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If by "surjective" you mean surjective in the usual sense (for example on $\overline{F}$-points) then maybe you have a problem, because $G_1(F)$ may not surject onto $G_2(F)$. So for example $SL(2)$ s …

You've essentially answered your own question. Let $G$ be the units of a division algebra of dimension $n^2$. Then $G$ is an inner form of a general linear group so the Galois action on the root datum …

I'm no expert, but I think the theorem is that (for $G$ reductive over a local field $F$) $G(F)$ is compact iff $G$ is $F$-anisotropic, that is, every $F$-torus in $G$ (or equivalently every maximal $ …

Let $F$ be a finite extension of $\mathbf{Q}_p$ with integers $\mathscr{O}$, let $\mathbb{G}$ be a connected reductive group over $F$ and let $G=\mathbb{G}(F)$ be its $F$-points. Let $X(G)=\operatorna …

Yes, the dual of $SL_2$ is $PGL_2$. But you're not going down the right track with $PSL_2$. The problem with $PSL_2$ is that it's not a variety at all! You can quotient out the variety $SL_2$ by the …

The fact that Hecke operators (double coset stuff coming from $SL_2(\mathbf{Z})$ acting on modular forms) and Hecke algebras (locally constant functions on $GL_2(\mathbf{Q}_p)$) are related has nothin …

Things are perhaps a bit messier than you hope. In particular it is not true that the unitary group is non-quasi-split if and only if $v$ ramifies. Disclaimer: I did not know the answer to this questi …

This question is a bit like saying "what's the point of the theory of bases for vector spaces -- this just gives you an isomorphism of your space with $\mathbb{R}^n$. What is the point of defining thi …