If $X$ is a connected algebraic variety of finite type over $k$ (with $k$ a field of positive characteristic) of dimension $d$, and if $\mathcal{F}$ and $\mathcal{G}$ are perverse sheaves on $X$ so $(\mathcal{F}\otimes\mathcal{G})[-d]$ is also a preverse sheaf?


In the same setting above if we have a unipotent connected group over $k$ acting on $X$ transitively what can we say about the equivariant derived category of $X$ (the category of equivariant complexes in $X$) ?



Perverse sheaves and tensor product
This is extremely false. Consider the skyscraper sheaf on a smooth point of a positive dimensional variety; this is always perverse (since it is Verdier self-dual). The tensor product of this with itself will be the same sheaf again, so when you shift, you mess up perversity.


