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?



What are the "special" strata of Sym^n(C^2)?
Just by coincidence, I was catching up on my arxiv reading and noticed that Losev's [paper](http://arxiv.org/abs/1001.0239) 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](http://arxiv.org/abs/0803.3639), 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](http://arxiv.org/abs/math/0208138) 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!)


