to ${\mathcal G}_s$.

\[Ealt\] If $char(F)=0$ then ${\mathcal E}$ admits an equivalent description as the space of invariant distributions $\alpha$ satisfying the following requirements:

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

b\) there exists a compact open subgroup $K\subset G$ such that for every element $z$ in the Bernstein center satisfying $\delta_{K}*z=0$ we have $\alpha*z=0$.

Equivalence of the two definitions of ${\mathcal E}$ follows from[^2] [@HC Theorem 16.2].

-   We define the space $ {\mathcal K}(G)$ of [*weightless*]{} functions as the subspace in ${\mathcal S}(G)$ of functions $f$ such that $\int _{u\in U_Q} f(lu)du=0, l\in L$ for all proper parabolic subgoups $Q=LU_Q\subset G$.

-   For a closed conjugation invariant subset $X$ of $G$ we define the space $ {\mathcal K}(X)={\mathcal K}_X\subset {\mathcal S}(X)$ as the subspace of functions $f$ such that $$\int _{u\in U_Q} f(lu)du=0,$$ for all proper parabolic subgroups $Q=LU_Q\subset G$ and $l\in L$ such that $lU_Q\subset X$.

\[diag\_rem\] For $f\in {{\mathcal H}}$ and a parabolic $P=LU_P\subset G$ let $A_P(f)\in {\mathcal S}(Y_P)$ denote its orishperic transform, i.e. the integral kernel of the action of $f$ on ${\mathcal S}(G/U_P)$. Let $\Delta_{Y_P}\subset Y_P$ be the preimage of diagonal under the projection $Y_P\to (G/P)^2$. Then 