Lattice points and Lie groups. I

Abstract

Assume that G is a compact semisimple Lie group and G \mathfrak {G} its associated Lie algebra. It is shown that the number of irreducible representations of G of dimension less than or equal to n is asymptotic to k n a / b k{n^{a/b}} , where a = the rank of G \mathfrak {G} and b = the number of positive roots of G \mathfrak {G} .