Invariant fundamental solutions and solvability for symmetric spaces of type \( G_C / G_ R \) with only one conjugacy class of Cartan subspaces
Let \(G\) be a~reductive complex connected Lie group with Lie algebra~\(\mathfrak g\) and let \(H\) be a~real form of~\(G\) with Lie algebra~\(\mathfrak h\). Let \(\sigma\) denote the conjugation of~\(\mathfrak g\) relative to~\(\mathfrak h\) and let also \(\sigma\) denote the involution of~\(G\) the differential of which is~\(\sigma\). Then \(H\)~is the open connected subgroup of~\(G^\sigma\), the fixed-point set of~\(\sigma\) in~\(G\). The space \(G/H\) is said to be a~reductive symmetric space of type \(G_{\mathbf C}/G_{\mathbf R}\). Let \(D:C^\infty(G/H)\to C^\infty(G/H)\) be a~nontrivial \(G\)-invariant differential operator. An invariant fundamental solution for~\(D\) is a~left-\(H\)-invariant distribution~\(T\) on~\(G/H\) solving the differential equation \(DT=\delta\) where \(\delta\)~is the Dirac measure at the origin of~\(G/H\). Let \(A\) denote the associated Cartan subset of~\(G/H\), identified with a~real abelian subgroup of~\(G\). Using results from the theory of orbit integrals on~\(G/H\) obtained by \textit{A. Bouaziz}, \textit{P. Harinck}, and \textit{S. Sano} the differential equation \(DT=\delta\) can be reduced to the differential equation \(\Gamma(D)T_A=\delta_A\) for some distribution~\(T_A\) on~\(A\) where \(\Gamma(D)\)~is a~uniquely defined differential operator with constant coefficients on~\(A\) and \(\delta_A\)~is the Dirac measure at the origin of~\(A\), i.e., \(T_A\)~is a~fundamental solution for~\(\Gamma(D)\). The main result of the paper says that, in the above context, \(D\)~has an invariant fundamental solution on~\(G/H\) if \(\Gamma(D)\)~has a~fundamental solution on~\(A\).
scientific article; zbMATH DE number 2057363