Ideals generated by traces in the symplectic reflection algebra $H_{1,\nu_1, \nu_2}(I_2(2m))$. II

Abstract

The associative algebra of symplectic reflections $\mathcal H:= H_{1,\nu_1, \nu_2}(I_2(2m))$ based on the group generated by the root system $I_2(2m)$ has two parameters, $\nu_1$ and $\nu_2$. For every value of these parameters, the algebra $\mathcal H$ has an $m$-dimensional space of traces. A given trace ${\rm tr}$ is called degenerate if the associated bilinear form $B_{\rm tr}(x,y)={\rm tr}(xy)$ is degenerate. Previously, there were found all values of $\nu_1$ and $\nu_2$ for which there are degenerate traces in the space of traces, and consequently the algebra $\mathcal H$ has a two-sided ideal. We proved earlier that any linear combination of degenerate traces is a degenerate trace. It turns out that for certain values of parameters $\nu_1$ and $\nu_2$, degenerate traces span a 2-dimensional space. We prove that non-zero traces in this $2d$ space generate three proper ideals of $\mathcal H$.