Root datum

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In mathematical group theory, the root datum (donnée radicielle in French) 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[edit]

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 \mathbb{Z} which we denote by ( , ) (in other words, each is identified with the dual lattice 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. The elements of X^\ast are sometimes called weights and those of X_\ast accordingly coweights.

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[edit]

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

  • X* is the lattice of characters of the maximal torus,
  • X* is the dual lattice (given by the 1-parameter subgroups),
  • Φ is a set of roots,
  • Φv 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*, 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[edit]