To a reductive group $G$, one can associate its "Langlands dual" group ${}^L G$. The Langlands dual group is notably important in the Langlands program. But it seems to me that the Langlands dual group itself should be something I can appreciate in a much more elementary way, without requiring me to develop any kind of understanding of the Langlands program.

**Question 1:** Is there some elementary (and in particular, not-requiring-an-understanding-of-the-Langlands-program) way to illustrate the significance of the Langlands dual group? (Ideally some kind of theorem which relates something about $G$ to something about ${}^L G$, or gives an action of data associated to one on the data associated to the other, or something like that. Perhaps any theorem of such a form *ipso facto* has something to do with the Langlands progam, but even in this case, such a theorem should be something I can appreciate without needing to understand the Langlands program as a whole.)

For context, the extent of my current understanding is basically that there is a canonical involution on the collection of all possible "root data", and this lifts to an involution on (say) reductive Lie groups over $\mathbb C$ which we call "Langlands duality".

**Question 2:** What kind of functoriality properties does $G \mapsto {}^L G$ have?

**Question 3:** I think the simplest case is supposed to be $G = GL_1$, which is Langlands self-dual. In this case, is there any way to see the self-duality at play? (for instance, if $G \mapsto {}^L G$ were contravariantly functorial, maybe this contravariance would carry through to some kind of representation-theoretic data associated to $G$ and ${}^L G$).

Bonus points if your answer can be used as an entry point toward getting an understanding of the Langlands program.

T-Duality for Langlands Dual Groupsarxiv.org/abs/1211.0763 $\endgroup$