I am trying to understand paragraph 1.6 of Lusztig's paper "Character Sheaves I". The basic setup is that $X$ is a smooth irreducible variety over a field $k=\overline{k}$, $D\_i, i=1,...,r$ are smooth divisors with normal crossings and $\mathcal{L}$ is $\overline{\mathbb{Q}}\_l$-local system of rank one such that the corresponding monodromy action factors through a finite quotient of order invertible in $k$. Now the IC sheaf $IC(X,\mathcal{L})$ is represented by a single constructible $\overline{\mathbb{Q}}\_l$-sheaf $\overline{\mathcal{L}}$.


Then he claims that if the local monodromy of $\mathcal{L}$ around one $D\_i$ is trivial, then $\overline{\mathcal{L}}$ restricted to $U\bigcup D\_i$ is a local system (this is a special case of his claim).
What is the local monodromy of $\mathcal{L}$ around a $D\_i$ and how to show his claim?



Extending IC sheaves across smooth divisors with normal crossings
The local monodromy around $D\_i$ can be obtained by taking a $\eta$ a geometric generic point of $D\_i$, $R$ the etale local ring of $X$ at $\eta$ with uniformizer $\pi$, then pulling $\mathcal L$ back to $\operatorname{Spec} R[ \pi^{-1} ]$. We then obtain a representation of the étale fundamental group of $\operatorname{Spec} R[ \pi^{-1} ]$. The image of this representation is called the local monodromy.


Rather than $U \cup D\_i$, it would be more precise to say $X \setminus \cup\_{j\neq i} D\_j$. To check that $\overline{\mathcal L}$ restricted to this space is a local system, we first check that for $j \colon U \to X \setminus \cup\_{j\neq i} D\_j$ the open immersion, $j\_\* \mathcal L$ is a local system, and then that $j\_\* \mathcal L$ is the restriction of $\overline{\mathcal L}$.


The pullback of $j\_\*\mathcal L$ to $\operatorname{Spec} R$ is the pushforward from $\operatorname{Spec} R[ \pi^{-1} ]$ to $\operatorname{Spec} R$ of the restriction of $\mathcal L$ to $\operatorname{Spec} R[ \pi^{-1} ]$. So it is the pushforward of a constant sheaf from the generic point to the whole spectrum and thus is a constant sheaf. Thus $j\_\* \mathcal L$ is locally constant at the generic point of $D\_i$. Thus it is locally constant on some neighborhood of the generic point. The complement of the largest open set on which $j\_\* \mathcal L$ is locally constant is a closed set contained in $D\_i$ but not containing the generic point of $D\_i$ and thus has codimension $\geq 2$.


If that complement is nonempty, consider a generic point $\eta'$ of that complement and a Henselian local ring $R'$ at $\eta$. The lisse sheaf gives a representation of the fundamental group of the punctured spectrum of $R'$, which by purity (since the codimension is $\geq 2$) is trivial. So again the sheaf is the constant sheaf and the pushforward is just the constant sheaf, contradicting the assumption that the sheaf is not locally constant at $\eta$, so the complement is indeed empty and $j\_\* \mathcal L$ is lisse.


Since the intermediate extension is the unique perverse extension with no irreducible components supported outside $U$, and $j\_\* \mathcal L$, being lisse, is perverse and has no irreducible components supported outside $U$, $j\_\* \mathcal L$ must give the intermediate extension.


