# What are the “special” strata of Sym^n(C^2)?

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

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".

### Edit

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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