Etingof's aspherical-locus conjecture

Let $H_{c,k}(\Gamma_n)$ be the symplectic reflection algebra with parameter $(\lambda,k)$, and let $\mathcal A(\Gamma_n)$ denote the aspherical locus: the set of parameters for which there is a nonzero module annihilated by the averaging idempotent. With

$$E_{m,N}: km+N=0$$

and

$$H_{\alpha,m,N}: (\lambda,\alpha)+km+N=0,$$

the parameters satisfy $1\le N\le m-1$ for the first family, and $|m|\le n-1$ and

$$0\le N\le \sqrt{n+\frac{m^2}{4}}+\frac{m}{2}-1$$

for the second family.

Etingof's aspherical-locus conjecture. The aspherical locus is the union of the hyperplanes $E_{m,N}$ for $1\le N\le m-1$ and the hyperplanes $H_{\alpha,m,N}$ satisfying the stated bounds.

This predicts the exact parameter locus where the spherical idempotent generates a proper ideal. The paper presents it as an open refinement of the description of singular parameters.