Dominant coweight - Knowlpedia

Let $G$ be a connected reductive group, choose a maximal torus $T\subseteq B$ in a Borel subgroup, and let $\Phi^+$ be the resulting positive roots. A coweight $\lambda\in X_*(T)$ is dominant if

\langle\alpha,\lambda\rangle\geq 0 \qquad\text{for every }\alpha\in\Phi^+.

Every Weyl-group orbit in $X_*(T)$ contains a unique dominant coweight. Dominant coweights index the positive-loop-group orbits on the affine Grassmannian.

References

T. A. Springer, Linear Algebraic Groups, 2nd ed., Birkhäuser, 1998, Chapters 7–8.