I wonder if there are any direct proof that $g_t=\text{diag}(e^{t/m}I_m,e^{-t/n}I_n)$'s action on $\operatorname{SL}(n,\mathbb R)/\operatorname{SL}(n,\mathbb Z)$ is ergodic (or even stronger, mixing). Here $I_n$ denote the $n \times n$ identity matrix.

The classical proof of this involves the Howe-Moore theorem, which uses Mautner's lemma and can be applied to much more general Lie groups. But I wonder in this concrete setting whether we could have a more simplified/elementary proof. Although I stated $(g_t)$ explicitly, I should mention that perhaps only the unboundedness of $(g_t)$ will be used, although $g_t$ being an unbounded diagonal flow might make the proof easier somehow.