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Let $G$ a split connected reductive group and $G(\mathbb{A})$ his points in the ring of adeles.

We have a degree map $G(\mathbb{A})\rightarrow X_{*}(Z)$ where $Z$ is the center of $G$.

Let $G(\mathbb{A})^{1}$ the degree zero elements and take an element $\gamma\in G(\mathbb{A})$

Do we have an isomorphism between $G_{\gamma}(\mathbb{A})^{1}\backslash G(\mathbb{A})^{1}$ and $G_{\gamma}(\mathbb{A})\backslash G(\mathbb{A})$ where $G_{\gamma}$ is the centralizer of $\gamma$?

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If the restriction of the degree map to the center surjects, then I'd think the answer is yes! e.g. Gl(n) and sl(n). –  plusepsilon.de Feb 19 '13 at 7:17
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