Coadjoint representation

Definition

Definition for a Lie group over a topological field

The coadjoint representation of a Lie group $G$ over a topological field $K$ is defined as the contragredient representation to the adjoint representation of $G$. Note that if ${\mathfrak {g}}$ denotes the Lie algebra of $G$, the adjoint representation of $G$ is on ${\mathfrak {g}}$ and the coadjoint representation is on the dual vector space ${\mathfrak {g}}^{*}$.

Definition for a Lazard Lie group

The coadjoint representation of a Lazard Lie group, viewed as simply being over $\mathbb {Z}$, is its natural representation on the Pontryagin dual of its Lie algebra that arises from the adjoint action.