Let $\mathcal F$ be a complex of constructible sheaves on a stratified algebraic variety $X$ of dimension $d$.
I read (*) that if $\mathcal F$ is perverse and $\mathscr H^j(\mathcal F) = 0$ for $j \neq -d$ (here $\mathscr H^*$ are the ordinary cohomology sheaves associated to a complex of sheaves), then $\mathcal F$ is a semisimple perverse sheaf.
However, if $X$ is smooth with a single strata, and $\mathcal F = \mathcal L[-1]$ where $\mathcal L$ is a not semisimple local system (for example, $X = \mathbb C^*$, $\mathcal L$ of rank $2$ associated to the endomorphism $e_1 \mapsto e_1, e_2 \mapsto e_1 + e_2$ ). It seems to me that $\mathcal F$ is still concentrated in a single degree by definition, and is also not a semisimple object of $\mathscr P(X)$ by definition. Where is my mistake ?
(*) Reference : Göttsche, Lothar and Soergel, Wolfgang - "Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces", (see the proof of proposition $3$ for the claim)
the paper you mentioned does not make this claim. in that paper you have a sheaf $F$ on $X$,$G$ is a finite group acting on $X$,and $Y$ is $X/G$. now if you look at $\pi^{*}(L)[d]$ where $L$ is a local system on $X$ then the fact about vanishing of the cohomology of$\pi_{*}L$ (and self-duality) just means that $\pi_{*}L$ is perverse.
semisimplicity deduced from the relation between sheaves on $Y$ and $G$ invariant sheaves on $X$,and the fact that $G$ is finite and hence all its representations are semisimple, and the relation between local systems on $X$ and representations of the fundamental group.
in fact you start with a representation of fundamental group $V$. then you can associate to this a representation $V\otimes F[G]$ of $G$ and then this representation is semisimple because $G$ is finite.