Root datum

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

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*, Φ, X*, Φv),

where

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*, Φ,X*, Φv), we can define a dual root datum (X*, Φv,X*, Φ) 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 LG is the complex connected reductive group whose root datum is dual to that of G.

References

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