General hypergeometric functions on complex Grassmannians.

*(Russian)*Zbl 0614.33008This 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.

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