On a matrix element representation of the GKZ hypergeometric functions

TL;DR

This paper introduces a novel matrix element representation for GKZ hypergeometric functions using non-reductive Lie algebras, expanding the understanding of their algebraic structure and connections to Whittaker functions.

Contribution

It develops a new representation theory approach, identifying GKZ hypergeometric functions with matrix elements of non-reductive Lie algebras of oscillator type.

Findings

GKZ hypergeometric functions can be represented as matrix elements of non-reductive Lie algebras.

Whittaker functions are special cases of GKZ functions with multiple matrix element representations.

The approach links hypergeometric functions to Lie algebra representations, broadening their algebraic interpretation.

Abstract

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