linear spaces, and as commutative algebras we have $$\label{structureofgrs} gr(\cH(n))\cong Sym \bigoplus_{k\geq 0} V_k.$$ In this notation, $V_k$ is the linear span of $\{\alpha^{i}_k |~ i=1,\dots,n\}$, where $\alpha^i_k$ is the image of $\partial^k \alpha^i(z)$ in $gr(\cH(n))$ under the projection $\phi_1: \cH(n)_{(1)}\ra \cH(n)_{(1)}/\cH(n)_{(0)}\subset gr(\cH(n))$. The action of $O(n)$ on $\cH(n)$ preserves this filtration, and induces an action of $O(n)$ on $gr(\cH(n))$ by algebra automorphisms. For all $k\geq 0$ we have isomorphisms of $O(n)$-modules $V_k\cong \mathbb{C}^n$. Finally, for any reductive subgroup $G\subset O(n)$, $\cH(n)^G\cong gr(\cH(n)^G)$ as linear spaces, and $$\label{structureofgrsinv} gr(\cH(n)^G )\cong (gr(\cH(n))^G \cong (Sym \bigoplus_{k\geq 0} V_k)^G$$ as commutative algebras. In the case $G=O(n)$, the following classical theorem of Weyl [@W] describes the generators and relations of the ring $(Sym \bigoplus_{k\geq 0} V_k)^{O(n)}$:

\[weylfft\] (Weyl) For $k\geq 0$, let $V_k$ be the copy of the standard $O(n)$-module $\mathbb{C}^n$ with orthonormal basis $\{x_{i,k}| ~i=1,\dots,n\}$. The invariant ring $(Sym \bigoplus_{k\geq 0} V_k )^{O(n)}$ is generated by the quadratics $$\label{weylgenerators} q_{a,b} = \sum_{i=1}^n x_{i,a} x_{i,b},\ \ \ \ \ \  0\leq a\leq b.$$ For $a>b$, define $q_{a,b} = q_{b,a}$, and let $\{Q_{a,b}|\ a,b\geq 0\}$ be commuting indeterminates satisfying $Q_{a,b} = Q_{b,a}$ and no other algebraic relations. The kernel $I_n$ of the homomorphism $\mathbb{C}[Q_{a,b}]\ra (Sym 