I understand that via the Borel density theorem given a finite dimensional (polynomial) representation of the simple non-compact Lie groups $SL_n \mathbb R$ or $Sp_n \mathbb R$, I get an irreducible representation when I restrict to $SL_n \mathbb Z$ or $Sp_n \mathbb Z$, respectively.

I wonder if the restriction is injective on the set of isomorphism classes of irreducible polynomial representations. I.e. is it true that if $V$ and $W$ are irreducible polynomial $SL_n \mathbb R$-representations and $$Res_{SL_n \mathbb Z} V \cong Res_{SL_n \mathbb Z} W$$ then $V\cong W$?