  We study the Dirichlet eigenvalue problem of homogeneous H\"{o}rmander
operators $\triangle_{X}=\sum_{j=1}^{m}X_{j}^{2}$ on a bounded open domain
containing the origin, where $X_1, X_2, \ldots, X_m$ are linearly independent
smooth vector fields in $\mathbb{R}^n$ satisfying H\"{o}rmander's condition and
a suitable homogeneity property with respect to a family of non-isotropic
dilations. Suppose that $\Omega$ is an open bounded domain in $\mathbb{R}^n$
containing the origin. We use the Dirichlet form to study heat semigroups and
subelliptic heat kernels. Then, by utilizing subelliptic heat kernel estimates,
the resolution of singularities in algebraic geometry, and employing some
refined analysis involving convex geometry, we establish the explicit
asymptotic behavior $\lambda_k \approx k^{\frac{2}{Q_0}}(\ln
k)^{-\frac{2d_0}{Q_0}}$ as $k \to +\infty$, where $\lambda_k$ denotes the
$k$-th Dirichlet eigenvalue of $\triangle_{X}$ on $\Omega$, $Q_0$ is a positive
rational number, and $d_0$ is a non-negative integer. Furthermore, we provide
optimal bounds of index $Q_0$, which depend on the homogeneous dimension
associated with the vector fields $X_1, X_2, \ldots, X_m$.
