{\mathcal K}_{\Omega}))=1$.

Recall that $G_s\subset G$ is the subset of regular split semisimple elements, let ${\mathcal K}_s\subset {\mathcal K}$ be the subspace of functions in ${\mathcal K}$ supported on $G_s$. We fix a Cartan subgroup $T$, the Weyl group $W= {{\mathbb Z}}/2{{\mathbb Z}}$ acts on $T$ and on $G/T$ in the usual way. Then the map $$(T-\{ e\}) \times G/T \to G_s,
 (s, g)\to gsg^{-1}$$ induces an isomorphism $$\label{Phit}
{\mathcal S}(G_s)\to ({\mathcal S}(T-\{e\})\otimes {\mathcal S}(G/T))^W.$$

\[split\] a) For $f\in {\mathcal K}_s$ the distribution $\hat f$ is locally constant on $G_s$.

b\) The map $f\mapsto (t\mapsto \hat f(t))$ induces an isomorphism $$H_0(G,{\mathcal K}_s)\to   {\mathcal S}(T-\{e\})^W.$$

The isomorphism is clearly compatible with the averaging map $f\mapsto \hat f$, which implies $a)$. Likewise, restriction to an orbit is compatible with averaging, thus in view of it suffices to show that for a fixed orbit ${\Omega}\subset G_s$ the map $f\mapsto \hat f (t)$, $t\in {\Omega}$ induces an isomorphism $H_0(G,{\mathcal K}({\Omega}))\to {{\mathbb C}}$. By Proposition \[regular\] it suffices to see that this map is nonzero. This follows, for example, from the fact that the character of a cuspidal representation does not necessarily vanish on $G_s$, while the character of an irreducible cuspidal representation is obtained 