Consider the action of $GL_n(\mathbb{Q}_p)$ on $\mathbb{Q}_p^n$, and let $T$ be the diagonal torus. Let $\Lambda$ and $\Lambda'$ be full-rank sublattices ($\mathbb{Z}_p$-submodules of rank $n$) such $Stab_T(\Lambda) = Stab_T(\Lambda')$. Is it true that $\Lambda' = t\cdot \Lambda$ for some $t\in T$?
This is true when $n = 2$; to see this, if $\{e_1,\, e_2\}$ is the standard basis and $\Lambda$ any lattice, we can always take a basis of the form $$\{p^re_1,\, \alpha e_1 + p^s e_2\}$$ where $r,\, s\in \mathbb{Z}$ and $\alpha$ is chosen modulo $p^s$; at this point you can check directly that $\Lambda$ and $\Lambda'$ have the same stabilizer if and only if $v_p(\alpha) - r = v_p(\alpha') - r'$, and any two such lattices are in the same $T$-orbit.
(This has been crosslisted to MathOverflow: https://mathoverflow.net/questions/196474/when-do-two-lattices-have-the-same-stabilizer-in-the-diagonal-torus)