Hypergeometric functions on the external power \(\Lambda^k\mathbb{C}^n\) and on the Grassmannian \(G_{k,n}\), their relationships and integral representations (Q1264955)

\textit{I. M. Gel'fand} [Dokl. Akad. Nauk SSSR 288, No. 1, 14-18 (1986; Zbl 0645.33010)] defined the notion of general hypergeometric functions (HF) on \(\mathbb{C}^N\) associated with a lattice \(L\) in \(\mathbb{C}^N\), and he and several coworkers have investigated various aspects of this theory in a number of papers since that time. Here the authors describe relationships between the HF on the Grassmannian \(G_{k,N}\) of \(k\)-planes in \(\mathbb{C}^N\) (actually on the subvariety that is the image of the Plücker coordinates) and HF on the \(k\)th exterior power \(\Lambda^k \mathbb{C}^N\) of \(\mathbb{C}^N\), and use these relationships to reduce the study of HF on the Grassmannian to those on \(\Lambda^k \mathbb{C}^N\). The algebraic structure of \(\Lambda^k \mathbb{C}^N\) is somewhat simpler, particularly as concerns strata and the torus actions occurring in the definition of the HF. The authors also provide integral representations for the HF. The previous paper in the series appeared in 1992 [\textit{I. M. Gel'fand, M. I. Graev} and\textit{V. S. Retakh}, Russ. J. Math. Phys. 1, No. 1, 19-56 (1993; Zbl 0870.33008)]. The translation shows a lack of familiarity with terms in algebraic and analytic geometry.