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

${\displaystyle (X^{\ast },\Phi ,X_{\ast },\Phi ^{\vee })}$,

where

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

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

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

The root datum of an algebraic group

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

${\displaystyle (X^{*},\Phi ,X_{*},\Phi ^{\vee })}$,

where

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

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

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

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