Let consider the map $tr:\mathbb{G}_{m}^{n}\rightarrow\mathbb{A}^{1}_{\mathbb{F}_{q}}$ given by the sum of the coordinates and let $\psi:\mathbb{F}_{q}\rightarrow\mathbb{Q}_{l}^{*}$ a non trivial additive character.

We consider $\mathcal{L}_{\psi}$ the corresponding Artin-Schreier sheaf on $\mathbb{A}^{1}_{\mathbb{F}_{q}}$ and $p:\mathbb{G}_{m}^{n}\rightarrow T$ a surjective morphisms of groups where $T$ is a torus.

Is it true that $p_{!}tr^{*} \mathcal{L}_{\psi}$ is always perverse?

Moreover, we already know by Deligne SGA 4 1/2 that $H^{i}(tr^{*} \mathcal{L}_{\psi})=0$ $\forall~ i\neq n$.

Let consider the map $tr:\mathbb{G}_{m}^{n}\rightarrow\mathbb{A}^{1}_{\mathbb{F}_{q}}$ given by the sum of the coordinates and let $\psi:\mathbb{F}_{q}\rightarrow\mathbb{Q}_{l}^{*}$ a non trivial additive character.

We consider $\mathcal{L}_{\psi}$ the corresponding Artin-Schreier sheaf on $\mathbb{A}^{1}_{\mathbb{F}_{q}}$ and $p:\mathbb{G}_{m}^{n}\rightarrow T$ a surjective morphisms of groups where $T$ is a torus. Explicitely the map $p$ is given by $n$ cocharacters $\lambda_{1},...,\lambda_{n}$ and we assume that the trivial cocharacter doesn't appear.

Is it true that $p_{!}tr^{*} \mathcal{L}_{\psi}$ is always perverse?

Moreover, we already know by Deligne SGA 4 1/2 that $H^{i}(tr^{*} \mathcal{L}_{\psi})=0$ $\forall~ i\neq n$.