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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

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A root datum consists of a quadruple

,

where

  • and are free abelian groups of finite rank together with a perfect pairing between them with values in which we denote by ( , ) (in other words, each is identified with the dual of the other).
  • is a finite subset of and is a finite subset of and there is a bijection from onto , denoted by .
  • For each , .
  • For each , the map induces an automorphism of the root datum (in other words it maps to and the induced action on maps to )

The elements of are called the roots of the root datum, and the elements of are called the coroots.

If does not contain for any , then the root datum is called reduced.

The root datum of an algebraic group

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If is a reductive algebraic group over an algebraically closed field with a split maximal torus then its root datum is a quadruple

,

where

  • is the lattice of characters of the maximal torus,
  • is the dual lattice (given by the 1-parameter subgroups),
  • is a set of roots,
  • is the corresponding set of coroots.

A connected split reductive algebraic group over 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 , we can define a dual root datum by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots.

If is a connected reductive algebraic group over the algebraically closed field , then its Langlands dual group is the complex connected reductive group whose root datum is dual to that of .

References

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  • 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