Root datum

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

[math]\displaystyle{ (X^\ast, \Phi, X_\ast, \Phi^\vee) }[/math],

where

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

The elements of [math]\displaystyle{ \Phi }[/math] are called the roots of the root datum, and the elements of [math]\displaystyle{ \Phi^\vee }[/math] are called the coroots.

If [math]\displaystyle{ \Phi }[/math] does not contain [math]\displaystyle{ 2\alpha }[/math] for any [math]\displaystyle{ \alpha\in\Phi }[/math], then the root datum is called reduced.

The root datum of an algebraic group

If [math]\displaystyle{ G }[/math] is a reductive algebraic group over an algebraically closed field [math]\displaystyle{ K }[/math] with a split maximal torus [math]\displaystyle{ T }[/math] then its root datum is a quadruple

[math]\displaystyle{ (X^*, \Phi, X_*, \Phi^{\vee}) }[/math],

where

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

A connected split reductive algebraic group over [math]\displaystyle{ K }[/math] 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 [math]\displaystyle{ (X^*, \Phi, X_*, \Phi^{\vee}) }[/math], we can define a dual root datum [math]\displaystyle{ (X_*, \Phi^{\vee},X^*, \Phi) }[/math] by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots.

If [math]\displaystyle{ G }[/math] is a connected reductive algebraic group over the algebraically closed field [math]\displaystyle{ K }[/math], then its Langlands dual group [math]\displaystyle{ {}^L G }[/math] is the complex connected reductive group whose root datum is dual to that of [math]\displaystyle{ G }[/math].

References

• Michel 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