$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}$Motivation: I am trying to understand why the Deligne-Langlands conjectures are only stated for $p$-adic reductive groups with connected center. I know that in such a case, the Langlands dual has simply connected derived group. I am trying to understand why this is a desirable condition.
Suppose $G$ is a complex reductive group (playing the role of the Langlands dual in the statement of the Deligne-Langlands conjecture) with datum $(X,Y, R, R^{\vee}, \Pi)$. Suppose that the derived group of $G$ is simply connected. Is it the case that all root groups in $G$ are in fact $SL_2$$\SL_2$? More precisely, is it true that all root homomorphisms $h_{\alpha}: SL_2\to G$$h_{\alpha}: \SL_2\to G$ has image isomorphic to $SL_2$$\SL_2$? What is the precise obstruction for a root homomorphism taking image $PGL_2$$\PGL_2$?
Assuming that the answer to the answer to question 1 is affirmative, does this necessarily imply that for all $\alpha\in \Pi$, we have $\alpha^{\vee}\notin 2Y$? Probably this one is easy but I cannot see it.
It seems we need both of the above questions to have positive answer; otherwise, I am rather lost.
