Root datum: Difference between revisions

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In mathematics, 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 M. Demazure in [[Grothendieck's Séminaire de géométrie algébrique|SGA III]], published in 1970.

==Definition==

A '''root datum''' consists of a quadruple
:(''X''^*, &PsiDelta;, ''X''<supsub>&or;*</supsub>, &PsiDelta;^&or;v),
where:

*''X'',^* and ''X''<supsub>&or;*</supsub> are free abelian groups of finite [[rank]] together with a perfect pairing <math>\langlebetween ,them \ranglewith :values Xin \times X^{\vee} \rightarrow \mathbf{'''Z}</math> between them''' (in other words, each is identified with the [[dual lattice]] of the other).

* <math>\Psi </math>&Delta; is a finite subset of ''X''<mathsup>X*</mathsup> and &Delta;<mathsup>\Psi^{\vee} v</mathsup> is a finite subset of ''X''<mathsub>X^{\vee}*</mathsub> and there is a bijection from <math>\Psi</math>&Delta; onto &Delta;<mathsup>\Psi^{\vee}v</mathsup>, denoted by &alpha;&rarr;&alpha;<mathsup>\alpha \mapsto \alpha^{\vee}v</mathsup>.

*For each &alpha;, (&alpha;, &alpha;^v)=2

*For each <math>\&alpha</math>;, the map <math>x \mapstotaking ''x'' - \langleto ''x''&minus;(''x'',\&alpha^{\vee} \rangle \alpha ;<sup>v</mathsup>)&alpha; induces an automorphism of the root datum (in other words it maps <math>\Psi</math>&Delta; to <math>\Psi</math>&Delta; and the induced action on ''X''<mathsub>X^{\vee} *</mathsub> maps &Delta; <mathsup> \Psi^{\vee} v</mathsup> to itself.&Delta;^v)

The elements of <math>\Psi</math>&Delta; are called the '''roots''' of the root datum, and the elements of &Delta;<mathsup> \Psi^{\vee} v</mathsup> are called the '''coroots'''.

*''X'', ''X''^&or; are free abelian groups of finite [[rank]] together with a perfect pairing <math>\langle , \rangle : X \times X^{\vee} \rightarrow \mathbf{Z}</math> between them (in other words, each is identified with the [[dual lattice]] of the other).

If <math> \Psi </math>&Delta; does not contain <math>2 \&alpha</math>; for any <math>\ &alpha</math>; in <math> \Psi </math>&Delta; then the root datum is called '''reduced'''.

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

*For each <math>\alpha</math>, we have: <math> \langle \alpha, \alpha^{\vee}\rangle =2 </math>

*For each <math>\alpha</math>, the map <math>x \mapsto x - \langle x,\alpha^{\vee} \rangle \alpha </math> induces an automorphism of the root datum (in other words it maps <math>\Psi</math> to <math>\Psi</math> and the induced action on <math>X^{\vee} </math> maps <math> \Psi^{\vee} </math> to itself.

The elements of <math>\Psi</math> are called the '''roots''' of the root datum, and the elements of <math> \Psi^{\vee} </math> are called the '''coroots'''.

If <math> \Psi </math> does not contain <math>2 \alpha</math> for any <math>\alpha</math> in <math> \Psi </math> then the root datum is called '''reduced'''.

==The root datum of an algebraic group==

If ''G'' is a reductive algebraic group over a field ''K'' with a split maximal torus ''T'' then its '''root datum''' is a quadruple

If ''G'' is a reductive algebraic group over a field ''K'' with a split maximal torus ''T'' then its '''root datum''' is a quadruple :(''X''^*, &Delta;,''X''_*, &Delta;^&or;v), where

where

*''X''^* is the lattice of characters of the maximal torus,

*''X''_* is the dual lattice (given by the 1-parameter subgroups),

*&Delta; is a set of roots,

*&Delta;^&or;v is the corresponding set of coroots.

A connected split reductive algebraic group over an algebraically closed''K'' 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''^*, &Delta;,''X''_*, &Delta;^&or;v), we can define a '''dual root datum''' (''X''_*, &Delta;^v,''X''^*, &Delta;) by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots.

A connected reductive algebraic group over 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''^*, &Delta;,''X''_*, &Delta;^&or;), we can define a '''dual root datum''' (''X''_*, &Delta;^v,''X''^*, &Delta;) 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''.