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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).
Aug
15
comment Embed one Coxeter System into another
@Jim: I would be very interested in a reference earlier than the Lusztig ref (as would Lusztig I guess). I think it is clear that Lusztig is aware of the general pattern that such embeddings follow. I agree the history is murky! Also +1 for "misleadingly named"!
Aug
15
comment Embed one Coxeter System into another
I don't think that this is what he means. The embedding of e.g. H_3 into D_6 doubles lengths, and so doesn't send simple reflections to reflections.
Aug
15
comment Embed one Coxeter System into another
If I remember correctly these embeddings are discussed (for the first time?) in Lusztig, "Some examples of square integrable representations of a p-adic group". See discussion around p. 636 and lovely diagram...
Jul
4
awarded  Revival
Jul
2
awarded  Curious
May
7
awarded  Nice Answer
Apr
15
comment Do mutually dual finite vector spaces have the same orbit cardinalities under a linear group action?
One can also prove that the number of orbits are the same using the Fourier transform. It gives a canonical isomorphism $Fun(V) \to Fun(V^*)$ and hence $Fun_G(V) \to Fun_G(V^*)$. The dimensions of both sides is the number of $G$-orbits.
Apr
12
awarded  Nice Answer
Apr
8
comment Smooth mixed hodge modules - representations of fundamental group?
It is important to specify whether one considers an integral, real or complex mixed Hodge structure. For example, the integral lattice in a variation of pure Hodge structures of weight 1 is the same as a family of smooth curves which can be interesting (even over a disc).
Mar
6
comment Schubert varieties which admit small resolutions of singularities
It is a result of Billey and Warrington that "321 hexagon avoiding" permutations in the symmetric group admit small resolutions. See: Billey and Warrington, Kazhdan-Lusztig Polynomials for 321-hexagon-avoiding permutations, J. of Algebraic Combinatorics, v. 13 (2001), no. 2, 111--136.
Mar
2
revised Divisibility of all entries in an intersection form
added 4 characters in body
Feb
28
awarded  Civic Duty
Jan
13
revised Intersection homology for toric varieties
added 159 characters in body
Dec
22
answered Intersection homology for toric varieties
Dec
17
comment Explicit examples presheaves associated to higher direct images which fail to be sheaves
Sorry, I misread what you were saying. I agree with (i) now.
Dec
17
comment Explicit examples presheaves associated to higher direct images which fail to be sheaves
For (i), $f$ is a fibration with fibre $S^1$ and so $f^1_*(\underline{\mathbb{Z}})$ is a locally constant sheaf with fibre $\mathbb{Z}$. As $S^2$ is simply connected, we have $f_*^1$ is the constant sheaf, so I don't agree that (i) is an example. $f^3_*$ would however give an example for (ii).
Oct
30
awarded  Necromancer
Oct
30
answered What to do now that Lusztig's and James' conjectures have been shown to be false?