Root datum

In mathematics, 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 M. Demazure in SGA III, published in 1970.

Definition

A root datum consists of a quadruple ("X", Ψ, "X"^∨, Ψ^∨), where:

*"X", "X"^∨ are free abelian groups of finite rank together with a perfect pairing $langle\; ,\; angle\; :\; X\; imes\; X^\{vee\}\; ightarrow\; mathbf\{Z\}$ between them (in other words, each is identified with the dual lattice of the other).

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

*For each $alpha$, we have: $langle\; alpha,\; alpha^\{vee\}\; angle\; =2$

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

The elements of $Psi$ are called the roots of the root datum, and the elements of $Psi^\{vee\}$ are called the coroots.

If $Psi$ does not contain $2\; alpha$ for any $alpha$ in $Psi$ 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 :("X"^*, Δ,"X"_*, Δ^∨), where

*"X"^* is the lattice of characters of the maximal torus,
*"X"_* is the dual lattice (given by the 1-parameter subgroups),
*Δ is a set of roots,
*Δ^∨ is the corresponding set of 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"^*, Δ,"X"_*, Δ^∨), 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".

References

*M. Demazure, Exp. XXI in [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 0-8218-3347-2