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See at Langlands correspondence.
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Remark on terminology: Robert Langlands did not actually introduce or use the term “dual group”. The original texts just speak of the “L-group” (Gelbhart 84, page 29 (205)). Also, the role of ${}^L G$ in the number-theoretic Langlands program is not as symmetric as in the geometric Langlands program, it essenrially serves as an ingredient for the construction of automorphic L-functions from Galois representations in $G$. In the comment to a message to Sarnak (2014) Langlands writes:
This duality $[$ electric-magnetic duality/S-duality $]$ is quite different than the functoriality and reciprocity introduced in the arithmetic theory $[$ of the Langlands program $]$.
Further comments along these lines are in (Langlands 14, pages 6-7).
Stephen Gelbart, An elementary introduction to the Langlands program, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219 (web)
Robert Langlands, Problems in the theory of automorphic forms -- 45 years later, Oxford 2014
Wikipedia, Langlands dual