$n=2$ and $n=3$, the strong finite generation of $\cH(2)^G$ and $\cH(3)^G$ for an arbitrary reductive $G$ is an immediate consequence.

There is an application of these results to invariant subalgebras of affine vertex algebras which we develop in a separate paper [@LIII]. Let $\gg$ be a simple, finite-dimensional Lie algebra, and let $V_k(\gg)$ denote the universal affine vertex algebra at level $k$ associated to $\gg$. It is freely generated by vertex operators $X^{\xi}$, which are linear in $\xi\in\gg$ and satisfy the OPE relations $$X^{\xi}(z) X^{\eta}(w) \sim k\bra \xi,\eta\ket (z-w)^{-2} + X^{[\xi,\eta]}(w)(z-w)^{-1},$$ where $\bra ,\ket$ denotes the normalized Killing form $\frac{1}{2 h^{\vee}} \bra ,\ket_K$. Let $G$ be a reductive group of automorphisms of $V_k(\gg)$ for all $k\in \mathbb{C}$. In particular, $G$ acts on the weight-one subspace $V_k(\gg)[1]\cong \gg$, and $G$ preserves both the bracket and the bilinear form on $\gg$. Therefore $G$ lies in the orthogonal group $O(n)$ for $n= \text{dim}(\gg)$, so $G$ also acts on the Heisenberg algebra $\cH(n)$. As vector spaces, we have $V_k(\gg)^G \cong (Sym \oplus_{j\geq 0} V_j)^G\cong \cH(n)^G$, where $V_j\cong \mathbb{C}^n$ for all $j\geq 0$, and we regard $\cH(n)^G$ as a partial abelianization" of $V_k(\gg)^G$. Invariant subalgebras of $V_k(\gg)$ are much more complicated and difficult to study 