In mathematicalgroup theory, theroot datum of a connected splitreductivealgebraic group over a field is a generalization of aroot system that determines the group up to isomorphism. They were introduced byMichel Demazure inSGA III, published in 1970.

Aroot datum consists of a quadruple

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

where

• ${\displaystyle X^{\ast }}$ and${\displaystyle X_{\ast }}$ are free abelian groups of finiterank together with aperfect 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).
• ${\displaystyle \mathrm{\Phi }}$ is a finite subset of${\displaystyle X^{\ast }}$ and${\displaystyle {\mathrm{\Phi }}^{\vee }}$ is a finite subset of${\displaystyle X_{\ast }}$ and there is a bijection from${\displaystyle \mathrm{\Phi }}$ onto${\displaystyle {\mathrm{\Phi }}^{\vee }}$, 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${\displaystyle \mathrm{\Phi }}$ to${\displaystyle \mathrm{\Phi }}$ and the induced action on${\displaystyle X_{\ast }}$ maps${\displaystyle {\mathrm{\Phi }}^{\vee }}$ to${\displaystyle {\mathrm{\Phi }}^{\vee }}$)

The elements of${\displaystyle \mathrm{\Phi }}$ are called theroots of the root datum, and the elements of${\displaystyle {\mathrm{\Phi }}^{\vee }}$ are called thecoroots.

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

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

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

where

• ${\displaystyle X^{\ast }}$ is the lattice of characters of the maximal torus,
• ${\displaystyle X_{\ast }}$ is the dual lattice (given by the 1-parameter subgroups),
• ${\displaystyle \mathrm{\Phi }}$ is a set of roots,
• ${\displaystyle {\mathrm{\Phi }}^{\vee }}$ 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 theDynkin diagram, because it also determines the center of the group.

For any root datum${\displaystyle (X^{\ast },\mathrm{\Phi },X_{\ast },{\mathrm{\Phi }}^{\vee })}$, we can define adual root datum${\displaystyle (X_{\ast },{\mathrm{\Phi }}^{\vee },X^{\ast },\mathrm{\Phi })}$ 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 itsLanglands dual group${\displaystyle {}^{L}G}$ is the complex connected reductive group whose root datum is dual to that of${\displaystyle G}$.

• Michel Demazure, Exp. XXI inSGA 3 vol 3
• T. A. Springer,Reductive groups, inAutomorphic forms, representations, and L-functions vol 1ISBN 0-8218-3347-2

• Representation theory
• Algebraic groups