in section \[compcat\]. Recall that according to Bernstein [@centre], $Sm$ can be identified with the category of coherent sheaves of modules over a certain sheaf of algebras over a scheme which is an infinite union of affine algebraic varieties, the spectrum ${{\mathfrak Z}}$ of the Bernstein center of $G$. The category ${{\overline{Sm}}}$ can be described as the category of coherent sheaves of modules over a certain coherent sheaf of algebras over a (componentwise) compactification of ${{\mathfrak Z}}$. An admissible module $\rho$ can also be viewed as an object in ${{\overline{Sm}}}$, so we can apply the Chern character to the class of $\rho$ obtaining $\bar{ch}(\rho)\in HH_0({{\overline{Sm}}})$.

We also need another invariant of the category $Sm$, namely the [*compactly supported*]{} Hochschild homology $HH_*^c(Sm)$ which is the derived global sections with compact support in the sense of [@D] of localized Hochschild homology $R\underline{Hom}_{{{\mathcal H}}\otimes {{\mathcal H}}^{op}}({{\mathcal H}},{{\mathcal H}})$.

We have natural maps $HH_*^c(Sm)\to HH_*({{\overline{Sm}}})\to HH_*(Sm)$; for an admissible module $\rho$ we have its compactly supported Chern character $ch^c(\rho)\in HH_0^c(Sm)$, so that $\bar{ch}(\rho)$ and $ch(\rho)$ equal the images of $ch^c(\rho)$ under the corresponding maps.

The first statement in the main conjecture (a theorem for $SL(2)$) provides a natural isomorphism $${{\mathcal K}}^c_G \cong Im(HH_0^c(Sm)\to HH_0({{\overline{Sm}}})),$$ 