Root datum

In mathematics, the root datum of a connected reductive algebraic group over a separably closed field is a combinatorial object that determines the group up to isomorphism.

Definition

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^v),

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^v is the corresponding set of coroots.

A connected split reductive algebraic group over K is uniquely determined (up to isomorphism) by its root datum. 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^v), 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.

References

• A. Borel, Automorphic L-functions, in Automorphic forms, representations, and L-functions vol 2, ISBN 0-8218-1437-0
• Springer, Reductive groups, in Automorphic forms, representations, and L-functions vol 1