$V = U(\gh)\otimes_{U(\gh_{\geq 0})} C$, and let $\alpha(n)\in End(V)$ be the linear operator representing $t^n$ on $V$. Define $\alpha(z) = \sum_{n\in\mathbb{Z}} \alpha(n) z^{-n-1}$, which is easily seen to lie in $QO(V)$ and satisfy the OPE relation $$\alpha(z)\alpha(w)\sim (z-w)^{-2}.$$ The vertex algebra $\cH$ generated by $\alpha$ is known as the [*Heisenberg vertex algebra*]{}. The rank $n$ Heisenberg algebra $\cH(n)$ is just the tensor product of $n$ copies of $\cH$, with generators $\alpha^1,\dots,\alpha^n$. There is a natural conformal structure of central charge $n$ on $\cH(n)$, with Virasoro element $$\label{virasoro} L(z) = \frac{1}{2} \sum_{i=1}^n :\alpha^i(z) \alpha^i(z):,$$ under which each $\alpha^i$ is primary of weight one. The full automorphism group of $\cH(n)$ preserving $L(z)$ is easily seen to be the orthogonal group $O(n)$, which acts linearly on the vector space $U$ spanned by $\{\alpha^1,\dots,\alpha^n\}$. First, any conformal automorphism $\phi$ of $\cH(n)$ must lie in $GL(U)$ by weight considerations. Moreover, since $\alpha^i \circ_1 \alpha^j = \phi(\alpha^i)\circ_1 \phi(\alpha^j) = \delta_{i,j}$, $\phi$ must preserve the pairing $\bra,\ket$ on $U$ defined by $\bra \alpha^i,\alpha^j\ket = \delta_{i,j}$.

We define a good increasing filtration on $\cH(n)$ as follows: $\cH(n)_{(r)}$ is spanned by the set $$\label{goodsv} \{:\partial^{k_1} \alpha^{i_1}  \cdots \partial^{k_s}\alpha^{i_s} :| ~ k_j\geq 0,~ s \leq r\}.$$ Then $\cH(n)\cong gr(\cH(n))$ as 