I am a physicist so I apologize if something is off in this question.
I began reading "Lectures on the Orbit Method" by Kirillov. If you have the Lie algebra $\mathfrak{g}$ of a Lie group $G$, you can also consider $\mathfrak{g}^*$, which is just he dual space of $\mathfrak{g}$ as a vector space. In other words, for $b \in \mathfrak{g}^*$, $X \in \mathfrak{g}$, we have $b(X) \in \mathbb{R}$.
Lie group elements act on Lie group elements by the Adjoint representation. If $G$ is a matrix group this is just given by conjugation $X \mapsto gXg^{-1}$. You can similarly consider the action of $G$ on $\mathfrak{g}^*$, which is called the coadjoint action. If you take a point in $\mathfrak{g}^*$ and act on it by every element of $G$, you get a surface in $\mathfrak{g}^*$ called a "coadjoint orbit." Remarkably, each coadjoint orbit has a natural symplectic form. The symplectic form at the point $b \in \mathfrak{g}^*$, where $X, Y \in \mathfrak{g}$ can be thought of as tangent vectors on the coadjoint orbit, is just given by $$ \omega_b(X,Y) = b([X, Y]). $$ Amazingly, the book shows that (usually) the different coadjoint orbits each correspond to a unitary representation of $G$. (I think the orbits need to satisfy some sort of "integrality condition" though?) The book goes through a few examples to convince you of this fact.
Here is my question, and where my understanding starts fading: how exactly can you construct a unitary representation from a coadjoint orbit in general? What is the procedure? I'm sure the book mentions this but I can't figure out where.