  We use a method developed by Bj\"orklund and Gorodnik to show a central limit
theorem (as $T$ tends to $\infty$) for the counting functions $\# \left(
\Lambda \cap \Omega_T \right)$ where $\Lambda$ ranges over the space $Y_{2d}$
of symplectic lattices in $\mathbb{R}^{2d}$ ($d \geqslant 4$). Here $\lbrace
\Omega_T \rbrace_T$ is a certain family of bounded domains in $\mathbb{R}^{2d}$
that can be tessellated by means of the action of a diagonal semigroup
contained in $\mathrm{Sp}(2d, \mathbb{R})$. In the process we obtain new $L^p$
bounds on a certain height function on $Y_{2d}$ originally introduced by
Schmidt.
