Root datum: Difference between revisions

Revision as of 12:22, 13 January 2007

Revision as of 12:23, 13 January 2007

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*Δ^v is the corresponding set of coroots.

A connected split reductive algebraic group over ''K''an algebraically closed field 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''^*, Δ,''X''_*, Δ^v), we can define a '''dual root datum''' (''X''_*, Δ^v,''X''^*, Δ) 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''.