K-theory of affine actions

For a Lie group $G$ and a vector bundle $E$ we study those actions of the Lie group $TG$ on $E$ for which the action map $TG\times E\to E$ is a morphism of vector bundles, and call those \emph{affine actions}. We prove that the category ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{TG}^{\mathrm{a}\mathrm{f}\mathrm{f}}\left(X\right)$ of such actions over a fixed $G$-manifold $X$ is equivalent to a certain slice category $\mathfrak g_X \backslash $\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}$_G\left(X\right)$. We show that there is a monadic adjunction relating ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{TG}^{\mathrm{a}\mathrm{f}\mathrm{f}}\left(X\right)$ to ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_G\left(X\right)$, and the right adjoint of this adjunction induces an isomorphism of Grothendieck groups $K_{TG}^{\mathrm{a}\mathrm{f}\mathrm{f}}\left(X\right)\cong K{O}_G\left(X\right)$. Complexification produces analogous results involving $T_{\mathbb{C}}G$ and $K_G\left(X\right)$.