Let $S$ be a closed surface and $G$ be a reductive Lie group. Goldman ([Invariant functions on Lie groups and Hamiltonian flows of surface group representations](http://terpconnect.umd.edu/~wmg/InvariantFunctions.pdf)) proved that, for a fairly general class of groups $G$, $M=Hom(\pi\_1(S),G)/G$ admits a symplectic structure, where the quotient is by conjugation action. Hence the space of all "real valued" functions $C^\infty(M,\mathbb R)$ admits a Lie bracket $\{\,,\}$.


Suppose $G$ is one of the following Lie groups: $\operatorname{GL}\_n(\mathbb R)$, $\operatorname{GL}\_n(\mathbb C)$, $\operatorname{SL}\_n(\mathbb R)$, or $\operatorname{SL}\_n(\mathbb C)$. Given any $x\in\pi\_1(S)$, we define a function $f\_x:M\to \mathbb R$ by $f\_x(\rho)=\Re(\operatorname{tr}(\rho(x)))$, where $\Re$ is the real-part of a complex number. Given $x,y\in\pi\_1(S)$, Goldman gave explicit formulas for $\{f\_x, f\_y\}$. 


In some papers, the authors consider the Lie bracket between two complex-valued function on $M$, and use Goldman's formula and paper as a reference. For example Section 4 of [Andersen, Mattes, and Reshetikhin - The Poisson structure on the moduli space of flat connections and chord diagrams](https://core.ac.uk/download/pdf/82095891.pdf), Page 542 of [Bishwas and Guruprasad - Principal bundles on open surfaces and invariant functions on Lie groups](https://www.worldscientific.com/doi/abs/10.1142/S0129167X93000285), and [Etingof Casimirs of the Goldman Lie algebra of a closed surface](https://arxiv.org/abs/math/0407505), considered the Lie bracket of the trace functions (not just the real part) defined similarly as above. 


My question is: what is the Lie bracket of $C^\infty(M, \mathbb C)$, and how is it related to the Lie bracket of $C^\infty(M, \mathbb R)$?


Any kind of suggestion/reference/comment will be extremely helpful. Thanks in advance.



Lie bracket on the complex valued functions of the space of representations of a Riemann surface
In the complex case, Goldman's symplectic form is a $(2,0)$-form, and the Poisson bracket is in terms of that form (the bracket is determined by a bivector and the bivector corresponds to the form). 


The Goldman bracket formula works the same way as it does in the real case. 
 You can find a proof about this latter fact, and concrete examples, in my paper: 


*[Poisson Geometry of SL(3,C)-Character Varieties Relative to a Surface with Boundary](https://arxiv.org/abs/math/0703251)*.


As far as learning about general complex Poisson structures, relevant to this setting, I recommend:


*[Poisson Structures](https://www.springer.com/us/book/9783642310898)* by Laurent-Gengoux, Pichereau, & Vanhaecke.


