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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 non-trivial summand gives a counter-example 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 t-exact with respect to the perverse t-structure
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 3-folds
there has recently been big progress here. See Maciocia, Piyaratne "Fourier-Mukai transforms and Bridgeland stability conditions on abelian threefolds" I and II and "The space of stability conditions for abelian threefolds, and some Calabi-Yau 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 Shephard-Todd 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: Beilinson-Bernstein for finite-dimensional 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 Borel-Weil) one instead should consider twisted $D$-modules. One possible ref for all of this is Gaitsgory's notes on geometric rep theory (on his web-page I think).