1997/11/10 by Mikhail Kapranov, M. Kapranov, Kapranov, M. · 3 citations
Computer Science · Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation #alg-geom #math.AG
paper · pdf · doi:10.48550/arxiv.alg-geom/9711011
plain TEX, 43 pages. Final version, to appear in Proceedings of the Taniguchi Symposium "Algebraic geometry and integrable systems" (Kobe-Kyoto, July 1997)
openalex publication_date 1997/11/10 · arxiv created 1998/04/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The A-hypergeometric system studied by I.M. Gelfand, M.I. Graev, A.V. Zelevinsky and the author, is defined for a set A of characters of an algebraic torus. In this paper we propose a generalization of the theory where the torus is replaced by an arbitrary reductive group H and A is a set of irreducible representations of H. The functions are thus defined on the space MA of functions on H spanned by the matrix elements of representations from A. The properties of the system are related to the geometry of a certain algebraic variety XA, which belongs to the class of group compactifications studied by De Concini and Procesi. We develop the theory of Euler integral representations for these generalized hypergeometric functions (with integrals taken over cycles in H). We also construct the analogs of hypergeometric series, by expanding the delta-function along a subgroup into a power series and taking the termwise Fourier transform.