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Root datum

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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 ∗ , Φ , X ∗ , Φ ∨ ) {\displaystyle (X^{\ast },\Phi ,X_{\ast },\Phi ^{\vee })} ,

where

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

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

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

The root datum of an algebraic group

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

( X ∗ , Φ , X ∗ , Φ ∨ ) {\displaystyle (X^{*},\Phi ,X_{*},\Phi ^{\vee })} ,

where

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

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

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

References

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