Root datum

In mathematics, the root datum (donnée radicielle in French) 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 M. Demazure in SGA III, published in 1970.

Definition

A root datum consists of a quadruple

(X^*, Δ,X_*, Δ^v),

where

• X^* and X_* are free abelian groups of finite rank together with a perfect pairing between them with values in Z (in other words, each is identified with the dual lattice of the other).
• Δ is a finite subset of X^* and Δ^v is a finite subset of X_* and there is a bijection from Δ onto Δ^v, denoted by α→α^v.
• For each α, (α, α^v)=2
• For each α, the map taking x to x−(x,α^v)α induces an automorphism of the root datum (in other words it maps Δ to Δ and the induced action on X_* maps Δ ^v to Δ^v)

The elements of Δ are called the roots of the root datum, and the elements of Δ^v are called the coroots.

If Δ does not contain 2α for any α in Δ then the root datum is called reduced.

The root datum of an algebraic group

If G is a reductive algebraic group over a field K with a split maximal torus T then its root datum is a quadruple

(X^*, Δ,X_*, Δ^v),

where

• X^* is the lattice of characters of the maximal torus,
• X_* is the dual lattice (given by the 1-parameter subgroups),
• Δ is a set of roots,
• Δ^v 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^*, Δ,X_*, Δ^v), we can define a dual root datum (X_*, Δ^v,X^*, Δ) by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots.

References

• M. Demazure, Exp. XXI in SGA 3 vol 3
• T. A. Springer, Reductive groups, in Automorphic forms, representations, and L-functions vol 1 ISBN 0-8218-3347-2