Let $G$ be a Lie group (and to be safe, let's assume it is semisimple). Consider the action of $G$ on itself by conjugation, and form the GIT (algebro-geometric) quotient $G/\\!/G$. Then let $\pi:G\to G/\\!/G$ denote the quotient map. For example, if $G=SL(2,\mathbb C)$, then $G=\operatorname{Spec}\mathbb C[a,b,c,d]/(ad-bc-1)$ and $G/\\!/G=\operatorname{Spec}\mathbb C[t]$, and the projection $\pi$ is induced from the map of rings which sends $t$ to $a+d$ (all I'm saying is that the trace determines the conjugacy class in $SL(2,\mathbb C)$, at least for semisimple ones).

There is a well-known closed $2$-form on $G$ whose symplectic leaves are exactly the fibers of $\pi$ (i.e. the conjugacy classes). It can be defined, for example, as follows. Let $g\in G$, and denote its conjugacy class by $G_g$. Then $T_gG_g$ is exactly the image of the map $R:\mathfrak g\to\mathfrak g$ given by $\alpha\mapsto\alpha-\operatorname{ad}_g\alpha$. Now the map $\mathfrak g\otimes\mathfrak g\to\mathbb C$ given by $\alpha\otimes\beta\mapsto\operatorname{tr}(g[\alpha,\beta])$ has exactly the same kernel as $R$, and thus it descends to give an alternating bilinear form on $T_gG_g$ (by $\operatorname{tr}$ I mean trace in the adjoint representation). Some additional arguments show that this form is closed.

This construction is sometimes done instead with $\mathfrak g^\ast\to\mathfrak g^\ast/\\!G$, though I'm more interested in the case $G\to G/\\!/G$.

There are many papers which study the deformation quantization (or geometric quantization) of (the leaves of) this Poisson structure on $G$, but everything is very VERY abstract. Is there any place where I can find explicit formulae for things like:

1. the Poisson bracket of standard functions on $G$ (e.g. the coordinates $a,b,c,d$ in the case $G=SL(2,\mathbb C)$).

or even better

2. the deformed product (star product) on $\mathcal O(G)$.

?

I feel like these should be known or easy to calculate, but an answer has eluded my search.

I've tried doing some explicit calculations myself, but everything seems to hinge on getting a succinct formula for the $2$-form on $G$, and I always end up with something horrendous.

Let $G$ be a Lie group (and to be safe, let's assume it is semisimple). Consider the action of $G$ on itself by conjugation, and form the GIT (algebro-geometric) quotient $G/\!/G$. Then let $\pi:G\to G/\!/G$ denote the quotient map. For example, if $G=SL(2,\mathbb C)$, then $G=\operatorname{Spec}\mathbb C[a,b,c,d]/(ad-bc-1)$ and $G/\!/G=\operatorname{Spec}\mathbb C[t]$, and the projection $\pi$ is induced from the map of rings which sends $t$ to $a+d$ (all I'm saying is that the trace determines the conjugacy class in $SL(2,\mathbb C)$, at least for semisimple ones).

There is a well-known closed $2$-form on $G$ whose symplectic leaves are exactly the fibers of $\pi$ (i.e. the conjugacy classes). It can be defined, for example, as follows. Let $g\in G$, and denote its conjugacy class by $G_g$. Then $T_gG_g$ is exactly the image of the map $R:\mathfrak g\to\mathfrak g$ given by $\alpha\mapsto\alpha-\operatorname{ad}_g\alpha$. Now the map $\mathfrak g\otimes\mathfrak g\to\mathbb C$ given by $\alpha\otimes\beta\mapsto\operatorname{tr}(g[\alpha,\beta])$ has exactly the same kernel as $R$, and thus it descends to give an alternating bilinear form on $T_gG_g$ (by $\operatorname{tr}$ I mean trace in the adjoint representation). Some additional arguments show that this form is closed.

This construction is sometimes done instead with $\mathfrak g^\ast\to\mathfrak g^\ast/\!/G$, though I'm more interested in the case $G\to G/\!/G$.

There are many papers which study the deformation quantization (or geometric quantization) of (the leaves of) this Poisson structure on $G$, but everything is very VERY abstract. Is there any place where I can find explicit formulae for things like:

1. the Poisson bracket of standard functions on $G$ (e.g. the coordinates $a,b,c,d$ in the case $G=SL(2,\mathbb C)$).

or even better

2. the deformed product (star product) on $\mathcal O(G)$?

I feel like these should be known or easy to calculate, but an answer has eluded my search.

I've tried doing some explicit calculations myself, but everything seems to hinge on getting a succinct formula for the $2$-form on $G$, and I always end up with something horrendous.