Root data

Last updated: October 16, 2005

Recall that a root datum is a quadruple (X,R,X,R) where X, X are two lattices in duality, R ⊂ X and R ⊂ X are finite subsets, together with a bijection α → α from R to R such that:

  1. for each α ∈ R, ⟨ α,α ⟩ = 2
  2. for each α ∈ R, the reflection of X (resp. X) defined by α and α induces a permutation of R (resp. R)

If G is a connected complex reductive group, T a maximal torus in G, X = X(T) the character lattice of T, X its cocharacter lattice, and if R, R are respectively the roots and coroots of (G,T), then (X,R,X,R) is a root datum, and this root datum determines G up to isomorphism. Conversely, one can show (cf. [1] for instance) that all root data arise in this fashion.

In actual practice, things are even much simpler. The lattices X and X are always implicit, and equal to Zn. Instead of giving the full sets of roots and coroots, we just give bases: thus we give two subsets Π and Π of Zn, of same cardinality ≤ n (the second one lying in the "dual" Z∧n), and a bijection α → α from Π to Π such that the matrix (⟨ α,β ⟩) is a Cartan matrix (i.e., a matrix that up to permutation of the indices has a block decomposition where each block is one of the irreducible Cartan matrices of the familiar types A1–G2.) From there, we construct R and R by saturating Π and Π through the group generated by the reflections corresponding to the elements of Π and Π. These simpler data correspond to what is usually called a based root datum.

Example: root data of rank 2 and semisimple rank 1. This amounts to the datum of a single vector α in Z2, for which there exists a vector α in Z2 such that ⟨ α,α ⟩ = 2. Up to GL(2,Z)-conjugacy there are exactly three such pairs: α = (1,0), α = (2,0); α = (2,0), α = (1,0); α = (1,1), α = (1,1); they correspond respectively to the groups PSL(2).T(1), SL(2).T(1) and GL(2).

This is the way reductive groups are handled by the program. In one way or another, we have to construct a root datum. This is currently done through the initial user interaction, but it might just as well be input from a file, or be handed by another program. Everything else is computed from there.

It is obvious that for any root datum (X,R,X,R), (X,R,X,R) is also a root datum, called the dual of (X,R,X,R). If (X,R,X,R) corresponded to a reductive group G, the group corresponding to the dual root datum is called the dual reductive group, and denoted G. Duality interchanges the root systems of types B and C, and preserves the other types; it also "reverses" coverings: the dual of a semisimple adjoint group is the simply connected group for the dual root system, and conversely. When we fix an inner class, we should also go over to the dual inner class (this is the inner class defined at the torus level by the negative transpose of any involution representing the chosen inner class in G.) This interchanges the "split" and "compact" inner classes, for instance.

[1] T.A. Springer, Linear algebraic groups. Second edition. Progress in Mathematics, 9. Birkhäuser Boston, Inc., Boston, MA, 1998.

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