Let f:X→Y be a semismall resolution of singularities. Then the pushforward of the constant sheaf on X is a semisimple perverse sheaf on Y. Under these conditions, I know how to calculate the direct summands of the pushforward f\*ℚX[dim X].


My question is as follows: What more general statements are there that enable us to explicitly calculate the direct summands of the pushforward? I'm thinking especially of the case where f:X→Y is as above (so in particular semismall), but we replace the constant sheaf ℚX with an arbitrary perverse sheaf (of geometric origin).



Explicit Direct Summands in the Decomposition Theorem 
I'm a little confused about your question. If you just want to know what the summands are there's nothing special about f\*ℚX[dim X]. The only way I know of understanding that sheaf is a general algorithm for understanding all semi-simple perverse sheaves. The only fact you use is that a simple perverse sheaf is concentrated in the highest degree allowed by perversity on the largest stratum in its support, and strictly below this degree on all smaller strata (do I need some hypothesis for this? This tends to be the sort of thing I forget). 


So, if I have a semi-simple perverse sheaf F, all I have to is look at the restriction of F to each stratum. This will be a complex, whose cohomology in each term is a local system. Perversity includes an upper bound on the degrees that this cohomology can be non-zero. I take the local system in the highest degree allowed by perversity on each stratum, and take the IC sheaves of all those local systems. It happens that in the case of f\*ℚX[dim X], this local system has a geometric interpretation (it's the highest degree cohomology of the fiber allowed by semi-smallness), but that's the only thing that's special about this case.


