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.
- Absolutely continuous measure
- Compact Lie algebra
- Orbital measure
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