Intertwining Operator for $Sp(4,\mathbb{R})$ and Orthogonal Polynomials

TL;DR

This paper computes the module structure of principal series representations of $Sp(4,\mathbb{R})$, introduces a hypergeometric generating function method, and shows that intertwining operator entries are Hahn polynomials or hypergeometric function expansions.

Contribution

It introduces a new hypergeometric generating function and inverse Mellin transform technique to compute intertwining operators for $Sp(4,\mathbb{R})$ representations, revealing their polynomial structure.

Findings

Matrix entries of simple intertwining operators are Hahn polynomials.

Long intertwining operator entries are constant terms of hypergeometric function Laurent expansions.

The paper advances methods for calculating intertwining operators in representation theory.

Abstract

We calculate the $(\mathfrak{g},K)$ module structure for the principal series representation of $Sp(4,\mathbb{R})$. Furthermore, we introduced a hypergeometric generating function together with an inverse Mellin transform technique as an improvement to the method to calculate the intertwining operators. We have shown that the matrix entries of the simple intertwining operators for $Sp(4,\mathbb{R})$-principal series are Hahn polynomials, and the matrix entries of the long intertwining operator can be expressed as the constant term of the Laurent expansion of some hypergeometric function.