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Let $\mathcal{L}=(L,[\cdot\,,\cdot],\delta)$ be an algebraic Lie algebroidover a smooth projective curve $X$ of genus $g\geq 2$ such that $L$ is a linebundle whose degree is less than $2-2g$. Let $r$ and $d$ be coprime numbers. Weprove that the motivic class of the moduli space of $\mathcal{L}$-connectionsof rank $r$ and degree $d$ over $X$ does not depend on the Lie algebroidstructure $[\cdot\,,\cdot]$ and $\delta$ of $\mathcal{L}$ and neither on theline bundle $L$ itself, but only on the degree of $L$ (and of course on $r$,$d$ and $X$). In particular it is equal to the motivic class of the modulispace of $K_X(D)$-twisted Higgs bundles of rank $r$ and degree $d$, for $D$ anyeffective divisor with the appropriate degree. As a consequence, similarresults (actually slightly stronger) are obtained for the corresponding$E$-polynomials. Some applications of these results are then deduced.

Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces