My question is about the extention of kirillov's symplectic structure on coadjoint orbits. The most remarkable feature
of the coadjoint representation is the fact that all coadjoint orbits possess a
canonical G-invariant symplectic structure.
Kirillov defined the coadjoint orbit by the natural way as follows,
Let $G$ be a Lie Group and $\mathfrak{g}$ be its lie algebra,and also $\mathfrak{g^*}$ be the dual of Lie algebra
$\mathfrak{G}=\{Ad^*(g)F, g\in G\}$ where $F\in\mathfrak{g^*}$.
In fact, Kirillov introduced an antisymmetric bilinear form $B_F$ on $\mathfrak{g}$ by
$B_F(X,Y)=<F,[X,Y]>$ and showed that $B_F$ is invariant under $Stab(F)=\{g\in G: Ad^*(g)F=F \}$ and by using this fact, he introduced a $G$-Invariant symplectic structure $\omega_{\mathfrak{G}}(F)(ad^*(X)F,ad^*(Y)F)=B_F(X,Y)$ on $\mathfrak{G}$ which is now known as Kirillov-Kostant-Souriau Theorem .
Now, I am trying to extend this result for $\mathfrak{g}\oplus \mathfrak{g^*} $ instead of $\mathfrak{g^*} $and try to find a G-invariant symplectic structure.
The fact is that $\mathfrak{g}\oplus \mathfrak{g^*}$ is exactly "equal"(in algebraic and geometric sense) to $Lie(T^*G)$, and we have a symplectic structure on $T^*G$ so we will have a symplectic structure on $Lie(T^*G)$, because we just can restrict the symplectic structure of $(T^*G)$ to $T_e(T^*G)$ and get a symplectic structure on $Lie(T^*G)$.(Also we can define a bilinear symmetric and antisymmetric form on $\mathfrak{g}\oplus \mathfrak{g^*}$ and define a generalized complex structure on it and so we will have a symplectic, poisson, kahler structure on $\mathfrak{g}\oplus \mathfrak{g^*}$ by this way. )
But it would be a good question , if we define the action of $g\in G$ on $X+F\in \mathfrak{g}\oplus \mathfrak{g^*} $ by $g.(X+F)=Ad(g)X+Ad^*(g)F$.
and define an orbit as same method of kirillov by $\mathfrak{O}=\{Ad(g)X+Ad^*(g)F, g\in G\}$.
So, by this definition,we will have $Stab(X+F)=\{g\in G: Ad^*(g)F=F , gX=Xg \}$, and then $G$ is a fibre bundle over the base $\mathfrak{O}$.
Now, if we construct the symplectic structure
$\omega_{X+F}:T_{X+F}{\mathfrak{O}}\times T_{X+F}{\mathfrak{O}}\to R$
$\omega_{X+F}([Y,X]+ad^*(Y)F,[Z,X]+ad^*(Z)F)$=?
THE FACT IS "?" (I mean right hand sight of equality) should be an invariant bilinear form $B_{X+F}(Y,Z)$ and we should find it. So, my question is how can we find this guy?
PS:I edited my question after comments of Ben and Mariano .
Best Answer
There is no hope for such a construction. The orbits of $G$ on $\frak{g}\oplus\frak{g}^\ast$ are not always of even dimension, so they will not support symplectic structures.
For example, take $G=\mathrm{SO}(3)$. Because $G$ is simple, $\frak{g}^\ast$ is isomorphic to $\frak{g}$ as a $G$-module, so the action of $G$ on $\frak{g}\oplus\frak{g}^\ast$ is just the diagonal action of $G$ on $\frak{g}\oplus\frak{g}$. Now, if $x, y\in\frak{g}$ are linearly independent, then the stabilizer of $(x,y)\in \frak{g}\oplus\frak{g}$ is trivial, so its orbit is just a copy of $G$, which has dimension $3$.
A similar phenomenon happens for any odd-dimensional compact simple Lie group.