Lie bracket on the complex valued functions of the space of representations of a Riemann surface

Question

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) 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, Page 542 of Bishwas and Guruprasad - Principal bundles on open surfaces and invariant functions on Lie groups, and Etingof Casimirs of the Goldman Lie algebra of a closed surface, 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.

Answer 1

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.

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

Poisson Structures by Laurent-Gengoux, Pichereau, & Vanhaecke.