vector bundle associated to the representation of a lattice

Question

Suppose that $G$ is a connected semisimple Lie group and that $\Gamma$ is a cocompact lattice in $G$. Given a complex-linear representation $\chi$ of $\Gamma$ on a finite-dimensional complex vector space $V_{\chi}$, apparently there is an associated vector bundle on $\Gamma\setminus G$. I was wondering where I could see a detailed construction of this vector bundle in terms of the usual definition involving a family of local trivialisations.

Answer 1

Define a right action of $\Gamma$ on $G \times V_{\chi}$ by $$ (x, v)g = \bigl(xg, \chi(g^{-1})v\bigr),\qquad x \in G,\quad v \in V_{\chi}. $$ The quotient $E = (G \times V_{\chi})/\Gamma$ is the total space of a vector bundle over $\Gamma\backslash G$, with fibre $V_{\chi}$ and projection map induced by projection on the first factor, $G \times V_{\chi} \to G$.

A fancy way to say this is: Right multiplication makes $G$ the total space of a principal $\Gamma$-bundle over $\Gamma\backslash G$, and $E$ is the vector bundle associated to the representation $\chi$.

Now to your question: If $U$ is a coordinate neighborhood for the coset space $\Gamma \setminus G$, a.k.a., a coordinate neighborhood of the Lie group $G$ whose translates by $\Gamma$ are pairwise disjoint, then $U \times V_{\chi}$ is a vector bundle chart for $E$.

The gluing is given by the global construction above. Explicitly, if $U_{1} \times V_{\chi}$ and $U_{2} \times V_{\chi}$ are bundle charts, and if $x_{i}$ in $U_{i}$ are points such that $x_{2} = x_{1}g$ for some $g$ in $\Gamma$, then $$ (x_{1}, v) \sim \bigl(x_{2}, \chi(g^{-1})v\bigr). $$