= Triviality of finite-dimensional unitary representations of SL2R
The diagonal element $\operatorname{diag}(m,m^{-1})$ conjugates $U(t)$ to $U(m^2t)=U(t)^{m^2}$. In a finite-dimensional <unitary representation>, conjugacy therefore makes its eigenvalue multiset invariant under every square power. <Square-power rigidity of a finite nonzero spectrum> makes the image of every upper unipotent the identity. Its kernel is a <normal subgroup>; <elementary unipotent generators of SL2R> then force the entire <group representation> to be trivial. This algebraic argument does not require continuity.
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