Let $\Gamma$ be a finite subgroup of SU(2) and consider the quotient of $S^3$ by $\Gamma$ via its left action. Pick a simply connected compact Lie group $G$ and take a flat connection on this quotient. Or equivalently, choose a homomorphism $\Gamma \to G$. The Chern-Simons action functional should assign a value in $\mathbb{R}/\mathbb{Z}$. How do I compute it?


The paper <http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002334585> by Kronheimer and Nakajima has the formula for the eta invariant which is closely related, but I think it gives the CS functional only as a value in $\mathbb{R}/(\mathbb{Z}/h^\vee(G))$.



Value of the Chern-Simons functional for flat connections on $S^3/\Gamma$
For $G=SU(N)$, there is a paper [*SU(n)–Chern–Simons invariants of Seifert fibered 3–manifolds*](http://www.worldscientific.com/doi/abs/10.1142/S0129167X98000130) (Int. J. Math., 09, 295-330 (1998))


