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The affine variety $Sym^n(\mathbb{C}^2)$ has a natural quantization as a spherical rational Cherednik algebra. Thus, any primitive ideal of the rational Cherednik algebra has an corresponding ideal in $\mathbb{C}[X_1,Y_1..,X_n,Y_n]^{S_n}$. By a theorem of Ginzburg, the zero-set of this ideal is the closure of the elements where the partition given by looking at which points coincide is a particular fixed one.

My question: Which paritions can come up this way?

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I'm slightly confused about the notation---is the ring $C[X_1,...,X_n]^{S_n}$ above supposed to have 2 sets of variables in it, or are you interested in what varieties can appear as supports in the ``base'' also? Bezrukavnikov-Etingof's recent paper answers the latter question (for each given parameter c of the RCA), but I'm not sure right now how far that goes to answering your question about subvarieties of $Sym^n(C^2)$. – GS Jan 6 '10 at 15:20
up vote 3 down vote accepted

Just by coincidence, I was catching up on my arxiv reading and noticed that Losev's paper appears to answer your question. See part (3) of Theorem 4.3.1 there. He says it was "known previously".


To be kinder to the reader: the Cherednik algebra is really a family of algebras depending on a parameter $c$; it looks like Losev's result is that at $c=k/m$ for relatively prime integers $k$ and $m$ the varieties we want correspond to partitions of $n$ which are of the form $(m,m,\dots,m,1,1,\dots,1)$ for some number of $m$'s and some number of $1$'s.

2nd Edit

Having looked more carefully at Losev's paper, it seems the idea of the proof is this: in the spirit of Bezrukavnikov-Etingof, having fixed a partition $\lambda$ Losev gives (his Theorem 1.3.1) an inductive description of primitive ideals with associated variety corresponding to $\lambda$. The partition $\lambda$ corresponds to a "parabolic" subgroup $W$ of $S_n$ consisting of those permutations fixing a generic point of the corresponding variety, and the set of primitive ideals corresponding to $\lambda$ is in bijection with the set of primitive ideals of finite codimension in the Cherednik algebra attached to $W$ at the same parameter $c=k/m$. But $W$ is a product of symmetric groups, so the classification due to Berest-Etingof-Ginzburg of finite dimensional modules for the type $S_n$ Cherednik algebra (they exist exactly when $c$ has denominator $n$) implies the result.

It appears that ideas from Losev's work on finite $W$-algebras also work for symplectic reflection algebras. It would therefore be nice to understand the finite dimensional modules outside of the symmetric group case a little better. (And now I'm more motivated to learn about $W$-algebras!)

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