ag.algebraic geometry - Langlands Dual Groups

As Peter mentioned, reductive groups are determined by their root data, and the Langlands dual is given by switching weights with coweights, and roots with coroots.

There is a "construction" of a group from a root datum in SGA III Exp 25 (in vol 3). It starts by reducing to the simply connected semisimple case (meaning there are ways of going from this case to the general case). The weights give you a torus T, and the positive/negative roots give you unipotent groups U+ and U-. You form a scheme Omega = U- x T x U+, and create G by gluing a few disjoint copies of Omega together, and writing down a composition law. This yields a split group over an arbitrary base scheme.