Proposition 22

If $g \in G$, the transform of $D_t$ under g is $D_{\varepsilon_g * t * \varepsilon_g^{-1}}$.

The value of this transform at $gx$ is

Let $(x, g) \mapsto \mu(x, g) = xg$ be a law of right operation of class $C'$ of G on X. Let $s \leq r$ and $t \in U_s(G)$. For all $x \in X$, $\varepsilon_x * t \in T^{(s)}_x(X)$. The mapping $x \mapsto \varepsilon_x * t$ is called the field of distributions defined by t and the action of G on X and is sometimes denoted by $D_t^u$ or simply $D_t$. Let $\Omega$ be an open subset of X. If $f : \Omega \to F$ is of class $C'$, the function $f * t^\vee$ is denoted by $D_t f$. Then

and, in the obvious notation,

Proposition 21 remains valid. Let $g \in G$. The transform of $D_t$ under g (that is under the automorphism $x \mapsto xg$ of X) is $D_{\varepsilon_g^{-1} * t * \varepsilon_g}$.