A Gelfand-Tsetlin type base for the algebra $\mathfrak{sp}_4$ and hypergeometric functions

TL;DR

This paper constructs a Gelfand-Tsetlin type basis for the Lie algebra rak{sp}_4, expressing the basis functions via hypergeometric functions and deriving explicit formulas for algebra generators' actions, advancing representation theory techniques.

Contribution

It introduces a new Gelfand-Tsetlin type basis for rak{sp}_4 and develops methods to explicitly compute generator actions in this basis, which were previously unknown.

Findings

Explicit formulas for generator actions in the new basis

Representation of basis functions through hypergeometric functions

Development of new analytical techniques for Lie algebra representations

Abstract

In the paper a realization of representation of a Lie algebra $\mathfrak{sp}_4$ in the space of function on the Lie group $Sp_4$ is considered. We find a function corresponding to a Gelfand-Tsetlin type vector for $\mathfrak{sp}_4$ constructed by D.P. Zhelobenko. This function is expressed though a $A$-hypergeometric function. After developing some new technique we derive analytically formulas for the action of generators of algebra in this base (the were not known before). These formula turn out to be much more difficult than the formulas for the action of generators in the Gelfand-Tsetlin type base constructed by Molev.