In mathematical group theory, the root datum of a connected split reductive algebraic group over a field is a generalization of a root system that determines the group up to isomorphism. They were introduced by Michel Demazure in SGA III, published in 1970.

Definition

A root datum consists of a quadruple

1. ,

where

• are free abelian groups of finite rank together with a perfect pairing which we denote by ( , ) (in other words, each is identified with the dual of the other).

• .

• .

• )

are called the roots are called the coroots.

, then the root datum is called reduced.

The root datum of an algebraic group

is a reductive algebraic group over an algebraically closed field then its root datum is a quadruple

1. ,

where

• is the lattice of characters of the maximal torus,

• is the dual lattice (given by the 1-parameter subgroups),

• is a set of roots,

• is the corresponding set of coroots.

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.

, we can define a dual root datum by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots.

, then its Langlands dual group.

References

• Michel Demazure, Exp. XXI in SGA 3 vol 3

• T. A. Springer, , in