Let $G$ be a commutative connected algebraic group over a field $k$, with group operation $m:G\times G\to G$. If $k=\mathbb{F}\_q$, we may use a character $\varphi:G(k)\to\overline{\mathbb{Q}}\_\ell^\times$ to construct the "Lang sheaf" $\mathscr{L}\_\varphi$ [$\S$1.4, SGA 4 $\frac{1}{2}$, Sommes trig.]. This is a rank 1 local system (= lisse sheaf) on $G$ satisfying "the character condition" $m^\* \mathscr{L}\_\varphi=\mathscr{L}\_\varphi\boxtimes \mathscr{L}\_\varphi$.


**Question 1:** is every rank 1 local system on $G$ satisfying the character condition a Lang sheaf?


If $k=\mathbb{C}$ and $G=\mathbb{G}\_a$ (resp. $G=\mathbb{G}\_m$), there's a rank 1 vector bundle with connection over $G$ satisfying the character condition. It's the exponential (resp. Kummer) D-module $e^{\alpha x}:=\mathcal{D}/\mathcal{D}(\partial -\alpha)$ (resp. $x^\alpha:=\mathcal{D}/\mathcal{D}(x\partial -\alpha)$).


**Question 2:** are those the only rank 1 vector bundles with connection satisfying the character condition on $G$?


**Question 3:** can we construct rank 1 vector bundles with connection satisfying the character condition on other groups $G$?



Every rank 1 local system $L$ satisfying $m^*L=L\boxtimes L$ comes from the Lang torsor? The same holds for D-modules?
Question 1: Yes. See Lemma 2.14 of my paper [On the Ramanujan conjecture for automorphic forms over function fields](https://arxiv.org/abs/1805.12231) with Nicolas Templier, although this simple argument is surely not original to us.


Question 2: Any rank one connection has the form $\partial-f$ for a function $f$ on $\mathbb G\_a$ or $x \partial -f$ for a function $f$ on $\mathbb G\_a$. (This uses crucially the fact that every rank one vector bundle on $G$ is trivial.) In both cases, the first term is the unique-up-to-scaling invariant vector field. Two different functions give the same connection if and only if there difference is the logarithmic derivative of a nowhere vanishing function. On $\mathbb G\_a$ this occurs only if their difference vanishes and on $\mathbb G\_m$ only if their difference is an integer multiple of $1$. In either case, this is a discrete set. Thus, a function gives a translation-invariant connection if and only if the function itself is translation-invariant, i.e. is a constant $\alpha$.


Question 3: Sure, just pick any basis $\nabla\_1,\dots, \nabla\_n$ for the $G$-invariant vector fields on $G$ and then mod out by $\nabla\_i-\alpha\_i$ for any $\alpha\_1,\dots, \alpha\_n \in \mathbb C$ (in other words, take the connection associated to a translation-invariant 1-form).


