Characterizing the absolute continuity of the convolution of orbital measures in a classical Lie algebra

Let g be a compact simple Lie algebra of dimension d. It is a classical result that the convolution of any d non-trivial, G-invariant, orbital measures is absolutely continuous with respect to Lebesgue measure on g, and the sum of any d non-trivial orbits has non-empty interior. The number d was later reduced to the rank of the Lie algebra (or rank +1 in the case of type A_n). More recently, the minimal integer k = k(X) such that the k-fold convolution of the orbital measure supported on the orbit generated by X is an absolutely continuous measure was calculated for each X ∈ g. In this paper g is any of the classical, compact, simple Lie algebras. We characterize the tuples (X₁, . . . , X_L), with X_i ∈ g, which have the property that the convolution of the L-orbital measures supported on the orbits generated by the X_i is absolutely continuous, and, equivalently, the sum of their orbits has non-empty interior. The characterization depends on the Lie type of g and the structure of the annihilating roots of the X_i. Such a characterization was previously known only for type A_n.