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

(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^*, Φ, X_*, Φ^v),

where

X^* is the lattice of characters of the maximal torus,
X_* is the dual lattice (given by the 1-parameter subgroups),
Φ is a set of roots,
Φ^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^*, Φ,X_*, Φ^v), we can define a dual root datum (X_*, Φ^v,X^*, Φ) 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 ^LG 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 ISBN 0-8218-3347-2

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