Etingof's finite-dimensional representation-count conjecture

Let $H_{c,k}(\Gamma_n)$ be the symplectic reflection algebra, and let $\mathfrak a$ and $P_+^{\mathfrak a}$ be as above. For each $\mu\in P_+^{\mathfrak a}$, let $L_\mu[\omega_0-n\delta]$ denote the indicated weight space of the corresponding irreducible integrable $\mathfrak a$-module.

Etingof's representation-count conjecture. The number of isomorphism classes of finite-dimensional irreducible representations of $H_{c,k}(\Gamma_n)$ is

$$\sum_{\mu\in P_+^{\mathfrak a}:\,\mu^2=0}\dim L_\mu[\omega_0-n\delta].$$

This is the $i=0$ consequence of the main filtration conjecture and gives a representation-theoretic prediction for the finite-dimensional simples. It is verified in the examples discussed in the paper but is open in general.