Questions about the proof of Chen's $\pi_1$-de Rham theorem in Hain's Hodge Structure on $\pi_1$ paper

Question

I read a proof of Chen's $\pi_1$-de Rham theorem in Hain's paper The Geometry of the Mixed Hodge Structure on the Fundamental Group. The proof is very elegant, but there are a few (hopefully trivial) things near the end that are confusing me.

Let me recall some notation and details in the proof. Let $M$ be a smooth manifold and $x\in M$ a basepoint. Let $G=\pi_1 (M,x)$. We denote the vector space of of iterated integrals on $M$ of length (number of differentials in the integrand) less than or equal to an integer $s$ by $B_s (M)$, and the vector subspace of those that are homotopy functionals by $H^0 (B_s (M),x) = H^0 (B_s (M))$ (all of this notation comes from the bar construction). Let $J$ be the augmentation ideal of the group ring $\mathbb{Z}[G]$. Chen's theorem asserts that for every $s\geq 0$ the integration map

$$H^0 (B_s (M)) \to \text{Hom}_{\mathbb{Z}} (\mathbb{Z}[G]/J^{s+1}, \mathbb{R}), \quad [\omega_1 \vert \dots \vert \omega_r]\mapsto \left(\gamma\mapsto \int_\gamma \omega_1\dots\omega_r \right)$$

is an isomorphism. (This can be interpreted as giving a canonical isomorphism between Betti and de Rham Tannakian fundamental groups after base change).

Let $R$ be a ring and $V = R[G]/J^{s+1}$. Assuming $G$ is finitely generated, $V$ is finite-dimensional. Hain shows that right translation on $V$ is a unipotent representation of $G$. We have a filtration of subspaces

$$V\supseteq J/J^{s+1}\supseteq J^2/J^{s+1}\supseteq \dots \supseteq J^s/J^{s+1}\supseteq 0, \quad (*)$$

and $G$ acts trivially on the graded quotients $J^t/J^{t+1}$. We form a flat line bundle $E\to M$ where $E=(V\times \tilde{M})/G$, and since $G$ stabilises $(*)$, this bundle inherits a filtration by flat subbundles

$$E\supseteq E^1\supseteq E^2 \supseteq \dots \supseteq E^s \supseteq 0,$$

with the fiber of $E^t$ equal to $J^t/J^{s+1}$. Hain shows that each of these flat connections can be trivialised.

Now comes the part I am confused about. Define $\text{End}_J (V)$ to be the Lie algebra of endomorphisms of $V$ that preserve the flag $(*)$. There are two confusing things:

1. Hain claims that every element of $\text{Hom}_J (V)$ satisfies $A^{s+1}=0$. I don't see why this is the case - for example, why is the identity map not in $\text{End}_J (V)$?
2. Secondly, parallel transport $T$ of the connection on $E$ is shown to be an element of $H^0(B_s (M))\otimes \mathbb{R}[G]/J^{s+1}$. Then it is claimed that integration $\gamma\mapsto T(\gamma)$ (parallel transport of the connection along $\gamma$) induces the identity map on $\mathbb{R}[G]/J^{s+1}$. I really don't understand where this step came from, and would be grateful if anyone can shed some light on this.

Many thanks!

Answer 1

For 1:

Stabilize means for $\phi \in End_{J}(V) \ : \phi(E^{i}) \subseteq E^{i+1} \ $ therefore $\ \phi^{k}(E^{i}) \subseteq E^{i+k}$ with $E^{0}:=E$ and $E^{s+1}=0$
So $ \phi^{s+1}(E) =0$

I am reading the paper right now and I also couldn't figure out 2. Were you able to figure it out by now?