than invariant subalgebras of $\cH(n)$, but for generic values of $k$, a strong generating set for $\cH(n)^G$ gives rise to a strong generating set for $V_k(\gg)^G$. Therefore the conjectured strong finite generation of $\cH(n)^{O(n)}$ has a far-reaching consequence; it implies that $V_k(\gg)^G$ is strongly finitely generated for generic values of $k$. Finally, since our conjecture holds for $n=3$, this statement is true in the case $\gg = \gs\gl_2$.

Vertex algebras
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In this section, we define vertex algebras, which have been discussed from various different points of view in the literature [@B][@FBZ][@FHL][@FLM][@K][@LiI][@LZ]. We will follow the formalism developed in [@LZ] and partly in [@LiI]. Let $V=V_0\oplus V_1$ be a super vector space over $\mathbb{C}$, and let $z,w$ be formal variables. By $QO(V)$, we mean the space of all linear maps $$V\ra V((z)):=\{\sum_{n\in\mathbb{Z}} v(n) z^{-n-1}|
v(n)\in V,\ v(n)=0\ \text{for} \ n>>0 \}.$$ Each element $a\in QO(V)$ can be uniquely represented as a power series $$a=a(z):=\sum_{n\in\mathbb{Z}}a(n)z^{-n-1}\in End(V)[[z,z^{-1}]].$$ We refer to $a(n)$ as the $n$th Fourier mode of $a(z)$. Each $a\in
QO(V)$ is of the shape $a=a_0+a_1$ where $a_i:V_j\ra V_{i+j}((z))$ for $i,j\in\mathbb{Z}/2\mathbb{Z}$, and we write $|a_i| = i$.

On $QO(V)$ there is a set of nonassociative bilinear operations $\circ_n$, indexed by $n\in\mathbb{Z}$, which 