Can someone explain, explicitly, how to, given a reductive complex algebraic group construct the Langlands dual group? I know it is a group with the cocharacters of G as its characters, but how does one go about writing down what group it is?



Langlands Dual Groups
You can construct the dual group in a combinatorial manner as follows: Reductive groups are classified by their root datum. There is an obvious duality on the set of all root data, and the dual group is the reductive group with the dual root datum.


You can see [Wikipedia](http://en.wikipedia.org/wiki/Root_datum) for the notion of the root datum.


