The number of independent Traces and Supertraces on the Symplectic Reflection Algebra $H_{1,\nu}(\Gamma \wr S_N)$

Abstract

Symplectic reflection algebra $ H_{1, \,\nu}(G)$ has a $T(G)$-dimensional space of traces whereas, when considered as a superalgebra with a natural parity, it has an $S(G)$-dimensional space of supertraces. The values of $T(G)$ and $S(G)$ depend on the symplectic reflection group $G$ and do not depend on the parameter $\nu$. In this paper, the values $T(G)$ and $S(G)$ are explicitly calculated for the groups $G= \Gamma \wr S_N$, where $\Gamma$ is a finite subgroup of $Sp(2,\mathbb C)$.