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.)

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