Generators of relations for annihilating fields

For an untwisted affine Kac-Moody Lie algebra $\tilde{\mathfrak g}$, and a given positive integer level $k$, vertex operators $x(z)=\sum x(n)z^{-n-1}$, $x\in\mathfrak g$, generate a vertex operator algebra $V$. For the maximal root $\theta$ and a root vector $x_\theta$ of the corresponding finite-dimensional $\mathfrak g$, the field $x_\theta(z)^{k+1}$ generates all annihilating fields of level $k$ standard $\tilde{\mathfrak g}$-modules. In this paper we study the kernel of the normal order product map $r(z)\otimes Y(v,z)\mapsto :r(z) Y(v,z):$ for $v\in V$ and $r(z)$ in the space of annihilating fields generated by the action of $\tfrac{d}{dz}$ and $\mathfrak g$ on $x_\theta(z)^{k+1}$. We call the elements of this kernel the relations for annihilating fields, and the main result is that this kernel is generated, in certain sense, by the relation $x_\theta(z)\tfrac{d}{dz}(x_\theta(z)^{k+1})= (k+1)x_\theta(z)^{k+1}\tfrac{d}{dz}x_\theta(z)$. This study is motivated by Lepowsky-Wilson's approach to combinatorial Rogers-Ramanujan type identities, and many ideas used here stem from a joint work with Arne Meurman.