Root datum

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.

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 ${\displaystyle G}$ .

• 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