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Dec 16, 2017 at 9:05 comment added user1688 One mods out the centre because it is not interesting. As for the last question: this is not a condition on $K_f$, but on the linear group you start with. For $GL_n$ the quotient is non-compact, if you take an anisotropic form of $GL_n$, for instance, the unit group of a division algebra, then, again after modding out the centre, the quotient will be compact.
Dec 16, 2017 at 8:13 history asked Vanya CC BY-SA 3.0