We study the analysis of a probability density $K$ on a Lie group $G$, where $G$ is a semidirect product of a compact group $M$ with a nilpotent group $N$. To approximate analysis on $G$ with analysis on $N$, it is natural to consider certain maps (“realizations”) of $G$ onto $N$. In this paper, we prove the existence of a realization of $G$ in $N$ which is $K$-harmonic (modulo the commutator subgroup of $N$). By utilizing this result and extending some ideas of Alexopoulos, we can prove the boundedness in $L^p$ spaces of some new Riesz transforms associated with $K$, and obtain new regularity estimates for the convolution powers of $K$.