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I think you're getting confused about duality (in the sense of Schur duality here). The endomorphism algebra $E$ of $L$ is just an algebra, and its irreducible representations are just vector spaces. …

The problem is your statement: "The ordinary ( = non-mixed) $Ext^1$ group should be the extensions between the restrictions of $M_1$ and $M_2$ to the intersection $Y_1\cap Y_2$." This is absolutely n …

Both of these are facts which developed gradually over the course of several papers, so it's hard to give a definitive reference. For the first, all the calculations one would need in order to esta …

The stalk of the IC sheaf you're interested in looking at is the intersection cohomology of an actual space, given by the intersection of the Schubert variety $\bar{O}_w$ with an orbit of an opposite …

The short answer is that in general its very hard. For special classes of maps like semi-small ones, it's not so bad (see the book of Chriss and Ginzburg), but for an arbitrary projective map, I don' …

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 i …