Let $G$ be a Hausdorff locally compact group and $H$ a Banach space. Let $\pi:G\rightarrow \operatorname{GL}(H)$ be a representation and define
$$\pi^1(\phi)v = \int_G\phi(x)\pi(x)vdx$$
for $v\in H$ and $\phi$ a compactly supported function on $G$. Then, Lang shows in his book $\operatorname{SL}_2(\Bbb R)$, p.5, that $\pi^1$ is a homomorphism of the convolution product with the following lines.
$$\begin{aligned} \pi^1(\phi \ast \psi) &= \int_G(\phi\ast\psi) (x)\pi(x)dx \\ &= \int_G\int_G\phi(xy^{-1})\psi(y)\pi(x)dydx \\ &= \int_G\int_G\phi(x)\psi(y)\pi(x)\pi(y)dxdy \\ &= \pi^1(\phi)\pi^1(\psi)\end{aligned}$$
Where from the second to third line he swapped the order of integration and $x\mapsto xy$. But this only works because $G$ is unimodular, right? Otherwise, shouldn't we get a factor of $\Delta(y)$?