  We study a set $\mathcal{M}_{K,N}$ parameterizing filtered $SL(K)$-Higgs
bundles over $\mathbb{CP}^1$ with an irregular singularity at $z = \infty$,
such that the eigenvalues of the Higgs field grow like $\lvert \lambda \rvert
\sim \lvert z ^{N/K} \mathrm{d} z \rvert$, where $K$ and $N$ are coprime.
$\mathcal{M}_{K,N}$ carries a $\mathbb{C}^\times$-action analogous to the
famous $\mathbb{C}^\times$-action introduced by Hitchin on the moduli spaces of
Higgs bundles over compact curves. The construction of this
$\mathbb{C}^\times$-action on $\mathcal{M}_{K,N}$ involves the rotation
automorphism of the base $\mathbb{CP}^1$. We classify the fixed points of this
$\mathbb{C}^\times$-action, and exhibit a curious $1$-$1$ correspondence
between these fixed points and certain representations of the vertex algebra
$\mathcal{W}_K$; in particular we have the relation $\mu = \frac{1}{12} \left(K
- 1 - c_{\mathrm{eff}} \right)$, where $\mu$ is a regulated version of the
$L^2$ norm of the Higgs field, and $c_{\mathrm{eff}}$ is the effective Virasoro
central charge of the corresponding $W$-algebra representation. We also discuss
a Bialynicki-Birula-type stratification of $\mathcal{M}_{K,N}$, where the
strata are labeled by isomorphism classes of the underlying filtered vector
bundles.
