Root datum explained

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 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 ^{*,} \Phi, X _{*,} \Phi ^{\vee} ) , where X ^{*} is the lattice of characters of the maximal torus, X _{*} is the dual lattice (given by the 1-parameter subgroups), \Phi is a set of roots, \Phi ^{\vee} 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 ^{*,} \Phi, X _{*,} \Phi ^{\vee} ) , we can define a dual root datum (X _{*,} \Phi ^{\vee} ,X ^{*,} \Phi) 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 {} ^{L} G 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