I came across this problem from an old problem set on topics in Lie theory but I don't really know how to approach
Let G be a compact connected Lie group and let $\nu$ be the normalized Haar measure on G ($\nu$ is unique up to multiplication by positive constants), show that, for each element $g\in G$, there exists a homotopy between the identity map and the operator $\omega \mapsto \int_GR^*_{g}(\omega)d\nu $ acting on left-invariant differential forms.
I first thought of the right action induced by the elements in the codomain of the exponential map from $Lie(G)$ to $G$, but I am not quite sure how can this map induce an endomorphism on differential forms that is homotopic to the identity? (in other words, I want to somehow come up with a linear map T such that $R_g + \{d,T\} = Id$ , where $R_g$ denotes right-translation, $\{,\}$ is the anti-commutator operation). Even then, I don't know how to proceed.
Any help is tremendously appreciated :)