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:=H1,η(G) has Lie algebra sl₂ of inner derivations and can be decomposed under spin: H=H₀ ⊕ H1/2 ⊕ H₁ ⊕ H3/2 ⊕ .... We show that if the ideals Iᵢ ($i=1,2$) of all the vectors from the kernel of degenerate bilinear forms Bᵢ(x,y):=spᵢ(x· y), where spᵢ are (super)traces on H, do exist, then I₁= I₂ if and only if I₁ H₀= I₂ H₀.