Assume I successfully classified the integrable representations of a certain semi-simple Lie group $G$. Given this information, what do I know about the integrable representations of $G^\vee$, the Langlans dual of $G$? More generally, is there any direct information I can learn about $G^\vee$ given the integrable reps. of $G$? Or is there no straightforward relation between these objects?

  • $\begingroup$ I'm new to these concepts, so I hope the question is not too vague or naïve. I apologise in advance if it is. Also, I'm a physicist, so feel free to use physicists language if convenient. $\endgroup$ – AccidentalFourierTransform May 25 '18 at 23:40

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