bio | website | people.maths.ox.ac.uk/… |
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visits | member for | 4 years, 11 months |
seen | 10 hours ago | |
stats | profile views | 1,617 |
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? |