Buzzard–Gee conjecture on C- and L-arithmetic automorphic representations

Let $G$ be a connected reductive algebraic group defined over a number field $F$, and let $\pi=\bigotimes_v\pi_v$ be an irreducible automorphic representation of $G(\mathbb{A}_F)$. The representation $\pi$ is C-algebraic or L-algebraic according to the corresponding condition on its Archimedean infinitesimal character, and it is C-arithmetic or L-arithmetic when the relevant unramified Hecke eigenvalues or Satake parameters are defined over a number field. Buzzard–Gee's arithmeticity conjecture. The representation $\pi$ is C-algebraic if and only if it is C-arithmetic, and it is L-algebraic if and only if it is L-arithmetic. This conjecture proposes that the Archimedean algebraicity conditions exactly match the corresponding arithmeticity conditions at unramified finite places; the source investigates the C-algebraic/C-arithmetic direction and reduction to cuspidal representations.