suppose we have a map $f:\mathbb{Z}\longrightarrow\mathbb{C}[t^{\pm}]$ with property that $f(i)=-f(-i)$.

The algebra $\mathcal{T}=\mathbb{C}[t^{\pm}][L^{\pm},M^{\pm}]/[LM=tML]$ acts on $f$ by $(Lf)(i)=f(i+1),(Mf)(i)=t^if(i)$.

Should the annihilating ideal of $f$ generated by annihilators in symmetric part of $\mathcal{T}$?

Symmetric part means the subset with all element like $aL^xM^y+aL^{-x}M^{-y}$ and their multiplication or linear combination.

Thanks.