rt.representation theory - Polytopes related to the conjugation action of a Lie group on multiple copies of itself?

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 $G/G$.