Let $V=\bigoplus\_{d\in\mathbb N}V(d)$ be a Möbius-covariant vertex algebra with $V(0)=\mathbb C$.  

Recall that a vector $v\in V$ is called *quasi-primary* if $L\_1v=0$.


For $v\in V(d)$, we write $Y(v,z)=\sum\_{n\in\mathbb Z} z^{-n-d}v\_{(n)}$.  

With that convention, $v\_{(n)}$ is an operator $V(k)\to V(k-n)$.



> 
> Let $d>n$.  
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>  Is it true that for any quasi-primary $v\in V(d)$ and any vector $w\in V(n)$, we have $v\_{(n)}w=0$?
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> 
> 


If the above relation does not always hold, are there reasonable extra assumptions that one can impose on $V$ that imply it?


Do the above relations hold when $v$ is required to be primary instead of quasi-primary?  
 (Add the assumption that $V$ is a VOA so that the notion of a primary vector make sense)



Annihilation operators in a vertex algebra
The answer seems to be yes for quasi-primary $v$ if $V$ has a suitable invariant bilinear form. Then one can identify $v\_{(n)} w$ with its pairing with the vacuum, and obtains it as the appropriate coefficient of
$$(\mathbf{1}, Y(v, x)w) = (-x^{-2})^d (Y(v, x^{-1})\mathbf{1}, w) = (-x^{-2})^d (e^{x^{-1} L(-1)} v, w) = 0$$
since the weight of $v$ is greater than that of $w$.


Of course, this argument does not work if $v$ is not quasi-primary since then $v$ must be replaced with $e^{x L(1)} v$ after the first equality in the calculation.


Note that it isn’t necessary to assume the bilinear form is nondegenerate, just nondegenerate on the one-dimensional vacuum space. 


