Character estimates of adjoint simple Lie groups

Abstract

Abstract Call a compact, connected, simple Lie group G adjoint simple if it has trivial center. Let C ⊂ G be a nontrivial conjugacy class, e ∈ G the identity element of G. We prove the existence of a bound N ∈ ℕ, depending on G but not C, such that e lies in the interior of Cn for all n ≥ N. We then prove that a disk D ⊂ ℂ of radius less than 1, properly contained in the unit disk D 1 and tangent to D 1 at z = 1, contains the image of every normalized character χ ( e ) - 1 χ ${\chi (e)^{-1}\chi }$ of G.