Real horo-hypercircle coordinate (Q1919017)

In this short note the author describes ``horo-hypercircle'' coordinates \((u_1, \dots, u_m,u_{m+1}, \dots, u_n)\) on Riemannian symmetric spaces of type III. They lead to a simple expression of its Laplace-Beltrami operator as \(\Delta_m+ \Sigma\partial_j \circ L_j\), where \(\Delta_m\) is a constant coefficients elliptic operator in \((u_1, \dots, u_m)\), \(L_j\) is a first order differential operator, and \(j\) runs over \(m+1, \dots, n\) only. One can then deduce the heat kernel of the symmetric space from the classical heat kernel of \(\Delta_m\).