map $ch:K^0(Sm)\to C({{\mathcal H}})$. For our present purposes we need to modify both the source and the target of this map. We replace the target $C({{\mathcal H}})={{\mathcal H}}_G$ by ${{\mathcal K}}_G$ where ${{\mathcal K}}\subset {{\mathcal H}}$ is a subspace invariant under the conjugation action of $G$, the so called space of “weightless” functions.[^1] Definition and some properties of ${{\mathcal K}}$ are discussed in section \[sec\_K\]. The key property is that $WO_g|_{{\mathcal K}}$ is a $G$-invariant functional for any regular semisimple element $g\in G$; furthermore, there is a well defined averaging map $Av$ from ${{\mathcal K}}$ to the space of invariant generalized functions on $G$ and for $\phi\in {{\mathcal K}}$ the value of $WO_g(\phi)$ coincides with $Av(\phi)(g)$ (the latter is well defined since $Av(\phi)$ is in fact a locally constant function on the set of regular elements in $G$). We also provide a conjecture with a proof for $PGL(2,F)$, $char(F)=0$ describing the image of the averaging map from ${{\mathcal K}}$ to the space of invariant distributions.

To describe the source of the map generalizing $ch$ we need some new ingredients. One of them is the so called [*compactified category*]{} of smooth (finitely generated) representations ${{\overline{Sm}}}$.

The abelian category ${{\overline{Sm}}}$ is defined 