I see, thank you for the reference!

@MikhailBorovoi Thank you! I now understand the classification of sl_2-triples is much easier (in terms of reps of sl_2).

Thank you! If S is a Dedekind scheme say Spec O for an order O in number field, do we still have an obstruction theory (without referring to the moduli space)?

@Kimball Thank you! I have a check. Their construction in the case of discrete series is quite beautiful and they explain the case of $SO_2n$ and $SO_{2n+1}$. But they start with a compact Lie group with finite center, and there is no description of characters (as in the usual Jacquet-Langlands transfers). I am also interested in the limits of discrete series (which appear naturally) and the case of Spin groups (e.g. $Sp_4=Spin(3,2)$).

@SpencerLeslie You are right. I have some examples in mind e.g. direct sum of two irreducible representations. In general, if we do not fix $G$, any component group will occur. If we fix $G$, then we are classifying component groups of subgroup $H$ of $G$ such that $G/H$ is quasi-affine.

Thank you for the excellent answer. Are there any geometric Langlands / TQFT patterns for representations of finite simple groups of Lie type (generalizing Deligne-Lusztig representations)?