Roots, Weyl group, and dominant weights

Let $G$ be a split connected reductive group with split maximal torus $T$ . The adjoint action of $T$ on $\operatorname{Lie}(G)$ gives

\operatorname{Lie}(G) = \operatorname{Lie}(T) \oplus \bigoplus_{\alpha\in\Phi}\mathfrak g_\alpha.

The nonzero characters $\alpha\in X^*(T)$ occurring here are the roots.

Weyl group and positive roots

The Weyl group is

W=N_G(T)/T

where $N_G(T)$ is the normalizer. The group $W$ acts on $X^*(T)$ and is generated by the root reflections. Choosing a Borel subgroup $B\supset T$ selects positive roots $\Phi^+$ and simple roots $\Delta$ .

A character $\lambda\in X^*(T)$ is dominant when

\langle\lambda,\alpha^\vee\rangle\geq0

for every $\alpha\in\Delta$ , using the root–coroot pairing.

Highest weights

Over an algebraically closed field of characteristic $0$ , irreducible finite-dimensional algebraic representations of $G$ are classified by dominant characters in $X^*(T)$ . The abstract weight lattice of the root system can be larger than $X^*(T)$ ; only the characters allowed by the isogeny form occur as representations of $G$ .

For $\operatorname{GL}_n$ , the roots are $e_i-e_j$ , the Weyl group is $S_n$ , and the standard Borel gives dominant weights $\lambda_1\geq\cdots\geq\lambda_n$ .

Langlands role

Langlands duality exchanges roots and coroots. Dominant coweights of $G$ become dominant weights of $\widehat G$ , indexing both the Satake basis and the Schubert strata used in geometric Satake.

References

J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, 1972.
A. Borel, Linear Algebraic Groups, second edition, Springer, 1991.