Let $K$ be a compact, connected (probably also simple) Lie group and with a maximal torus $T$. Regular coadjoint orbits $\mathcal{O}_{\lambda} \cong K/T$, parameterized by a regular element $\lambda \in \mathfrak{t}^*$, have several properties:

- They are symplectic, equipped with KKS forms.
- They are Hamiltonian $T$-manifolds with respect to the coadjoint action of $T$, generated by the canonical projection map $\mathcal{O}_{\lambda} \to \mathfrak{t}^*$.
- The fixed points of the $T$-action are precisely the elements of $\mathcal{O}_{\lambda} \cap \mathfrak{t}^*$ (where $\mathfrak{t}^*$ is identified with the subspace $(\mathfrak{k}^*)^T$).
- Using the killing form (or whatever scalar multiple of it that you like), we can make an identification $T_{\lambda}\mathcal{O}_{\lambda} \cong \mathfrak{t}^{\perp}$. One has that $\mathfrak{t}^{\perp}$ is invariant under the adjoint action of $T$ and coincides under this identification with the isotropy action of $T$ at the fixed point $\lambda$.

The local normal form theorem for Hamiltonian torus actions tells us that there exists a symplectomorphism $\varphi$ from a neighbourhood of 0 in the symplectic vector space $T_{\lambda}\mathcal{O}_{\lambda}$ to a neighbourhood of $\lambda$ in $\mathcal{O}_{\lambda}$ such that:

- $\varphi$ is $T$-equivariant.
- $\varphi$ intertwines the moment maps, i.e. if $\mu = \phi + \lambda$ denotes the quadratic moment map for the isotropy action of $T$ on $T_{\lambda}\mathcal{O}_{\lambda}$ shifted by $\lambda$, and $\text{pr}\colon \mathcal{O}_{\lambda} \to \mathfrak{t}^*$, then $$ \text{pr} \circ \varphi = \mu.$$

Since everything in this example is fairly explicit, it seems reasonable to ask for an explicit formula for the map $\varphi$.

What is a general formula for $\varphi$?

There is an ``obvious candidate'' that looks good at first: one can easily define the map $$\psi\colon \mathfrak{t}^{\perp} \to \mathcal{O}_{\lambda}, \, \psi(X) = Ad^*_{e^X}\lambda.$$ This map is $T$-equivariant, but it fails to be symplectic (which can be demonstrated easily for coadjoint orbits of $SU(2)$).

(As a side note, the killing form restricted to $\mathfrak{t}^{\perp}$ and then extended $\mathcal{O}_{\lambda}$ by $K$-invariance makes $\mathcal{O}_{\lambda}$ a reductive homogeneous space, and $\psi$ is in fact the exponential map for this metric.)

I believe that, essentially, the map $\psi$ fails to be symplectic because $e^Xe^Y \neq e^{X+Y}$ if $[X,Y]\neq 0$ (which, if you are not careful means that you can accidentally ``prove'' this map is symplectic and spend an afternoon being very confused), which suggests that maybe there is some sort of CBH-related correction to $\psi$ that makes it symplectic (I am really just taking a guess here).

Edit: It will be sufficient to find a $T$-equivariant map $\rho\colon \mathfrak{t}^{\perp} \to K$ such that $$\varphi(Y) := Ad_{\rho(Y)}\lambda$$ satisfies the condition $$ d(\varphi)_Y(ad_X\lambda) = ad_{Ad_{\rho(Y)}X}\varphi(Y).$$

Edit 2: Unpacking the proof of local normal forms, one sees that there exists a map $\phi\colon U \to \mathfrak{t}^{\perp}$ defined on a neighbourhood $U$ of the origin in $\mathfrak{t}^{\perp}$ that fixes the origin and is a diffeomorphism onto it's image, such that the map

$$\varphi(Y) := Ad_{e^{\phi(Y)}}\lambda$$

is a symplectomorphism. $\phi$ is the time-1 flow of a moser vector field. It's not clear from this perspective whether an explicit formula for $\phi$ is a reasonable thing to hope for.

Edit 3: I found a partial answer that I've posted below, but I am interested if anyone can expand on it.