I'm reading Simpson's paper "Higgs Bundles and Local Systems". There he defines for a local system $V$ of vector spaces on a compact Kahler manifold $M$ we have a representation of fundamental group $\pi_1(M,x) \to GL(V_x)$. Define the monodromy group $M(V,x)$ to be the Zariski closure of the image of $\pi_1(M,x)$.
He goes on to say that there's an alternative description of monodromy group.For each $T^{a,b}V = V^{\otimes a} \otimes V^{\otimes b}$ identify the set of sub local systems $W \subset T^{a,b}V$. Then $M(V,x)$ is the subgroup of $GL(V_x)$ of all $g$ such that $g(W_x) \subset W_x$ for all sub local systems $W$.
Here are my questions :
- How does the alternative description relate to the usual one?
- How exactly does $g$ act on $W_x$?
- He also says that if $V$ is semisimple local system (which is the same as the fundamental group representation being semisimple) then $M(V,x)$ is a reductive group. Why is that so?
