$\mathrm{GL}(n,\mathbb{Z}_p)$-invariant Gaussian measures on the space of $p$-adic polynomials

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 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 $\mathrm{GL}(n,R)$.