I am most probably wrong in asserting as follows.
Let $G$ be a connected reductive group over $\mathbb{Q}$, and $S_{K_f} = G(\mathbb{Q}) \backslash G(\mathbb{A}/K_\infty Z(\mathbb{A}) \cdot K_f $ be the adelic locally symmetric space of level $K_f$ attached to adelic points of $G$. Let $\mathcal{F}$ be a locally constant sheaf of $\mathbb{C}$-vector spaces.
The inclusion of $S_{K_f}$ into its Borel-Serre compactification $\widetilde{S}_{K_f}$ induces a corresponding inclusion of sheaves $\mathcal{F} \hookrightarrow \widetilde{\mathcal{F}}$ by extending by zero outside $S_{K_f}$.
Then since the target space is compact, its cohomology as a topological space coincides with its $L^2$-cohomology.
Moreover cohomologies of $S_{K_f}$ and $\widetilde{S}_{K_f}$ with coefficients in their respective sheaves coincide.
These two statements mean that the $H^{\bullet}(S_{K_f}, \mathcal{F}) \cong H^{\bullet}_{L^2}(\widetilde{S}_{K_f}, \widetilde{\mathcal{F}})$.
Could you please point out where I went wrong?
Thank you.
