The Lefschetz formula for locally symmetric spaces

Let $G$ be the reductive group, $\Gamma$ the arithmetic subgroup, $A$ the split torus, and $A^-$ its negative chamber. Let $\hat G_{\rm adm}$ denote the admissible dual, let $N_\Gamma(\pi)$ be the discrete multiplicity, and let $C^{\mu,j}(A^-)$ be the Banach space of functions with the stated support and growth conditions. For $\lambda\in\mathfrak a^*$ and $\pi\in\hat G_{\rm adm}$, write $L^\tau_\lambda(\pi)$ for the spectral Lefschetz contribution and $L^\tau(\gamma)$ for the local Lefschetz number attached to $[\gamma]\in{\cal E}_P(\Gamma)$. Lefschetz formula. For every $\lambda\in\mathfrak a^*$ and $\pi\in\hat G_{\rm adm}$, there is an integer $N_{\Gamma,\rm cont}(\pi,\lambda)$, vanishing when $\operatorname{Re}(\lambda)\notin\overline{\mathfrak a_{\mathbb R}^{*,+}}$, and there are $\mu\in\mathbb A^*$ and $j\in\mathbb N$ such that, with $m_\lambda(\pi)=N_\Gamma(\pi)+N_{\Gamma,\rm cont}(\pi,\lambda)$, every $\varphi\in C^{\mu,j}(A^-)$ satisfies

$$\sum_{\substack{\pi\in\hat G\lambda\in\mathfrak a^*}}m_\lambda(\pi)L^\tau_\lambda(\pi)\int_A\varphi(a)a^\lambda\,da=\sum_{[\gamma]\in{\cal E}_P(\Gamma)}L^\tau(\gamma)\varphi(a_\gamma).$$

Both sides define continuous functionals on $C^{\mu,j}(A^-)$. The formula is known for cocompact $\Gamma$, where all continuous-spectrum corrections vanish, and for $G={\rm PSL}_2(\mathbb R)$; the general case remains conjectural.