  Let $\Delta_{k,a}$ and $\Delta_k $ be the $(k,a)$-generalized Laguerre
operator and the Dunkl Laplacian operator on $\mathbb{R}^n$, respectively. The
aim of this article is twofold. First, we prove a restriction theorem for the
Fourier-$\Delta_{k,a}$ transform. Next, as an application of the restriction
problem, we establish Strichartz estimates for orthonormal families of initial
data for the Schr\"odinger propagator $e^{-i t \Delta_{k, a}} $ associated with
the operator $ \Delta_{k, a}$. Further, using the classical Strichartz
estimates for the free Schr\"odinger propagator $e^{-i t \Delta_{k, a}} $ for
orthonormal systems of initial data and the kernel relation between the
semigroups $e^{-i t \Delta_{k, a}}$ and $e^{i \frac{t}{a}\|x\|^{2-a}
\Delta_{k}},$ we prove Strichartz estimates for orthonormal systems of initial
data associated with the Dunkl operator $ \Delta_k $ on $\mathbb{R}^n$.
Finally, we present some applications to our aforementioned results.
