Homology of Hilbert schemes of reducible locally planar curves

Let $C$ be a complex, reduced, locally planar curve. We extend the results of Rennemo arXiv:1308.4104 to reducible curves by constructing an algebra $A$ acting on $V=\bigoplus_{n\geq 0} H_*(C^{[n]}, \mathbb{Q})$, where $C^{[n]}$ is the Hilbert scheme of $n$ points on $C$. If $m$ is the number of irreducible components of $C$, we realize $A$ as a subalgebra of the Weyl algebra of $\mathbb{A}^{2m}$. We also compute the representation $V$ in the simplest reducible example of a node.