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

Heckman, G. J.; Opdam, E. M.

Compos. Math. 64, 329-352 (1987).

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)

Keywords:

Weyl group; root system; hypergeometric functions; hypergeometric (partial) differential equations; monodromy; multivariable hypergeometric functions; commuting differential operators; integrable systems

Citations:

Zbl 0408.34008; Zbl 0359.14011; Zbl 0656.17007

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