# “geometric” description of the algebra of central functions on a Lie group

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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Isn't this "just" maximal torus quotiented by action of Weyl group? – Yemon Choi Jan 23 '13 at 16:43
(For you want a cts function constant on conjugacy classes, hence by max torus theory it can be viewed as a function on T/W. Then appeal to something like Peter-Weyl in form of combinations of group characters to show that the map from your algebra to C(T/W) has dense range, so by cstar theory is onto) – Yemon Choi Jan 23 '13 at 16:47
Yes, thank you! I was hoping for a more "geometric" description of this set. Say we are interested in convolution semigroups of central probability measures (or, equivalently, conjugate invariant Levy processes). For this I have to study how I can move move away from the identity. The descriptions above for SU(2) and SU(3) show that there is only one possible direction to leave the identity without jumping (the identity is the boundary point to the right, i.e. 2 for SU(2) and (3,0) for SU(3)). This explains intuitively why the diffusion part in the generator of conjugate invariant Levy – Uwe Franz Jan 23 '13 at 17:05
processes has only one parameter. Of course we already know that this is true more generally for simple compact Lie groups, it has to be a multiple of the Laplace-Beltrami operator. – Uwe Franz Jan 23 '13 at 17:13
The space is the Weyl alcoven, which is a simplex. This space is always homeomorphic to the quotient G/ad G. – Thomas Nikolaus Jan 23 '13 at 19:19

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.

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Just a comment on references: É. Cartan wrote down explicit descriptions of these quotient simplices for all of the compact simple, simply-connected Lie groups (including the exceptional ones) some time in the 1930s. These are exactly the kind of descriptions that the OP wants. I'm traveling right now and don't have a copy of his collected works handy, or else I'd supply it, but I'm sure that you can find the right references in his 1936 paper La topologie des espaces représentatifs des groupes de Lie (in Part I, volume 2 of Cartan's Collected Works). – Robert Bryant Jan 23 '13 at 19:57
Thank you very much, Allen and Robert, your explanations are very helpful for me! – Uwe Franz Jan 25 '13 at 8:54

Not an answer as such, but some extended comments with explicit references (posted as community-wiki).

1) As a non-specialist, I'm unsure whether you are asking for new information or for better references than you have. Compact Lie groups have been studied thoroughly for more than a century, going back (as Robert Bryant points out) to foundational work by E. Cartan, followed soon by Weyl and others. By now there are many textbook and lecture treatments of compact groups in the context of more general semisimple or reductive Lie groups, along with a few more targeted accounts of the compact groups by themselves: structure, classification, representations, topology. So it's difficult to find much new to say, though not impossible.

2) Concerning more recent references, I'd mention the Springer GTM 98 Representations of Compact Lie Groups by Brocker and tom Dieck, which lays out the basic theory with detailed examples. For instance, they deal explicitly with central functions in their Chapter IV, followed by more on root systems and representations. Another source is Bourbaki's Chapter IX on compact Lie groups, where the emphasis is often quite analytic and includes a treatment in section 8.3 of Fourier transforms of central functions.

3) Allen has provided a basic summary in terms of maximal tori and Weyl groups. The sources I've mentioned provide a lot of details about all of this, while the topology (Stiefel, Bott-Samelson, ... ) has been exposed in lecture notes by Bott. It's mostly a question of putting your own concerns in clear focus relative to all this literature. The drawback of Lie theory is the multitude of approaches taken to it, but that also reflects its importance.

4) I'd mention finally that there is a remarkable algebraic parallel for all these questions about class functions in the theory of semisimple algebraic groups. Although "tori" become algebraic tori, the Weyl groupa and root systems play much the same role. Here the Chevalley restriction to a maximal torus leads again to study of the Weyl group orbits on a fixed maximal torus. In this algebraic setting, the resulting geometry is that of an affine algebraic variety, which in the simply connected case is just affine space. Moreover, the characters of irreducible representations and "characters" of maximal tori come into play in a way completely parallel to the study of compact Lie groups (even without analysis being involved).

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Many thanks, Jim! My question was mainly for good references, to find out how well this spaces are understood. – Uwe Franz Jan 25 '13 at 8:56