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In this paper, following a similar procedure developed by Buttcane and Millerin \cite{MillerButtcane} for $SL(3,\RR)$, the $(\frakg,K)$-module structure ofthe minimal principal series of real reductive Lie groups $SU(2,1)$ isdescribed explicitly by realizing the representations in the space of$K$-finite functions on $U(2)$. Moreover, by combining combinatorial techniquesand contour integrations, this paper introduces a method of calculatingintertwining 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 begeneralized to real reductive Lie groups whose maximal compact subgroup is aproduct of $SU(2)$'s and $U(1)$'s.

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