Absolute continuity of convolutions of orbital measures on Riemannian symmetric spaces
Let \(G\) be a semisimple noncompact connected Lie group with finite center, and let \(K\) be a maximal compact subgroup of \(G,\) and let \(X=G\slash K\) be the corresponding symmetric space of noncompact type. Let \(\mathfrak g=\mathfrak k+\mathfrak p\) be the Cartan decomposition and let \(\mathfrak a\) be a chosen maximal abelian subalgebra of \(\mathfrak p.\) Let \(m_K\) denote the Haar measure on \(K,\) and let \(*\) denote the convolution on the group \(G.\) For \(X\in\mathfrak a\), define \(\delta_{e^X}^\natural=m_K*\delta_{e^X}*m_K\) and put \(m_{X_1,\dots,X_m}=\delta_{e^{X_1}}^\natural*\dots*\delta_{e^{X_m}}^\natural.\) The authors show that, for a Riemannian symmetric space of noncompact type of rank \(r,\) the measure \(m_{X_1,\dots,X_{r+1}}\) is absolutely continuous provided that, for all \(i,\) \(X_i\not=0,\) and they conjecture that \(r+1\) is optimal for this property (they prove that the conjecture is valid on \(X=SL(n,F)\slash SU(n,F),\) where \(F=\mathbb R,\) \(\mathbb C\) or \(\mathbb H\)). Some other properties of the support and the density of \(m_{X_1,\ldots,X_m}\) are given. Moreover, they give a sufficient and necessary condition on \(l\) such that \((\delta_{e^X}^\natural)^{*l}\) has a density.
scientific article; zbMATH DE number 5779349