I tried asking this question on stackexchange and received no response.

Given a semisimple Lie group, there is a symplectic structure on the coadjoint orbits arising from the Kirillov 2-form. Can I use this to define a volume form on the adjoint orbits, perhaps pulling it back via a homeomorphism between corresponding adjoint and coadjoint orbits?

I know there is another way to get a volume on adjoint orbits, via Haar measure on the Lie group and taking the quotient measure identifying the orbit with the quotient of $G$ by the stabilizer, but I would like to do it ignoring this identification entirely.

My end goal is this; given $f:\mathfrak{g} \to \mathbb{R}$, and $X \in \mathfrak{g}$ I would like to define some type of integration $\int_{O(X)} f \cdot d\omega$. As mentioned above, there is a way this is usually done by taking a Haar measure on $G$ and getting some normalized quotient measure $\dot{dg}$ over $G/C_G(X)$, giving the so-called orbital integrals $\int_{G/C_G(X)} f(gXg^{-1})\, \dot{dg}$. But I would like to instead have some general volume form on $O(X)$. The Kirillov-Kostant-Souriau symplectic structre on the coadjoint orbits seems to be the way to do this, but I am unsure about the technical details of such an approach.

I tried asking this question on stackexchange and received no response.

Given a semisimple Lie group, there is a symplectic structure on the coadjoint orbits arising from the Kirillov 2-form. Can I use this to define a volume form on the adjoint orbits, perhaps pulling it back via a homeomorphism between corresponding adjoint and coadjoint orbits?

I know there is another way to get a volume on adjoint orbits, via Haar measure on the Lie group and taking the quotient measure identifying the orbit with the quotient of $G$ by the stabilizer, but I would like to do it ignoring this identification entirely.

My end goal is this; given $f:\mathfrak{g} \to \mathbb{R}$, and $X \in \mathfrak{g}$ I would like to define some type of integration $\int_{O(X)} f \cdot d\omega$. As mentioned above, there is a way this is usually done by taking a Haar measure on $G$ and getting some normalized quotient measure $\dot{dg}$ over $G/C_G(X)$, giving the so-called orbital integrals $\int_{G/C_G(X)} f(gXg^{-1})\, \dot{dg}$. But I would like to instead have some general volume form on $O(X)$. The Kirillov-Kostant-Souriau symplectic structure on the coadjoint orbits seems to be the way to do this, but I am unsure about the technical details of such an approach.

I tried asking this question on stackexchange and received no response (link here: https://math.stackexchange.com/questions/2871644/can-i-bring-the-kirilov-2-form-on-coadjoint-orbits-to-adjoint-orbits )

Given a semisimple Lie group, there is a symplectic structure on the coadjoint orbits arising from the Kirilov 2-form. Can I use this to define a volume form on the adjoint orbits, perhaps pulling it back via the homeomorphism between corresponding adjoint and coadjoint orbits?

I know there is another way to get to volume on adjoint orbits via the Haar measure on the Lie group and taking the quotient measure identifying the orbit with the quotient of G by the stabilizer, but I would like to do it ignoring this identification entirely.

My end goal is this; given $f:\mathfrak{g} \to \mathbb{R}$, and $X \in \mathfrak{g}$ I would like to define some type of integration $\int_{O(X)} f \cdot d\omega$. As mentioned above, there is a way this is usually done by taking a Haar measure on $G$ and taking some normalized quotient measure $\dot{dg}$ over $G/C_G(X)$, giving the so-called orbital integrals $\int_{G/C_G(X)} f(gXg^{-1}) \dot{dg}$. But I would like to instead have some general volume form on $O(X)$. The Kirilov-Kostant-Souriau symplectic structre on the coadjoint orbits seems to be the way to do this, but I am unsure about the technical details of such an approach.

Thank you.

I tried asking this question on stackexchange and received no response.

Given a semisimple Lie group, there is a symplectic structure on the coadjoint orbits arising from the Kirillov 2-form. Can I use this to define a volume form on the adjoint orbits, perhaps pulling it back via a homeomorphism between corresponding adjoint and coadjoint orbits?

I know there is another way to get a volume on adjoint orbits, via Haar measure on the Lie group and taking the quotient measure identifying the orbit with the quotient of $G$ by the stabilizer, but I would like to do it ignoring this identification entirely.

My end goal is this; given $f:\mathfrak{g} \to \mathbb{R}$, and $X \in \mathfrak{g}$ I would like to define some type of integration $\int_{O(X)} f \cdot d\omega$. As mentioned above, there is a way this is usually done by taking a Haar measure on $G$ and getting some normalized quotient measure $\dot{dg}$ over $G/C_G(X)$, giving the so-called orbital integrals $\int_{G/C_G(X)} f(gXg^{-1})\, \dot{dg}$. But I would like to instead have some general volume form on $O(X)$. The Kirillov-Kostant-Souriau symplectic structre on the coadjoint orbits seems to be the way to do this, but I am unsure about the technical details of such an approach.