any $f\in {\mathcal K}$ we denote by $[f]$ it image in $H_0(G,{\mathcal K})$.

\[imb\] a) $\hat{f}\in {\mathcal E}$ for $f\in {\mathcal K}$.

b\) The map $\tau$ defines an isomorphism between $H_0(G,{\mathcal K})$ and ${\mathcal E}$.

c\) $\dim (H_0(G,{\mathcal K}({\Omega}) ))=1$ for any regular semisimple conjugacy class ${\Omega}\subset G$.

One can check that $\hat{f}$ satisfies the conditions of Remark \[Ealt\], thus if $char(F)=0$ then part (a) of the conjecture follows from Harish-Chandra’s Theorem [@HC Theorem 16.2]; see [@W0 Corollary 5.9], [@BP Proposition 5.6.1] for details.

It is clear that part $c)$ follows from $a)$ and $b)$.

\[c\_ell\] If the centralizer of an element $g\in {\Omega}$ is an anisotropic (compact) torus then statement (c) clearly follows from uniqueness (up to scaling) of a Haar measure on $G$. In the case when that centralizer has split rank one the statement is checked in the next subsection.

Almost elliptic orbits
----------------------

To simplify the wording we assume in this subsection that the center of $G$ is compact. A regular semisimple element $g\in G$ will be called almost elliptic if the split rank of its centralizer is at most one. We now prove Conjecture \[imb\](c) in the case when ${\Omega}$ consists of almost elliptic elements.

