\[distribution0\]

For any $ \bar \psi \in  {\mathcal G}_{Z_G(s),s}$ we choose a representative ${\tilde}\psi \in {\mathcal D}^{Z_G(s)}(Z_G(s))$ of $\bar \psi$.

-   For a function $f\in {\mathcal S}(G)$ and $z\in R\subset X_s$ we define $f_z \in {\mathcal S}(Z_G(s))$ by $f_ z:=f({\kappa}(z,x))$.

-   We define a function $\bar f$ on $R$ by $\bar f(z):={\tilde}\psi (f_z)$.

-   We define a distribution $\psi ({\tilde}\psi) $ on $G$ by $\psi (f):=\int _{X_s}\bar f(z)$.

-   We denote by $[\psi ({\tilde}\psi)]$ the germ of the distribution $\psi$ at $s$.

It is clear that for any two choices ${\tilde}\psi , {\tilde}\psi '$ of representatives of $\bar \psi$ the difference $\psi ({\tilde}\psi) -\psi ({\tilde}\psi ')$ vanishes on a $G$-invariant open neighborhood of $s.$ Therefore the germ $[\bar \psi]$ does not depend on a choice of a representative of $\bar \psi$.

\[distributions’\]

-   We denote by ${\mathcal G}_s$ the space of germs at $s$ of $Ad$-invariant distributions of the form $[\psi ], \psi \in {\mathcal G}_{Z_G(s),s}$.

-   We denote by ${\mathcal E}\subset  {\mathcal D}^G$ the subspace of distributions $\alpha$ such that

    a\) there exists a compact subset $C$ in $G$ such that $supp
     (\alpha)\subset G(C)$ and

    b\) for any semisimple $s\in G $ the germ of $\alpha$ at $s$ belongs 