image of ${\kappa}(f)$ in $H_0(G, {\mathcal K}_u)$.

We start with the following geometric statement.

\[extension\] Let $f\in {\mathcal S}(G)$ be such that $f|_{\bar N}\in {\mathcal K}_u$. Then there exists ${\epsilon}>0$ such that $f|_{G^{\epsilon}}\in {\mathcal K}$.

Recall notations of Remark \[diag\_rem\]. We have $\Delta_{Y_B}=G/B\times T$, where $T$ is the (abstract) Cartan subgroup of $G$. It is easy to see that condition $f|_{\bar N}\in {\mathcal K}_u$ is equivalent to vanishing of the restriction of $A_B(f)$ to $G/B\times \{1\}\subset \Delta_{Y_B}$. Also, condition $f|_{G^{\epsilon}}\in {\mathcal K}$ is equivalent to vanishing of $A_B(f)$ on $G/B\times T_{\epsilon}\subset \Delta_{Y_B}$, where $T_{\epsilon}=G_{\epsilon}\cap T$ (here we abuse notation by identifying the abstract Cartan subgroup $T$ with an arbitrarily chosen Cartan subgroup). Since $A_B(f)$ is locally constant for $f\in {{\mathcal H}}$, the statement follows from compactness of $G/B$.

\[extensions\] For any $f\in {\mathcal K}$ such that $[{\kappa}(f)]=0$ there exists $f'\in {\mathcal K}$ with the same image in $H_0(G,{\mathcal K})$ and such that $f'|_{\bar N}=0$.

Since $[f| _{\bar N}]=0$ we can write the restriction of $f$ to $\bar N$ as a finite sum $\sum _i ({\tilde}f_i^{g_i}-{\tilde}f_i), {\tilde}f_i \in  {\mathcal K}_0', g_i\in G$. As follows from the Lemma \[extension\] we can choose $f_i\in {\mathcal K}$ such that ${\tilde}f_i={f_i}|_{\bar N}$. Then the function 