What follows is a question that’s probably well-known to experts, but I haven’t been able to find a reference.

Let $ \mathrm G$ be a connected, semisimple $ \mathbb Q$ -group. Let $ K \subset \mathrm G(\mathbb A_f)$ be a good open compact subgroup of the points of $ \mathrm G$ over the finite adeles. By good, I mean small enough for all necessary purposes. Let $ S(K) = \mathrm G(\mathbb Q) K_{\infty} \backslash \mathrm G(\mathbb A) / K$ be the usual arithmetic manifold, where $ K_{\infty} \subset \mathrm G(\mathbb R)$ is a maximal compact subgroup of the real points of $ \mathrm G$ .

Then we have a map between (singular) compactly supported cohomology of $ S(K)$ and (singular) cohomology of $ S(K)$ , say with complex coefficients: \begin{equation} \iota: H^*_c \left( S(K) , \mathbb C \right) \longrightarrow H^* \left( S(K), \mathbb C \right) \end{equation} Let $ \pi$ be a cuspidal automorphic representation such that $ \pi^K \neq 0$ , so that $ \pi$ appears in $ H^* \left( S(K), \mathbb C \right)$ . Letting $ S$ be the usual finite set of bad places (together with the archimedean place) we have the global Hecke algebra $ \mathbf H = C_c^{\infty} \left( \mathrm G(\mathbb A_f), K^S \right)$ acting compatibly (via $ \iota$ ) on $ H^*_c \left( S(K), \mathbb C \right)$ and $ H^* \left( S(K), \mathbb C \right)$ . Moreover, $ \pi$ defines a character $ \chi_{\pi}: \mathbf H \longrightarrow \mathbb C$ of the global Hecke algebra, and we can localize $ \iota$ at this character to compare the Hecke eigensystems at $ \pi$ between compactly supported cohomology and cohomology: \begin{equation} \iota_{\pi}: H^*_c \left( S(K), \mathbb C \right)_{\pi} \longrightarrow H^* \left( S(K), \mathbb C \right)_{\pi} \end{equation}

**Question:** under what conditions is $ \iota_{\pi}$ an isomorphism?

My heuristic is that $ \pi$ tempered and cuspidal should indeed yield an isomorphism $ \iota_{\pi}$ , roughly because $ \pi$ corresponds to cuspidal automorphic forms which should thus be zero `at the cusps’ – the cusps correspond to the boundary of $ S(K)$ , and this is usually where the difference between compactly supported cohomology and cohomology lies.

I do not know how sketchy my heuristic is, or if it is just totally wrong, but I would appreciate any insight about the question. Specific examples are very welcome, as well as references.