where ${{\mathcal K}}^c\subset {{\mathcal K}}$ is the subspace of measures supported on compact elements. By the previous paragraph, $\bar{ch}(\rho)$ belongs to that image, thus we obtain a homological construction of an element in ${{\mathcal K}}^c_G$ from an admissible representation, the so called “compactified pseudo-matrix coefficient” of the representation.

The second main statement (proved for $SL(2)$) asserts that for a compact regular element $g$ and an admissible representation $\rho$ we have $WO_g(\bar{ch}(\rho))=\chi_\rho(g)$. Notice that for a noncompact regular element $g$ the value of $\chi_\rho(g)$ coincides with a character value of the Jacquet functor applied to $\rho$ [@DCas], in particular it vanishes for a cuspidal module.

We view Conjecture \[char\_form\_conj\] as an algebraic statement underlying some aspects of Arthur’s local trace formula [@Ar], while equality underlies the elliptic part of the local trace formula (see also Remark \[int\_rem\] below). We plan to develop this theme in a future publication.

[**[Acknowledgements.]{}** ]{} We thank Joseph Bernstein, Vladimir Drinfeld, Dmitry Kaledin and Dmitry Vaintrob for many useful discussions over the years. In particular, the definition of the compactified category was conceived as a result of discussions with Kaledin and it took its present form partly due to discussions with Drinfeld.

We also thank Rapha" 