Connection between the ideals generated by traces and by supertraces in the superalgebras of observables of Calogero models

If $G$ is a finite Coxeter group, then symplectic reflection algebra $H:=H_{1,eta}(G)$ has Lie algebra $mathfrak {sl}_2$ of inner derivations and can be decomposed under spin: $H=H_0 oplus H_{1/2} oplus H_{1} oplus H_{3/2} oplus ...$. We show that if the ideals $mathcal I_i$ ($i=1,2$) of all the vectors from the kernel of degenerate bilinear forms $B_i(x,y):=sp_i(xcdot y)$, where $sp_i$ are (super)traces on $H$, do exist, then $mathcal I_1=mathcal I_2$ if and only if $mathcal I_1 bigcap H_0=mathcal I_2 bigcap H_0$.