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Root systems and hypergeometric functions. I. (English) Zbl 0656.17006

Let E be an n-dimensional Euclidean space with the inner product (, ), and W be the Weyl group generated by reflections \(r_{\alpha}(\lambda)=\lambda -(\lambda,\alpha)\alpha /2(\alpha,\alpha)\) where the \(\alpha\) ’s are taken from a root system R of rank n. The authors propose first an operator L which leaves the space of Weyl group invariant exponential polynomials on H invariant, where H is the complex torus with the Lie algebra \({\mathfrak h}=E^*\otimes_{{\mathbb{R}}}{\mathbb{C}}\). Under a crucial conjecture on the form of the commuting algebra of differential operators which contains L they then study the equation (1) \(L\phi =(\lambda -\rho,\lambda +\rho)\phi\), where \(\rho\) is a suitable linear combination of \(\alpha \in R_+\). In the case that R is of type \(A_ 1\) (or \(BC_ 1)\), the equation (1) reduces to the well-known equation for the ordinary hypergeometric functions. In the general case when R is an arbitrary root system the equation (1) leads to a system of PDEs called by the authors the system of hypergeometric (partial) differential equations. The monodromy of this system is explicitly determined which implies the existence of multivariable hypergeometric functions.
This paper deals with theory of commuting differential operators which is closely related to the theory of integrable systems as mentioned by the authors, and the reviewer hopes thus that the results and the method of this paper could be used to generate integrable systems as has been done by Soviet mathematicians [see e.g. I. M. Krichever, Funkts. Anal. Prilozh. 12, No.3, 20-31 (1978; Zbl 0408.34008), V. G. Drinfel’d, ibid. 11, No.1, 11-14 (1977; Zbl 0359.14011)].
[For part II, see the following review (Zbl 0656.17007).]
Reviewer: Tu Guizhang

MSC:

17B20 Simple, semisimple, reductive (super)algebras
33C05 Classical hypergeometric functions, \({}_2F_1\)
35C10 Series solutions to PDEs
13N05 Modules of differentials
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
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