TITLE: extensions of IC sheaves
QUESTION [5 upvotes]: Let $X$ be a connected smooth algebraic variety (say over $\mathbb{C}$) and let $L$ be a local system on an open subvariety. Suppose that we know $H^*(X)$. How can we compute $Ext^i(IC(L),IC(L))$ for $i\ge  1$ ? Is there a general method or only ad-hoc arguments specific to examples? 
A source of examples: let $p:X'\to X$ be a small map. Then the sheaf $\mathcal{F} = Rp_*\mathbb{C}_{X'}$ is an IC sheaf of the above form. (and here suppose we also know  $H^*(X')$).
I would be interested to know if for this class of examples one can say something precise about the dimensions of the Ext groups. 
One could also replace cohomology with equivariant cohomology, or more generaly consider the question for algebraic stacks.
Any references where this is computed in specific examples would be also very useful.

REPLY [5 votes]: Not sure what to say in general, but in practice one often uses a semismall resolution. For example if your sheaf $IC(L)$ is given as $\pi_* C_Z$ (pushforward of constant sheaf - I'm assuming $Z$ smooth and oriented) for a proper $Z\to X$ you can calculate derived self-maps $$Hom(\pi_*C_Z,\pi_*C_Z)=
Hom(C_Z, \pi^!\pi_*C_Z).$$ Now write $\pi_1,\pi_2$ for the two projections from $Z\times_X Z\to Z$. Then we calculate further 
$$Hom(C_Z, \pi^!\pi_*C_Z)=R\Gamma_Z(\pi_{1,*}\pi_2^!C_Z)=R\Gamma_{Z\times_X Z}(\omega_{Z\times_X Z}),$$
i.e., Borel-Moore homology of $Z\times_X Z$ (up to a shift I've ignored). This works in the equivariant setting (i.e. for sheaves on stacks) equally well, giving equivariant Borel-Moore homology. The famous example of this is the case where $X$ is the nilpotent cone [or its quotient by the group], $Z$ the [equivariant] Springer resolution, and $Z\times_X Z$ the [equivariant] Steinberg variety, cf. the book of Chriss-Ginzburg. In this case the equivariant BM homology (=Ext algebra of the Springer sheaf) is the degenerate affine Hecke algebra. This calculation works as is in the dg setting --- for the statement on the level of derived categories you need formality, which in the Springer case is a result of Laura Rider here.