by unipotent orbits.

Recall that ${\mathcal E}\subset  {\mathcal D}$ is the subspace of distributions $\alpha$ satisfying the following three conditions:

a\) There exists a compact subset $C$ in $G$ such that $supp (\alpha)\subset C^G$.

b\) The restriction of $\alpha$ on $G-\{ e\}$ is given by a locally constant function.

c\) The germ of $\alpha$ at $e$ belongs to ${\mathcal D}_e$.

$\tau ({\mathcal K})\subset {\mathcal E}$.

Fix $f\in {\mathcal K}$. It is clear that the distribution $\hat f$ satisfies condition $a)$.

To prove that $\hat f$ satisfies condition $b)$ we have to show that for any semisimple element $s\in G-\{ e\}$ there exists an open neighborhood $R\subset G$ of $s$ such that the restriction $\hat f|_{R}$ is a constant. If $s$ is split then this follows from Corollary \[split\](a), if $s$ is elliptic the proof is similar.

To prove that $\hat f$ satisfies condition $c)$ we observe that Corollary \[cusp\] implies existence of $f_{cusp}\in {\mathcal K}_{cusp}$ such that $[{\kappa}(f)]=[{\kappa}(f_{cusp})]$.

It is easy to see that when $f$ is a matrix coefficient of an irreducible cuspidal representation $\rho$ then $\hat f$ is proportional to the character of $\rho$. Thus condition $c)$ is satisfied by $\alpha_{cusp}=\hat{f_{cusp}}$. However, by Lemma \[extension\] and Corollary \[extensions\] 