of identity are not proportional. This is easily done by inspecting the character tables, see e.g. [@Sil §2.6].

\[cusp\]

a\) For any $f\in {\mathcal K}$ there exists $f_{cusp} \in {\mathcal K}_{cusp}$ such that $[{\kappa}(f)]=[{\kappa}(f_{cusp'})]$.

b\) For any $f_0\in {\mathcal K}_u$ there exists $f\in {\mathcal K}_{cusp}$ such that $[f_0]=[{\kappa}(f)]$.

Let $s\in G$ be a regular split semisimple element and ${\Omega}\subset G$ be the conjugacy class of $s$.

\[regular\] $\dim  (H_0(G, {\mathcal K}_{\Omega}))=1$.

Let $T=Z_G(t)$ be the split torus and $B,B'\subset G$ be Borel subgroups containing $T$. Since ${\Omega}=G/T$ we have maps $r:{\Omega}\to G/B$ and $r':{\Omega}\to G/B'$ and therefore morphisms $r_\star :{\mathcal S}({\Omega})\to {\mathcal S}(G/B)$ and $r'_\star :{\mathcal S}({\Omega})\to {\mathcal S}(G/B')$.

As a special case of Lemma \[compl\_ae\_lem\] we get:

\[SL2ex\] The sequence $$\label{star}
 0\to {\mathcal K}_{\Omega}\to {\mathcal S}({\Omega}) \to {\mathcal S}(G/B) \oplus  {\mathcal S}(G/B') \to {{\mathbb C}}\to 0,$$ where the last map $l$ is given by $(\nu ,\nu ')\mapsto \int \nu -\int \nu '$, is exact.

Let $L:=ker (l)$. We have an exact sequence $$0 \to L \to {\mathcal S}(G/B) \oplus  {\mathcal S}(G/B') \to {{\mathbb C}}\to 0.$$ Using that $G$ has homological dimension one, we get that the corresponding long exact sequence of homology contains the following fragment: $$0   \to H_1(G,L)\to 