On a matrix element representation of the GKZ hypergeometric functions

Abstract We develop a representation theory approach to the study of generalized hypergeometric functions Gelfand, Kapranov and Zelevisnky (GKZ). show that GKZ may be identified with matrix elements non-reductive Lie algebras $${\mathfrak L}_N$$ L N oscillator type. The Whittaker associated principal series representations $$\mathfrak {gl}_{\ell +1}({\mathbb {R}})$$ gl ℓ + 1 ( R ) being special cases thus admit along standard element reductive algebra , another in terms L}_{\ell (\ell +1)}$$ .