Knowlpedia

Coroots and the character-cocharacter pairing

Coroots and their integral pairing with characters, controlling dominance and dual root data.

For a TT, its characters and cocharacters have the perfect pairing

⟨ , ⟩:X∗(T)×X∗(T)⟶Z\langle\ ,\ \rangle: X^*(T)\times X_*(T)\longrightarrow\mathbb Z

defined by

λ∘μ(z)=z⟨λ,μ⟩.\lambda\circ\mu(z)=z^{\langle\lambda,\mu\rangle}.

For every α∈Φ⊂X∗(T)\alpha\in\Phi\subset X^*(T), the root datum specifies a coroot α∨∈X∗(T)\alpha^\vee\in X_*(T) satisfying

⟨α,α∨⟩=2.\langle\alpha,\alpha^\vee\rangle=2.
Reflections and dominance

The root and coroot define reflections

sα(λ)=λ−⟨λ,α∨⟩α,sα(μ)=μ−⟨α,μ⟩α∨.s_\alpha(\lambda) = \lambda-\langle\lambda,\alpha^\vee\rangle\alpha, \qquad s_\alpha(\mu) = \mu-\langle\alpha,\mu\rangle\alpha^\vee.

A character λ\lambda is dominant relative to a if ⟨λ,α∨⟩≥0\langle\lambda,\alpha^\vee\rangle\geq0 for every α\alpha.

Lie-algebra interpretation

Differentiating α∨\alpha^\vee gives a Cartan element HαH_\alpha. With the standard normalization,

dλ(Hα)=⟨λ,α∨⟩.d\lambda(H_\alpha) = \langle\lambda,\alpha^\vee\rangle.

In positive characteristic, remain the safer definition because differential maps can lose information.

Langlands duality

The dual based root datum exchanges

(X∗(T),Φ)with(X∗(T),Φ∨).(X^*(T),\Phi) \quad\text{with}\quad (X_*(T),\Phi^\vee).

This exchange defines the and turns of GG into dominant weights of G^\widehat G.

References
  1. A. Borel, Linear Algebraic Groups, second edition, Springer, 1991.