  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 M_A 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 X_A, 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.
