Singularity of orbital measures in SU(n)

We show that the minimal k such that μ^k ∈ L¹ (SU(n)) for all central, continuous measures μ on SU(n) is k = n. We do this by exhibiting an element g ∈ SU(n) for which the (n - 1)-fold product of its conjugacy class has zero Haar measure. This ensures that if μ_g is the corresponding orbital measure supported on the conjugacy class, then μ_g^n-1 is singular to L¹.