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General hypergeometric functions on complex Grassmannians. (Russian) Zbl 0614.33008
This article is a sequel of studies by the Gel’fand school on generalized hypergeometric functions, which in turn continues K. Aomoto’s work. It gives a new construction of stratification of the Grassmann manifold $${\mathbb{C}}P^ n$$ by means of the dimensions of the intersections with the coordinate planes. When $$\Gamma$$ is a fundamental stratum (i.e. for $$\forall \zeta \in \Gamma$$ the coordinate hyperplanes $$\sigma_ j$$ are normally crossing in $$\zeta$$ outside one particular $$\sigma_ 0)$$, putting $$K=\zeta -\sigma_ 0$$, $$S=K\cap \cup \sigma_ j$$, it calculates the homology groups $$H_ r(K-S,L_{\tau})$$, $$H_ r^{\ell f}(K- S,L_{\tau})$$ with coefficients in a local system $$L_{\tau}$$. These remain only at $$r=k$$, whose dimension is equal to the number of connected components of $$K_{{\mathbb{R}}}-S$$. Then it picks up three types of special k-cycles: imaginary cones, visible cycles (form $$H_ k^{\ell f}(K- S,L_{\tau}))$$, double knots (from $$H_ k(K-S,L_{\tau}))$$, which form bases of these groups under some additional condition.
Generalized hypergeometric functions are defined as the integrals of the fractional function $$\prod_{0\leq j\leq n}x_ j^{\alpha_ j-1}$$ on these k-cycles as an analogy of the corresponding integral representations of Gauss’ classical hypergeometric function. It also gives another representation of hypergeometric functions by means of piecewise polynomial functions defined via a continuous analogue of partition functions.
Reviewer: A.Kaneko

MSC:
 33C05 Classical hypergeometric functions, $${}_2F_1$$ 33C60 Hypergeometric integrals and functions defined by them ($$E$$, $$G$$, $$H$$ and $$I$$ functions)