Weil representations of finite symplectic groups, and Gow lattices

A study is made of the positive-definite integral lattices $\Lambda$ introduced by Gow and contained in the space of the faithful rational Weil representation of the finite symplectic group $S=\operatorname{Sp}(2n,p)$ ($p$ a prime number, $p\equiv -1$ (mod 4)) and invariant under the action of this group. In the special case $n=2$, $p=3$ all such lattices are found, up to similarity. In the general case the group $G=\operatorname{Aut}(\Lambda)$ of all automorphisms of such lattices is computed. In particular, it is determined that in most cases $G$ coincides with $\operatorname{Aut}(S)$.