Let G be a closed subgroup of the orthogonal group O(V) of a
finite dimensional real Euclidean space V. The space of G-orbits X=V/G has
a natural structure of metric space simply by declaring the distance beween
two points in X to be the distance between the corresponding orbits in V.

In this talk, we would like to address the following question: "How much of
the G-action on V can be recovered from the metric space X?"
A class of examples that is at the origin and basis of our considerations
is the adjoint representation of a compact connected semisimple Lie group G
on its Lie algebra g. It is well know that the orbit space g/Ad_G can be
isometrically recovered as t/W, where it is the Lie algebra of a maximal
torus of G and W is the corresponding Weyl group.

The second presentation of the orbit space is simpler and easier to
understand because W is a finite group. (Based on joint work with A.
Lytchak (Cologne)).