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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.

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    $\begingroup$ Your assertion (in the fourth paragraph) that $\pi^K \ne 0$ implies that $\pi$ appears in $H^*(S(K), \mathbf{C})$ is not true. This fails even for $PGL_2 / \mathbf{Q}$. Some $\pi_\infty$ will contribute to cohomology, but only with non-trivial coefficients. Other $\pi_\infty$, e.g. holomorphic limits of discrete series for $PGL_2$ (corresponding to weight 1 modular forms), don't contribute to topological cohomology at all. $\endgroup$ – David Loeffler Nov 22 '17 at 6:42

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