Shamely I cannot answer by heart, later I'll ask some colleagues doing invariant theory.

But let me give some example and some comments (partly extending Victor's).

**Example:** consider g=sl(2), then if I understand correctly the factor will be C^3 and
invariant polynoms can be taken like this:

Let me denote by $(\Lambda_1, \Lambda_2)$ points in $sl(2) \oplus sl(2)$

$t_1 = Tr (\Lambda_1^2), ~~ t_2=Tr(\Lambda_1 \Lambda_2), ~~ t_3=Tr(\Lambda_2^2)$

Clearly these invariant polynoms and clearly they are algebraically independent.

**(Probably??????)** They generate the coordinate ring in invariants.

Some comment on this coordinates. See our paper with Dmitry Talalaev

http://arxiv.org/abs/hep-th/0303069 Hitchin system on singular curves I
page 30.

The the space $(Mat_n^n)^{gl(n)}$ can be identified as a moduli space of vector bundles on the very singular curve which is $P^1$ with n-cusps.
In particular if n=2 we get the curve with 2 cusps:
$y^2=(x-a)^3 (x-b)^3$.
This is degeneration of the curve of genus 2.
**For smooth curves of genus 2 there is well-known result by Narasimhan and Ramanan that moduli space of rank 2 bundles is $CP^3$**

Our curve is degeneration of the smooth curve, nevertheless we may try to find analogs
of Narasimhan-Ramanan coordinates. So in this paper we argued that coordinates above are in some sense analogs of their coordinates. There is description of Hitchin system in this coordinates in that paper.

Next two comments related to the following questions $g\oplus g$ can be endowed with Poisson and symplectic structure. You may see $g\oplus g$ as a Lie algebra, and also as $T^*g$ (the first is Poisson, the second is symplectic).
As vector spaces they are the same.

So having poisson manifolds you may want to study the SYMPLECTIC reduction.

The case $T^*g$ and its quantum version $Diff(g)$ is subject of famous paper by Etingof and Ginzburg

http://arxiv.org/abs/math/0011114
Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism

Where the moral is roughly speaking the rational Cherednik algebra in A_n case is quite related to $Diff(g)//g$.

The case of reduction $U(g)\oplus U(g) // g$ has been studied quite recently by
Sergey Khoroshkin, Oleg Ogievetsky

http://arxiv.org/abs/0912.4055
Diagonal reduction algebras of $gl$ type

They called these algebras Mickelson-Zhlobenko algebras (as far as I understand) and they were used in several subsequent works by different authors

polarization operatorscorresponding to the commuting $GL_2$-action. If $V$ is a vector representation of $G=GL_n$ or its contragredient (and in a few other cases), because of the Howe duality, the commuting algebra is indeed generated by the polarization operators. However, this is very far from being true when $V$ is the "matrix" representation. $\endgroup$1more comment