by averaging from its matrix coefficient.

Now we can finish the proof of Proposition \[inj\]. Let $f\in {\mathcal K}$ be such that $\hat f=0$. It follows from Corollary \[extensions\] and Lemma \[extension\] that we can can find $f'$ with the same image in $H_0(G,{\mathcal K})$ such that $f'=f_s+f_e$ where $f_s$ is supported on regular split semisiple elements and $f_e$ on regular elliptic elements. The condition $\hat f=0$ implies that $\hat{f'}=0$, hence $\hat f_s=0$ and $\hat f_e=0$. It is easy to see that the condition $\hat f_e=0$ implies that $[f_e]=0$. So we may assume that the support of $f$ is contained in the subset $G_s\subset G$ of regular split semisimple elements. Now Proposition \[inj\] follows from Corollary \[split\].

Let ${\tilde}{\mathcal D}_e$ be the space of germs of distributions at $e$ and $ {\mathcal D}_e\subset {\tilde}{\mathcal D}_e$ be the subspaces spanned by germs of characters of irreducible representations.

\[germs\] The space ${\mathcal D}_e$ is $2$-dimensional. It is spanned by germs of characters of irreducible cuspidal representations.

The second statement is a special case of a theorem of Harish-Chandra [@HC]. The first one also follows from [*loc. cit.*]{}, as it shown there that more generally the space ${\mathcal D}_e$ has a basis indexed 