$\Delta_{Y_P}\cong (G/U_P\times L)/L$, where $L$ acts on the first factor by right translations and on the second one by conjugation. It is easy to see that for $f\in {{\mathcal H}}$ we have $f\in {{\mathcal K}}$ iff for any parabolic subgroup $P\subsetneq G$ we have $A_P(f)|_{\Delta_{Y_P}}=0$.

The following result is due to J.-L. Waldspurger (see [@W Lemma 9]).

\[Waldspurger\] For $f\in {{\mathcal H}}(G)$ the following are equivalent:

a\) $f\in {{\mathcal K}}$.

b\) For any $h\in {{\mathcal H}}(G)$ the function $g\mapsto \langle ^gf, h\rangle$ has compact support.

For any $f\in {\mathcal S}(G)$ we define distribution $\hat f $ by: $$\langle \hat f ,h\rangle :=\int _{g\in  G}\langle ^gf, h  \rangle dg.$$

For future reference we mention the following.

For $f\in {{\mathcal K}}$ the distribution $\hat{f}|_{G^{rs}}$ (where $G^{rs}$ is the open set of regular semisimple elements) is a locally constant function. For $g\in G^{rs}$ we have $$\hat f (g) =WO_g(f),$$ where $WO_g$ denotes the [*weighted orbital integral*]{}.

[*Proof *]{} follows from the definition and basic properties of the weighted orbital integral, see e.g. [@ArIn §I.11].

The group $G$ acts on ${\mathcal K}$ by conjugation. It is clear that the map $f\to \hat f$ factors through a map $\tau : H_0(G,{\mathcal K})\to  {\mathcal D}^G$. For 