• problem with torsion in $\pi_1(G)$.

So, what's the difference between $GL_2, PGL_2, SL_2$? $GL_2$ is the father of all, $PGL_2$ and $SL_2$ each take out some abelian part out of it. $PGL_2$ totally killed the central subgroup, by quotienting out it, whereas $SL_2$ tries to take a slice, that intersects minimally with it.

Then, $T_{SL} \to T_{GL} \to T_{PGL}$ the composition is a finite cover. This has many immediate consequences

• The cocharacter lattice $Hom(\C^*,T_{SL}) \to Hom(\C^*, T_{PGL})$ is injective. This is because $Hom(S^1,-)$ is only left exact, not right exact, hence loses surjectivity, we are led to $Hom^1(S^1, - )$. If the target is a discrete group $\Gamma$, then $Hom^1(S^1, \Gamma) = \Gamma$.
• The character lattice, $Hom(T_{PGL}, \C^*) \to \Hom(T_{SL}, \C^*)$, seems to be injective. Is it because $Hom(-, \C^*)$ is exact? Here, if $\Gamma$ is a finite abelian group, then $Hom(\Gamma, \C^*)$ is just the 'Pontryagin dual'.