See at Langlands correspondence.
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 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 . In the comment to a message to Sarnak (2014) Langlands writes:
Further comments along these lines are in (Langlands 14, pages 6-7).
Wikipedia, Langlands dual