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Let G be a finite dimensional real Lie group. As I understand it, the quotient space of G acting on itself by conjugation is a well studied polytope which can be identified with the fundamental alcove of G. It has all kinds of uses and consequences for the representation theory of G.

Is there a similar interpretation for the diagonal conjugation action of G on $G^n$? Have these spaces been studied? and if so, what sort of applications or uses do they have?

I'm not a representation theorist, so I apologize if my question is well-known or naive. I'm hoping that these or similar spaces will have interesting combinatorial/representation theoretic properties to the space $G/G$.

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    $\begingroup$ Where is there a reference for G acting on itself, even? $\endgroup$ Commented Jan 23, 2010 at 22:41

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One application is that this is the space of representations of the fundamental group of an $n+1$-punctured sphere, and those have been studied a lot; one of the most interesting questions is if I fix the conjugacy classes of the n elements, what can the conjugacy class of their product be? This question has been studied a lot and has led to interesting developments like quasi-Hamiltonian G-spaces.

I was once very interested in this stuff (Allen Knutson once suggested I study the relationship of these spaces to fusion of quantum group representations as a thesis problem), but I haven't been keeping up with recent developments.

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