Principal Series Representation of $SU(2,1)$ and Its Intertwining Operator

TL;DR

This paper explicitly describes the structure of principal series representations of SU(2,1) and introduces a method to compute intertwining operators using combinatorial and contour integration techniques, with results expressed via special functions.

Contribution

It provides a detailed realization of the $(\mathfrak{g},K)$-module structure for SU(2,1) and a novel method for calculating intertwining operators applicable to similar Lie groups.

Findings

Explicit $(\mathfrak{g},K)$-module structure for SU(2,1)

Method to compute intertwining operators using gamma functions and hypergeometric series

Generalization potential to other real reductive Lie groups

Abstract

In this paper, following a similar procedure developed by Buttcane and Miller in \cite{MillerButtcane} for $SL(3,\RR)$, the $(\frakg,K)$-module structure of the minimal principal series of real reductive Lie groups $SU(2,1)$ is described explicitly by realizing the representations in the space of $K$-finite functions on $U(2)$. Moreover, by combining combinatorial techniques and contour integrations, this paper introduces a method of calculating intertwining operators on the principal series. Upon restriction to each $K$-type, the matrix entries of intertwining operators are represented by $\Gamma$-functions and Laurent series coefficients of hypergeometric series. The calculation of the $(\frakg,K)$-module structure of principal series can be generalized to real reductive Lie groups whose maximal compact subgroup is a product of $SU(2)$'s and $U(1)$'s.