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

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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 Aₙ). 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,...,Xₗ), with Xᵢ∈ g, which have the property that the convolution of the L -orbital measures supported on the orbits generated by the Xᵢ 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 Xᵢ. Such a characterization was previously known only for type Aₙ.