|
In mathematicsmathematical [[group theory]], the '''root datum''' ('''donnée radicielle''' in French) of a connected split [[reductive group|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 [[Grothendieck's Séminaire de géométrie algébrique|SGA III]], published in 1970.
==Definition==
A '''root datum''' consists of a quadruple
:<math>(X^\ast, \Phi, X_\ast, \Phi^\vee)</math>,
:(''X''<sup>*</sup>, Δ, ''X''<sub>*</sub>, Δ<sup>v</sup>),
where
*''X'' <supmath>*X^\ast</supmath> and ''X''<submath>*X_\ast</submath> are free abelian groups of finite [[Rank (linear algebra)|rank]] together with a [[perfect pairing]] between them with values in '''<math>\mathbb{Z'''}</math> which we denote by ( , ) (in other words, each is identified with the [[dual lattice]] of the other).
*Δ <math>\Phi</math> is a finite subset of ''X''<supmath>*X^\ast</supmath> and Δ<supmath>v\Phi^\vee</supmath> is a finite subset of ''X''<submath>*X_\ast</submath> and there is a bijection from Δ<math>\Phi</math> onto Δ<supmath>v\Phi^\vee</supmath>, denoted by α→α<supmath>v\alpha\mapsto\alpha^\vee</supmath>.
* For each &<math>\alpha;</math>, <math>(&\alpha;, &\alpha;<sup>v</sup>^\vee)=2</math>.
* For each &<math>\alpha;</math>, the map taking ''<math>x'' to\mapsto ''x''−-(''x'',&\alpha^\vee)\alpha;<sup>v</supmath>)α induces an automorphism of the root datum (in other words it maps Δ<math>\Phi</math> to Δ<math>\Phi</math> and the induced action on ''X''<submath>*X_\ast</submath> maps Δ <supmath>v\Phi^\vee</supmath> to Δ<supmath>v\Phi^\vee</supmath>)
The elements of Δ<math>\Phi</math> are called the '''roots''' of the root datum, and the elements of Δ<supmath>v\Phi^\vee</supmath> are called the '''coroots'''.
If Δ<math>\Phi</math> does not contain <math>2&\alpha;</math> for any &<math>\alpha; \in Δ\Phi</math>, then the root datum is called '''reduced'''.
==The root datum of an algebraic group==
If ''<math>G''</math> is a reductive algebraic group over aan field[[algebraically closed field]] ''<math>K''</math> with a split maximal torus ''<math>T''</math> then its '''root datum''' is a quadruple
:<math>(''X''<sup>^*</sup>, Δ\Phi, ''X''<sub>X_*</sub>, Δ<sup>v\Phi^{\vee})</supmath>),
where
*''X''<supmath>X^*</supmath> is the lattice of characters of the maximal torus,
*''X''<submath>X_*</submath> is the dual lattice (given by the 1-parameter subgroups),
*Δ<math>\Phi</math> is a set of roots,
*Δ<supmath>v\Phi^{\vee}</supmath> is the corresponding set of coroots.
A connected split reductive algebraic group over ''<math>K''</math> 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 <math>(''X''<sup>^*</sup>, Δ\Phi,''X''<sub> X_*</sub>, Δ<sup>v\Phi^{\vee})</supmath>), we can define a '''dual root datum''' (''X''<submath>(X_*</sub>, Δ<sup>v</sup>\Phi^{\vee},''X''<sup>^*</sup>, Δ\Phi)</math> by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots.
If ''<math>G''</math> is a connected reductive algebraic group over the algebraically closed field ''<math>K''</math>, then its [[Langlands dual group]] <supmath>''{}^L'' G</supmath>''G'' is the complex connected reductive group whose root datum is dual to that of ''<math>G''</math>.
==References==
*[[Michel Demazure]], Exp. XXI in [https://web.archive.org/web/20011126072304/http://modular.fas.harvard.edu/sga/sga/3-3/index.html SGA 3 vol 3]
*[[T. A. Springer]], [http://www.ams.org/online_bks/pspum331/pspum331-ptI-1.pdf ''Reductive groups''], in [http://www.ams.org/online_bks/pspum331/ ''Automorphic forms, representations, and L-functions'' vol 1] ISBN {{isbn|0-8218-3347-2 }}
[[Category:Representation theory]]
[[Category:Algebraic groups]]
[[zh:根資料]]
| 