Macdonald Formula, Ricci Curvature, and Concentration Locus for Classical Compact Lie Groups

We consider the phenomenon of concentration of measures, which is restricted to the case of families of compact connected Lie groups. While in the literature, powerful general results regarding the existence of concentration and its relations to extremal amenability of infinite dimensional groups have been determined, there are few explicit examples, specially regarding the determination of the region where the measure concentrates. Since they can be relevant for concrete applications, both in mathematics and in physics, in the present paper, we provide a number of such examples, using compact Lie groups as basic ingredients. In particular, our strategy is to employ the Macdonald’s formula, giving the volume of compact simple Lie groups, and Ricci curvature of the bi-invariant metric for analyzing a “concentration locus”, which is a tool to detect where a sequence of metric, Borel measurable spaces concentrates its measure.