Fourier-Mukai transformation and logarithmic Higgs bundles on punctual Hilbert schemes

Given a vector bundle $E$ on a smooth projective curve or surface $X$ carrying the structure of a $V$-twisted Hitchin pair for some vector bundle $V$, we observe that the associated tautological bundle $E^{[n]}$ on the punctual Hilbert scheme of points $X^{[n]}$ has an induced structure of a $((V^{\vee }{)}^{[n]}{)}^{\vee }$-twisted Hitchin pair, where $(V^{\vee }{)}^{[n]}$ is a vector bundle on $X^{[n]}$ constructed using the dual $V^{\vee }$ of $V$. In particular, a Higgs bundle on $X$ induces a logarithmic Higgs bundle on the Hilbert scheme $X^{[n]}$. We then show that the known results on stability of tautological bundles and reconstruction from tautological bundles generalize to tautological Hitchin pairs.