root datum

A root datum consists of a quadruple

:$(X^\backslash ast,\; \backslash Phi,\; X\_\backslash ast,\; \backslash Phi^\backslash vee)$,

where

• $X^\backslash ast$ and $X\_\backslash ast$ are free abelian groups of finite rank together with a perfect pairing between them with values in $\backslash mathbb\{Z\}$ which we denote by ( , ) (in other words, each is identified with the dual of the other).
• $\backslash Phi$ is a finite subset of $X^\backslash ast$ and $\backslash Phi^\backslash vee$ is a finite subset of $X\_\backslash ast$ and there is a bijection from $\backslash Phi$ onto $\backslash Phi^\backslash vee$, denoted by $\backslash alpha\backslash mapsto\backslash alpha^\backslash vee$.
• For each $\backslash alpha$, $(\backslash alpha,\; \backslash alpha^\backslash vee)=2$.
• For each $\backslash alpha$, the map $x\backslash mapsto\; x-(x,\backslash alpha^\backslash vee)\backslash alpha$ induces an automorphism of the root datum (in other words it maps $\backslash Phi$ to $\backslash Phi$ and the induced action on $X\_\backslash ast$ maps $\backslash Phi^\backslash vee$ to $\backslash Phi^\backslash vee$)

The elements of $\backslash Phi$ are called the roots of the root datum, and the elements of $\backslash Phi^\backslash vee$ are called the coroots.

If $\backslash Phi$ does not contain $2\backslash alpha$ for any $\backslash alpha\backslash in\backslash Phi$, then the root datum is called reduced.

If $G$ is a reductive algebraic group over an algebraically closed field $K$ with a split maximal torus $T$ then its root datum is a quadruple

:$(X^*,\; \backslash Phi,\; X\_*,\; \backslash Phi^\{\backslash vee\})$,

where

• $X^*$ is the lattice of characters of the maximal torus,
• $X\_*$ is the dual lattice (given by the 1-parameter subgroups),
• $\backslash Phi$ is a set of roots,
• $\backslash Phi^\{\backslash vee\}$ is the corresponding set of coroots.

A connected split reductive algebraic group over $K$ 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 $(X^*,\; \backslash Phi,\; X\_*,\; \backslash Phi^\{\backslash vee\})$, we can define a dual root datum $(X\_*,\; \backslash Phi^\{\backslash vee\},X^*,\; \backslash Phi)$ 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 $G$.

• 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|0-8218-3347-2}}

Category:Representation theory

Category:Algebraic groups