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awarded  Enthusiast 
Apr 5 
comment 
Cohomology of a local system and Deligne's weight filtration
@use54343: What do you mean by "your construction"? Restricting $IC$ to a strata? Doesn't taking $\mathcal{L}$ to the the local system I describe (an $IC$) recover what you want. (It is not clear to me how your second paragraph relates to the first. It seems your question is simply a question about variations of mixed Hodge structures on $\mathbb{C}^*$ and their cohomology. It would be much clearer if stated in this way.) 
Apr 4 
comment 
Cohomology of a local system and Deligne's weight filtration
You are right to be suspicious. If one takes the direct image of the constant sheaf under $z \mapsto z^2$ one gets a local system which splits into two pieces, both of which are pure, and one of which has no cohomology. The nontrivial summand gives a counterexample to your hopes. Making $1  f$ compatible with weight filtrations is a tricky business, in the mixed Hodge world it is given by the limit mixed Hodge structure... 
Apr 3 
comment 
Status of Borho and Brylinski's irreducibility conjectures?
@Webstermeister To be honest I don't see immediately why it is the same question... 
Apr 2 
comment 
Status of Borho and Brylinski's irreducibility conjectures?
This may or may not be relevant: front.math.ucdavis.edu/1405.3479 ! 
Feb 27 
revised 
Residual finiteness: why do we care?
added 1 character in body 
Dec 12 
answered  Example to show that the inverse image under a finite morphism is not texact with respect to the perverse tstructure 
Dec 9 
answered  Representation of GL(n, F_p) over F_p, for n small 
Dec 9 
answered  counting points on nilpotent Springer fiber 
Dec 9 
comment 
counting points on nilpotent Springer fiber
One should be a little careful here. De Concini, Lusztig and Procesi do not show the existence of an affine paving in exceptional type. Moreover I think that the argument with a torus having finitely many fixed points only works in type $A$. 
Dec 3 
awarded  Necromancer 
Dec 2 
revised 
Motivation behind the construction of Deligne and Lusztig
edited body 
Dec 2 
answered  Motivation behind the construction of Deligne and Lusztig 
Nov 24 
comment 
Stability conditions of coherent sheaves on abelian 3folds
there has recently been big progress here. See Maciocia, Piyaratne "FourierMukai transforms and Bridgeland stability conditions on abelian threefolds" I and II and "The space of stability conditions for abelian threefolds, and some CalabiYau threefolds" by Beyer, Macri and Stellari. (I am not an expert and so could have missed an important reference, but you certainly should be aware of these papers.) 
Nov 7 
comment 
Unitary dual of $Sp_4(\mathbb{R})$
Did you try google? Vogan, "The unitary dual of G2", Invent. math. (1994). 
Nov 5 
comment 
Why is Klein's representation of $PSL_2(\mathbb{F}_7)$ hard to obtain?
Perhaps section 11.4.4 in "Representations of $SL_2(\mathbb{F}_q)$" by Cédric Bonnafé is interesting to you. He identifies $PSL_2(\mathbb{F}_7) \times \mathbb{Z}/2\mathbb{Z}$ as an exceptional complex reflection group of rank 3 ($G_{24}$ on the ShephardTodd list). 
Oct 21 
comment 
Various definitions of the Bruhat decomposition of the affine Grassmannian
Yes I would guess these orbits are the same (not only topologically equivalent). As each $t^\lambda$ is a $T$fixed point the $J$ and $I$ orbit through it are the same. The finite dimensional version of your question is whether one considers $N^+$ orbits or $B$orbits. This seems to be a matter of taste (as long as you are not worrying about equivariant cohomology / sheaves, where the difference can matter.) 
Oct 21 
awarded  Yearling 
Sep 13 
awarded  Nice Answer 
Sep 12 
comment 
Reference request: BeilinsonBernstein for finitedimensional reps and category O
Having associated variety the zero section is the same thing as being a vector bundle with flat connection. Now $\pi_1(G/B)$ is trivial, and so there are only trivial bundles whose global sections give direct sums of the trivial rep. To get all reps (and recover BorelWeil) one instead should consider twisted $D$modules. One possible ref for all of this is Gaitsgory's notes on geometric rep theory (on his webpage I think). 