Root datum

Definition

A root datum consists of a quadruple :(X^\ast, \Phi, X_\ast, \Phi^\vee), where

• X^\ast and X_\ast are free abelian groups of finite rank together with a perfect pairing between them with values in \mathbb{Z} which we denote by ( , ) (in other words, each is identified with the dual of the other).
• \Phi is a finite subset of X^\ast and \Phi^\vee is a finite subset of X_\ast and there is a bijection from \Phi onto \Phi^\vee, denoted by \alpha\mapsto\alpha^\vee.
• For each \alpha, (\alpha, \alpha^\vee)=2.
• For each \alpha, the map x\mapsto x-(x,\alpha^\vee)\alpha induces an automorphism of the root datum (in other words it maps \Phi to \Phi and the induced action on X_\ast maps \Phi^\vee to \Phi^\vee)

The elements of \Phi are called the roots of the root datum, and the elements of \Phi^\vee are called the coroots.

If \Phi does not contain 2\alpha for any \alpha\in\Phi, then the root datum is called reduced.

The root datum of an algebraic group

If G is a reductive algebraic group over an algebraically closed field K with a split maximal torus T then its root datum is a quadruple :(X^, \Phi, X_, \Phi^{\vee}), where

• X^* is the lattice of characters of the maximal torus,
• X_* is the dual lattice (given by the 1-parameter subgroups),
• \Phi is a set of roots,
• \Phi^{\vee} is the corresponding set of coroots.

A connected split reductive algebraic group over K is uniquely determined (up to isomorphism) by its root datum, which is always reduced. Conversely for any root datum there is a reductive algebraic group. 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^, \Phi, X_, \Phi^{\vee}), we can define a dual root datum (X_, \Phi^{\vee},X^, \Phi) 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 {}^L G is the complex connected reductive group whose root datum is dual to that of G.

References

• Michel Demazure, Exp. XXI in SGA 3 vol 3
• T. A. Springer, Reductive groups, in Automorphic forms, representations, and L-functions vol 1