ag.algebraic geometry – Tensor product of perverse sheaves on flag varieties

I am interested in computing tensor products of perverse sheaves on (partial) flag varieties. For a specific example – consider the product of the big projective on $mathbb{P}^1$ with itself (This is the projective cover of the skyscraper sheaf on the 0-dimensional stratum). Does this have a simple description? How can I compute its cohomology?

Any general tips or computational tricks in this context are very welcome. I am especially interested in tilting perverse sheaves such as the one above.