Infinitely differentiable functions invariant on the tangent space of a symmetric space (Q1275260)

Let \(G\) be a connected reductive Lie group, \(\sigma\) an involution of \(G\) and \(H\) the identity component of the group of its fixed points under \(\sigma\). Let \({\mathfrak g}\) be the Lie algebra of \(G\) and \({\mathfrak g}= {\mathfrak h}+{\mathfrak q}\) its decomposition into \(+1\)(resp., \(-1)\)-eigenspaces for \(\sigma\). Use Harish-Chandra's descent method from the Lie algebra to the tangent space of a symmetric space. The author obtains a necessary and sufficient condition for a function \(f\) defined on \({\mathfrak q}\) to be \(C^\infty\) and \(H\)-invariant, that is: (1) the restriction of \(f\) on any Cartan subspace \({\mathfrak a}\) of \({\mathfrak q}\) remains \(C^\infty\) and is \(W(H,{\mathfrak a})\)-invariant, where \(W(H,{\mathfrak a})\) is the Weyl group of \(H\) in \({\mathfrak a}\); (2) roughly speaking, the restrictions of \(f\) on two different Cartan subspaces of \({\mathfrak q}\) have the same derivatives if these two Cartan subspaces are properly connected. - If \(G\) takes the form \(G'\times G'\), \(G'\) is a reductive Lie group, and the involution \(\sigma:G'\times G'\to G'\times G'\) is defined by \(\sigma (x,y)=(y,x)\), then the above characterization of the \(C^\infty\)-function \(f\) on \({\mathfrak q}\) reduces to a result of \textit{A. Bouaziz} [C. R. Acad. Sci., Paris, Sér. I, 314, 9-12 (1992; Zbl 0784.43007)].