  We prove that if $p>d$ there is a unique gaussian distribution (in the sense
of Evans) on the space $\mathbb{Q}_p[x_1, \ldots, x_n]_{(d)}$ which is
invariant under the action of $\mathrm{GL}(n, \mathbb{Z}_p)$ by change of
variables. This gives the nonarchimedean counterpart of Kostlan's Theorem
\cite{Kostlan93} on the classification of orthogonally (respectively unitarily)
invariant gaussian measures on the space $\mathbb{R}[x_1, \ldots, x_n]_{(d)}$
(respectively $\mathbb{C}[x_1, \ldots, x_n]_{(d)}$). More generally, if $V$ is
an $n$-dimensional vector space over a nonarchimedean local field $K$ with ring
of integers $R$, and if $\lambda$ is a partition of an integer $d$, we study
the problem of determining the invariant lattices in the Schur module
$S_\lambda(V)$ under the action of the group $\mathrm{GL}(n,R)$.
