Proposition 23

(i) The mapping $t \mapsto L_t$ (resp. $t \mapsto R_t$) is an isomorphism of the vector space $U(G)$ onto the vector field of left (resp. right) invariant distributions on $G$. (ii) For $t, t'$ in $U(G)$, $L_{t*t'} = L_t \circ L_{t'}$, $R_{t*t'} = R_{t'} \circ R_t$, $L_t \circ R_{t'} = R_{t'} \circ L_t$ (with the abuse of notation of no. 5). (iii) If $\theta$ is the mapping $g \mapsto g^{-1}$ of $G$ onto $G$, then $\theta(L_t) = R_{t^\vee}$. (iv) If $t \in U(G)$ and $g \in G$, then $(L_t)_g = (R_{\varepsilon_g * t * \varepsilon_g^{-1}})_g$.

In $G$ every right translation commutes with every left translation. By Proposition 21 of no. 5, $L_t$ is therefore left invariant. As $(L_t)_e = t$, the mapping $t \mapsto L_t$ is injective. Let $\Delta$ be a field of left invariant distributions on $G$; let $t = \Delta_e$; then $\Delta$ and $L_t$ have the same value at $e$ and are left invariant and hence $\Delta = L_t$. This proves (i) for $L_t$ and the argument is similar for $R_t$. The formulae $L_{t*t'} = L_t \circ L_{t'}$, $R_{t*t'} = R_{t'} \circ R_t$ follow from (21) and (18). Let $t \in U_s(G)$, $t' \in U_{s'}(G)$, $f \in C^r(\Omega, F)$, where $\Omega$ is open in $G$ and $s + s' \leq r$; then

and hence $L_t \circ R_{t'} = R_{t'} \circ L_t$. As $\theta$ is an isomorphism of $G$ onto $G^\vee$, $\theta(L_t)$ is a field of right invariant distributions on $G$; its value at $e$ is $\theta(t) = t^\vee$; hence $\theta(L_t) = R_{t^\vee}$. Finally,