Root datum: Difference between revisions

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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==

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:<math>(X^\ast, \Phi, X_\ast, \Phi^\vee)</math>,

where

* <math>X^\ast</math> and <math>X_\ast</math> 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).

* <math>\Phi</math> is a finite subset of <math>X^\ast</math> and <math>\Phi^\vee</math> is a finite subset of <math>X_\ast</math> and there is a bijection from <math>\Phi</math> onto <math>\Phi^\vee</math>, denoted by <math>\alpha\mapsto\alpha^\vee</math>.

* For each <math>\alpha</math>, <math>(\alpha, \alpha^\vee)=2</math>.

* For each <math>\alpha</math>, the map <math>x\mapsto x-(x,\alpha^\vee)\alpha</math> induces an automorphism of the root datum (in other words it maps <math>\Phi</math> to <math>\Phi</math> and the induced action on <math>X_\ast</math> maps <math>\Phi^\vee</math> to <math>\Phi^\vee</math>)

The elements of <math>\Phi</math> are called the '''roots''' of the root datum, and the elements of <math>\Phi^\vee</math> are called the '''coroots'''. The elements of <math>X^\ast</math> are sometimes called '''[[Weight (representation theory)|weights]]''' and those of <math>X_\ast</math> accordingly '''coweights'''.

If <math>\Phi</math> does not contain <math>2\alpha</math> for any <math>\alpha\in\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 an [[algebraically closed field]] ''<math>K''</math> with a split maximal torus ''<math>T''</math> then its '''root datum''' is a quadruple

:<math>(''X''^{^*}, &\Phi;, ''X''_{X_*}, &\Phi;<sup>v^{\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),

*&<math>\Phi;</math> is a set of roots,

*&Phi;<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''^{^*}, &\Phi;,''X''_{X_*}, &\Phi;<sup>v^{\vee})</supmath>), we can define a '''dual root datum''' (''X''<submath>(X_*</sub>, &\Phi;^v^{\vee},''X''^{^*}, &\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]]