I am looking for a a description of the algebra of continuous central functions on a group, say a compact simple Lie group $G$, as the algebra of all continuous functions on a "nice" compact Hausdorff space. By central I mean of course constant on the conjugacy classes, i.e.


$f(gxg^{-1}) = f(x)$ for all $x,g\in G$.


By Gelfand's theorem such a compact Hausdorff space exists, but I am looking for a more "geometrical" construction. Something like the following two examples.


$SU(2)$: Any central function on $SU(2)$ is a function of the trace, i.e. the sum of the two eigenvalues. Since the two eigenvalues of a matrix in $SU(2)$ multiply to one, we get the interval $[-2,2]$, i.e. $C\_{centr}(SU(2))\cong C([-2,2])$.


$SU(3)$: Central functions on $SU(3)$ are again functions of the trace of a matrix in $SU(3)$. Denote by $D$ the possible values of this trace, these are exactly the complex numbers that can be written as a sum of three complex numbers of modulus one that multiply to one. This set has a nice geometric description: $D$ is the closed domain in $\mathbb{C}$ bounded by the curve $2e^{i\phi}+e^{-2i\phi}$, $0\le \phi\le 2\pi$, which can be obtained by letting a disk of radius one rotate inside a disk of radius three centered at the origin, attaching a pen to a point on the boundary of the small disk and starting from the point $(3,0)$. We get $C\_{centr}(SU(3))\cong C(D)$.


One can now try to continue in a similar manner for general $SU(N)$, and look for a "nice" geometrical description of the quotient of


{$\{(\lambda\_1,\cdots,\lambda\_n)\in \mathbb{T}^n:\lambda\_1\cdots\lambda\_n=1\}$}


under permutations. 


I expect that this has already been done, but haven't been able to find any references (except for some related work in the case of $SU(3)$). Who can help me?


**Comment added:** First of all thanks Yemon for the quick answer. I see that I have to explain better what I mean here by "nice geometric" Hausdorff space. Is it possible to find coordinates for $T/W$ such that the characters if the irreps of $G$ are polynomials in these coordinates? Or, equivalently, a compact subset $D$ in $\mathbb{R}^n$ such that $C\_{centr}(G)\cong C(D)$ and such that the characters correspond to polynomials?


Note that $T/W$ (or $(D)$) inherits a measure from the Haar measure on $G$ and that characters of inequivalent irreps will be orthogonal w.r.t. this measure. (For $SU(2)$ the measure is the semi-circular distribution and the characters will be mapped to Chebyshev polynomials.)



"geometric" description of the algebra of central functions on a Lie group
If $G$ is compact, then as explained in comments $G/ad$ is $T/W$. If
further $G$ is simply-connected, hence a product of simple factors,
then $T/W$ is a corresponding product of simplices.


The point is that $T/W = ({\mathfrak t}/\Lambda)/W 
= {\mathfrak t}/(\Lambda \rtimes W)$, where $\Lambda$ is the kernel of the
exponential map ${\mathfrak t}\to T$, and when $G$ is simply connected this latter group is an affine reflection group (the affine Weyl group, or at least, the product of those of the simple factors) whose fundamental chamber is the product of simplices.


