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2d

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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). 
Aug 15 
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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 
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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 
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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 padic group". See discussion around p. 636 and lovely diagram... 
Jul 4 
awarded  Revival 
Jul 2 
awarded  Curious 
May 7 
awarded  Nice Answer 
Apr 15 
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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 
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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 
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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, KazhdanLusztig Polynomials for 321hexagonavoiding permutations, J. of Algebraic Combinatorics, v. 13 (2001), no. 2, 111136. 
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 
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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 
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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). 