# Colóquio MAP - 30/08/2013 - A METRIC APPROACH TO REPRESENTATIONS OF COMPACT LIE GROUPS

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)).