Let \(G\) be a compact, connected, semi-simple Lie group and suppose \(\theta\) is a Cartan involution that fixes the closed Lie subgroup \(K\). The quotient space \(G/K\) is known as a compact symmetric space. Denote by \(N_{G}(K)\) the normalizer of \(K\) in \(G\). Let \(\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}\) be the corresponding Cartan decomposition of the Lie algebra \(\mathfrak{g}\) of \(G\), \(\mathfrak{a}\subseteq\mathfrak{p}\) be a maximal abelian subspace. The \(K\)-orbital measure on \(G\) is defined by \(\mu_a=m_K\ast\delta_a\ast m_K\), where \(m_K\) denotes the normalized Haar measure on \(K\) and \(\delta_a\) denotes the point mass measure at \(a\). The \(K\)-orbital measure, \(\mu_a\), is a singular probability measure which is supported on \(KaK\), and is continuous (meaning nonatomic) if \(a\not\in N_{G}(K)\) when viewed as a measure on the symmetric space \(G/K\). These measures are the extreme points of the unit ball of the space of \(K\)-bi-invariant, continuous measures. The main result of the paper under review is the following.
Theorem 3.1. Suppose \(a_1,a_2\in \exp\mathfrak{a}\) are regular elements and \(\mu_{a_1}\), \(\mu_{a_2}\) are the associated \(K\)-orbital measures. Then \(\mu_{a_1}\ast\mu_{a_2}\) is absolutely continuous with respect to Haar measure on \(G\) and \(Ka_1Ka_2K\) has nonempty interior in \(G\).
Corollary 3.2. Suppose \(\mu_{1}\), \(\mu_{2}\) are \(K\)-bi-invariant measures, compactly supported on \(\bigcup_{a\in D}KaK\) where \(D\) is the dense set of regular elements. Then \(\mu_{1}\ast\mu_{2}\) is absolutely continuous.
Corollary 3.3. Suppose \(G/K\) is a compact symmetric space which admits only one positive restricted root. Then for any \(a_1,a_2\not\in N_{G}(K)\), \(\mu_{a_1}\ast\mu_{a_2}\) is absolutely continuous.
Some related results can be found in [C. F. Dunkl, Trans. Am. Math. Soc. 125, 250–263 (1966; Zbl 0161.33901) and D. L. Ragozin, J. Funct. Anal. 17, 355–376 (1974; Zbl 0297.43002)].