Additionally to what Joel was saying, Langlands conjectured the existence of a universal group $\widehat{G}$ (depending on the number field only) whose 2-dim'l representation correspond to automorphic representation of GL(2) in a suitable way. These would include automorphic forms with the archimedean factors not being algebraic, e.g., even Maass forms with Laplace eigenvalue $\neq 1/4$. From what I understand, Arthur suggests a definition of $\widehat{G}$ here: http://www.claymath.org/cw/arthur/pdf/automorphic-langlands-group.pdf. The situation is similar to case (1c) described above.

Additionally to what Joel was saying, Langlands conjectured the existence of a universal group $\widehat{G}$ (depending on the number field only) whose 2-dim'l representation correspond to automorphic representation of GL(2) in a suitable way. These would include automorphic forms with the archimedean factors not being algebraic, e.g., even Maass forms with Laplace eigenvalue $\neq 1/4$. From what I understand, Arthur suggests a definition of $\widehat{G}$ here:¹ https://www.claymath.org/library/cw/arthur/pdf/automorphic-langlands-group.pdf. The situation is similar to case (1c) described above.

¹Arthur J. A Note on the Automorphic Langlands Group. Canadian Mathematical Bulletin. 2002;45(4):466-482. doi:10.4153/CMB-2002-049-1, Wayback Machine

Additionally to what Joel was saying, Langlands conjectured the existence of a universal group $\widehat{G}$ (depending only on the number field only) whose 2-dim'l representation correspond to automorphic representation of GL(2) in a suitable way. These would include automorphic forms with the algebraic factor not being algebraic, e.g., even Maass forms with Laplace eigenvalue $\neq 1/4$. The situation is that nobody knows how to define $\widehat{G}$.

See e.g. http://www.claymath.org/cw/arthur/pdf/automorphic-langlands-group.pdf.

Additionally to what Joel was saying, Langlands conjectured the existence of a universal group $\widehat{G}$ (depending on the number field only) whose 2-dim'l representation correspond to automorphic representation of GL(2) in a suitable way. These would include automorphic forms with the archimedean factors not being algebraic, e.g., even Maass forms with Laplace eigenvalue $\neq 1/4$. From what I understand, Arthur suggests a definition of $\widehat{G}$ here: http://www.claymath.org/cw/arthur/pdf/automorphic-langlands-group.pdf. The situation is similar to case (1c) described above.