Let $GL^+_n \subset \mathbb{R}^{n \times n}$ be the subgroup of real matrices with positive determinant and $\widetilde{GL}^+_n$ be its universal cover. Why does $\widetilde{GL}^+_n$ not admit finite-dimensional representations? I would be happy with a proof or a reference.
This question is relevant in spin geometry. Normally, one uses $Spin_n$, which is the universal cover of $SO_n$ (if $n \geq 3$), and a finite-dimensional representation $\rho:Spin_n \to GL(V)$ to construct the spinor bundle of a Riemannian spin manifold $(M,g)$ with spin structure $\Theta^g: Spin^g M \to SO^g M$ via $\Sigma^g M := Spin^g \times _{\rho} V$. In texts like [1], the authors work with a topological spin structure $\widetilde{GL}^+ M \to GL^+ M$ up to the point where one has to construct the spinor bundle. I like this approach, which is why I would like to better understand why this part of the construction fails. Unfortunately, I do not know much about representation theory besides what is mentioned in [1].
[1] Bär, Gauduchon, Moroianu: Generlized cylinders in Semi-Riemannian and Spin Geometry
EDIT: One should require the representations to be faithful.
By [1, Lem. II.5.23], any finite-dimensional representation $\tilde \rho: \widetilde{GL}_n^+ \to GL(V)$, $n>2$, factors through $GL^+_n$, i.e. there exists $\rho:GL^+_n \to Aut(V)$ such that $\rho \circ \vartheta = \tilde \rho$, where $\vartheta:\widetilde{GL}^+_n \to GL^+_n$ is the two-fold cover. Consequently, $\ker \tilde \rho \supset \ker \vartheta \cong Z_2 $ and $\tilde \rho$ cannot be faithful.
[1] Lawson/Michelsohn: Spin Geometry