Wiki. “Langlands 对偶” [Langlands对偶]
Wiki. “Langlands 对偶” [Langlands对偶]
设 $G$ 为约化代数群, 其 Langlands 对偶 ${^LG}$ 是另一个代数群,
Given a reductive algebraic group $G$, one constructs its Langlands dual $^LG$ by applying an involution to its root data. Under the Langlands correspondence, automorphic representations of the group $G$ (其实是 $G(\mathbb A_F)$) correspond to Galois representations with values in $^LG$ (即同态 $\mathrm {Gal}(\overline{F}/F)\to {^LG}$).
规范理论中的 Langlands 对偶
A gauge theory has a coupling constant $g$, which plays the role of the electric charge $e$. The conjectural non-abelian electro-magnetic duality, which has later become known as $S$-duality, has the form $(G, g) \leftrightarrow (^LG, 1/g)$.