Definition For Separably Closed Fields
From a reductive algebraic group over a separably closed field K we can construct its root datum (X*, Δ,X*, Δv), where X* is the lattice of characters of a maximal torus, X* the dual lattice (given by the 1-parameter subgroups), Δ the roots, and Δv the coroots. A connected reductive algebraic group over K is uniquely determined (up to isomorphism) by its root datum. A root datum contains slightly more information than the Dynkin diagram, because it also determines the center of the group.
For any root datum (X*, Δ,X*, Δv), we can define a dual root datum (X*, Δv,X*, Δ) by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots.
If G is a connected reductive algebraic group over the algebraically closed field K, then its Langlands dual group LG is the complex connected reductive group whose root datum is dual to that of G.
Examples: The Langlands dual group LG has the same Dynkin diagram as G, except that components of type Bn are changed to components of type Cn and vice versa. If G has trivial center then LG is simply connected, and if G is simply connected then LG has trivial center. The Langlands dual of GLn(K) is GLn(C).
Read more about this topic: Langlands Dual
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