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 $\operatorname{SL}(d,\mathbb R)/\operatorname{SL}(d,\mathbb Z)$ 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. 

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Update: this should be true for any $t\ne 0$. So we can actually just fix $t=1$ and just prove that $g_1$ acts ergodically, which is equivalent to saying that the subgroup $(g_n)_{n\in \mathbb Z}$ acts ergodically.