Spectral theory for non-unitary twists

Let $G$ be a Lie-group and $\text{Ga}\subset G$ a cocompact lattice. For a finite-dimensional, not necessarily unitary representation $\text{om}$ of $\text{Ga}$ we show that the $G$-representation on $L^2(\text{Ga}\text{bs}G,\text{om})$ admits a complete filtration with irreducible quotients. As a consequence, we show the trace formula for non-unitary twists and arbitrary locally compact groups.