Let $G = GL_n$. In the literature, I see that the symmetric space associated to $G$ is of the form $G(\mathbb{R})/K_\infty$ with $K_\infty = O(n) Z(\mathbb{R})^\circ = O(n) \mathbb{R}^\times_{+}$. Could you please tell me why one needs to `thicken' the maximal compact subgroup $O(n)$ with the identity component $\mathbb{R}^{+}_{>0}$ of the center $Z(\mathbb{R})$.
Secondly, the adelic locally symmetric space is defined by $S_{K_f} := G(\mathbb{Q}) \backslash G(\mathbb{A})/ K_\infty K_f $ for some compact open (neat) subgroup $K_f \subset G(\mathbb{A}_f)$; here $\mathbb{A} = \mathbb{R} \times \mathbb{A}_f$ is the ring of adeles over $\mathbb{Q}$. As a topological space this is known to be homeomorphic to the union of finitely many connected components of the form $\Gamma \backslash G(\mathbb{R})/K_\infty$ for some arithmetic subgroup $\Gamma \subset G(\mathbb{Q})$. What conditions on $K_f$ ensure that the components $\Gamma \backslash G(\mathbb{R})/K_\infty$ are compact?
Thank you.
