zbl 0656.17007
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\textit{E. M. Opdam}, Compos. Math. 67, No. 1, 21--49 (1988; Zbl 0669.33007)
Full Text: Numdam EuDML

References:

[1] R. Beerends : On the Abel transform and its inversion , thesis (1987). · Zbl 0597.43006
[2] G.J. Heckman : Root systems and hypergeometric functions II . Comp. Math. 64 (1987) 353-373. · Zbl 0656.17007
[3] Harish Chandra : Differential operators on a semisimple Lie algebra . A.J.M. 79 (1957) 87-120. · Zbl 0072.01901
[4] G.J. Heckman and E.M. Opdam : Root systems and hypergeometric functions I . Comp. Math. 64 (1987) 329-352. · Zbl 0656.17006
[5] T.H. Koornwinder : Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent differential operators, I-IV . Indag. Math 36 (1974) 48-66 and 358-381. · Zbl 0267.33008
[6] H. V.D.Lek : The homotopy type of complex hyperplane complements . Thesis, Nijmegen (1983).
[7] I.G. Sprinkhuizen-Kuyper : Orthogonal polynomials in two variables . A further analysis of the polynomials orthogonal over a region bounded by two lines and a parabola. SIAM 7 (1976). · Zbl 0332.33011
[8] J. Sekiguchi : Zonal spherical functions on some symmetric spaces . Publ. RMS Kyoto Univ, 12 Suppl. 455-459 (1977). · Zbl 0383.43005
[9] L. Vretare : Formulas for elementary spherical functions and generalized Jacobi polynomials . SIAM 15 (1984). · Zbl 0549.43006
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