Perverse sheaves and tensor product 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 $X$) ?
 A: 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.
A: I don't really know much about perverse sheaves, but the answer for the first general question is "No", I think, even for curves.  
Here is a simple example for $\bar{\mathbf{Q}}_{\ell}$-sheaves, which is the context I understand a bit. Take a Kummer sheaf on the multiplicative group over a finite field associated to a non-trivial character $\chi$ (often denoted $\mathcal{L}_{\chi}$), and define a perverse sheaf $F$ on the projective line as the middle-extension of this sheaf to the projective line (suitably normalized by shifting); this is just the extension by zero. Let $G$ be its dual; it is the perverse sheaf associated in the same way to $\mathcal{L}_{\bar{\chi}}$. Then $F\otimes G[-1]$, unless I misunderstand the definitions, is the extension by zero of the lisse sheaf $\mathcal{L}_{1}$ on the multiplicative group, i.e., the extension by zero to the projective line of the trivial rank $1$ sheaf.  This is not perverse (because it would be simple perverse, being of rank $1$, and therefore, because of the classification of these and because it is not punctual, it would be the middle extension of its restriction to the multiplicative group, but the latter is the trivial sheaf on the projective line).
A: I know next to nothing about perverse sheaves, except that they are often used as a geometric construction of highest weight categories from Lie theory. In a highest weight category, you have things like simple objects, standard objects, projectives, injectives, and tiltings. Most of these classes will not be preserved under tensor product. However, the tensor product of tilting modules is a tilting module (in the context of finite-dimensional reps of reductive algebraic groups in positive characteristic). Thus your statement ought to be true if you make it about tilting perverse sheaves, instead of all perverse sheaves, but under an appropriate multiplication for perverse sheaves, such as convolution, and for appropriate X. I don't know in what generality this would hold.
Notes on tilting modules in positive characteristic representation theory: http://www.imsc.res.in/~knr/vdk/mathieu.pdf
Geometric survey of tilting theory: http://arxiv.org/pdf/math/0301098.pdf
