  In this paper we will define and investigate the imaginary powers
$\left(-\triangle_{k,1}\right)^{-i\sigma},\sigma\in\mathbb{R}$ of the
$(k,1)$-generalized harmonic oscillator
$-\triangle_{k,1}=-\left\|x\right\|\triangle_k+\left\|x\right\|$ and prove the
$L^p$-boundedness $(1<p<\infty)$ and weak $L^1$-boundedness of such operators.
It is a parallel result to the $L^p$-boundedness $(1<p<\infty)$ and weak
$L^1$-boundedness of the imaginary powers of the Dunkl harmonic oscillator
$-\triangle_k+\left\|x\right\|^2$. To prove this result, we develop the
Calder\'on--Zygmund theory adapted to the $(k,1)$-generalized setting by
constructing the metric space of homogeneous type corresponding to the
$(k,1)$-generalized setting, and show that
$\left(-\triangle_{k,1}\right)^{-i\sigma}$ are singular integral operators
satisfying the corresponding H\"ormander type condition.
