homology of an abelian category. Since ${{\mathcal H}}$ is Noetherian and has finite homological dimension, there is a well defined Chern character (also called the Hattori-Stallings or Dennis trace) map $ch: K^0(Sm)\to C({{\mathcal H}})$ (we will abbreviate $ch([M])$ to $ch(M)$). It has been conjectured in [@cusp] and proven in [@SS], [@thes] that for an [*elliptic*]{} regular semisimple element $g\in G$ and an admissible representation $\rho$ we have $$\label{ell}
\chi_\rho(g)=O_g(ch(\rho)),$$ where $O_g$ denotes the orbital integral. Here we use that $O_g:{{\mathcal H}}\to {{\mathbb C}}$ being conjugation invariant factors through $C({{\mathcal H}})$.

If $\rho$ is a cuspidal irreducible representation then (assuming that $G$ has a compact center) a matrix coefficient $m_\rho\in {{\mathcal H}}$ is a representative of the class $ch(\rho)\in C({{\mathcal H}})$. Thus in this case reduces to an earlier result of Arthur [@Ar1]. However, the latter applies also to nonelliptic regular semisimple elements: for such an element $g$ and a cuspidal irreducible representation $\rho$ Arthur has proved that $$\label{Ar_f}
 \chi_\rho(g)=WO_g(m_\rho),$$ where $WO_g$ denotes the [*weighted orbital integral*]{}. Our Conjecture \[char\_form\_conj\] provides a generalization of to all regular semisimple compact elements $g$, which for a cuspidal representation $\rho$ reduces to .

The first step in this direction is a generalization of the 