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In Nakajima's Geometric construction of algebras(pages 3-7), he uses the subalgebra of the convolution algebra of $Gr(k^N)\times Gr(k^N)$ invariant under $GL_N$ action to construct $U_q(sl_2)$. To do this, he finds imposes relations to form some special operators, which he shows are isomorphic to the generators of $U_q(sl_2)$. In particular, he uses characteristic functions on certain subsets of $k^N$ with dimensionality conditions. These conditions smell like Schubert conditions.

Can we see these as the Schubert conditions in some way?

I have compared them, but I don't see the link.

EDIT: Read the comments below for a related paper.

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This might be spoken about in "a geometric setting for quantum groups" by Beilinson, Lusztig, and MacPhearson, but I can't seem to get a copy to check. –  B. Bischof Oct 29 '10 at 7:27
@Bischof This is a 1990 Duke J. paper: MR1074310 (91m:17012). Be˘ılinson, A. A. [Beilinson, Aleksandr A.] (1-MIT); Lusztig, G. (1-MIT); MacPherson, R. [MacPherson, RobertD.] (1-MIT). A geometric setting for the quantum deformation of GLn. Duke Math. J. 61 (1990), no. 2, 655–677. –  Jim Humphreys Oct 29 '10 at 13:31
This Duke J. paper may require the help of a library to access online, but one line from the introduction sums it up: "In this paper we construct the entire algebra (not only the + part) assuming that we are in type A, using the geometry of relative positions of pairs of flags in infinite dimensional space." This may not be close enough to Nakajima's formulation involving $\mathfrak{sl}_2$ to be helpful. –  Jim Humphreys Oct 29 '10 at 13:51
@Bischof I'm definitely not an expert on these things, but keep in mind that quantum groups (except at a root of unity) tend to have an infinite dimensional flavor in terms of representations and such. So anything like Schubert conditions here would likely have that flavor. –  Jim Humphreys Oct 29 '10 at 16:39
Nakajima is exactly talking about the BLM construction; there's no real significance to the fact that Nakajima uses finite dimensional things in his paper and BLM use infinite dimensional. You really just take the limit as N goes to infinity. If you don't take this limit, you only get the quotient of sl(2) that only acts faithfully on representations of highest weight $\leq N$, so you have to send $N$ to infinity to get all of the algebra. –  Ben Webster Oct 29 '10 at 17:11

1 Answer 1

up vote 1 down vote accepted

Well, it depends what you mean; there's certainly a close link.

Any $GL_N$ invariant closed subset of the product of two Grassmannians can be described as a family of Schubert varieties. Consider the projection to the first factor. This must be surjective, since $GL_n$ acts transitively on the Grassmannian. The fiber over a subspace $V\subset \mathbb{C}^N$ must be a closed subset of the other Grassmannian which is preserved by the subgroup stabilizing $V$; that is, it must be a Schubert variety of the form $\{U\subset \mathbb{C}^N | \dim(U)=m, \dim(U\cap V)\geq h\}$.

The same argument can easily be applied to any product of two partial flag varieties with the same conclusion (unfortunately, it completely breaks down for more factors).

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Great, thank you. In the higher dim case of flag varieties, can we see the Schubert cells in this way? –  B. Bischof Oct 29 '10 at 17:48
Sure. The point is just that for any $G$-space $X$, $G$-orbits on $G/H \times X$ are in canonical bijection with $H$-orbits on $X$, by taking the fiber over the identity coset. –  Ben Webster Oct 29 '10 at 19:51
Again, thank you very much! –  B. Bischof Oct 29 '10 at 20:55

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