The stack of equivariant local system is quasi-smooth

Question

Let $G$ be a (connected ?) algebraic group and $X$ a smooth, projective, and connected algebraic curve, both over an algebraically closed field $k$ of characteristic $0$.

My questions are then as follows:

1. By "local systems" in this context, do we mean lisse $\ell$-adic étale sheaves on $X$ (with coefficients in $\bar{\mathbb{Q}}_{\ell}$ ?), for some prime $\ell$ ?
2. It is apparently well-known to experts that the dg stack $LocSys(X)^G$ of $G$-equivariant local systems on $X$ is quasi-smooth (cf. subsubsection 1.1.5 of Arinkin–Gaitsgory, Singular support of coherent sheaves, and the geometric Langlands conjecture, arXiv:1201.6343); recall that a dg algebraic stack $\mathscr{Z}$ is quasi-smooth iff for all points $z \in |\mathscr{Z}|$, $H^i(T_z\mathscr{Z}) = 0$ whenever $i < 0$. Where might I find a reference for this fact ?

Thanks in advance!

Edit: Thanks to Will Samin for pointing out to me that it should've been $i < 0$ instead of $i \not = -1, 0$.

Answer 1

These are both answered in section 10 of the linked paper but since this is done at a very high level I will try to provide a down-to-earth explanation.

1 No, absolutely not. We mean local systems in the D-module sense, i.e. $G$-torsors with flat connection. So for $G =G L_n$ these are $D$-modules that, restricted to $\mathcal O_X$, are rank $n$ vector bundles.

This is crucial as the base field for the moduli space of local systems must be the coefficient field of these local systems. For coherent sheaves on this space to be equivalent to $D$-modules on $Bun_G$, the coefficient field of this space must match the coefficient field of $Bun_G$, which is the base field $k$. The only sheaf theory with coefficient field naturally the base field $k$ is $D$-module theory.

2 This is proven in Proposition 10.4.5 of the linked paper. I think you have the definition wrong and they only demand $H^i$ vanish for $i<0$. The calculation they do shows that $H^i(T_z \mathcal L)$ is $H^{i+1}$ of $X$ with coefficients in the adjoint representation composed with $\mathcal L$. This is believable as it's not too hard to see (in any sheaf theory - $\ell$-adic, Betti, $D$-module, whatever) that deformations of a local system are controlled by $H^1$ of the adjoint. Since curves have de Rham cohomology concentrated in degrees $0,1,2$, we have $H^i(T_z \mathcal L) \neq 0$ only for $i=-1, 0,1$.