Integration on the \(n\)th power of a hyperbolic space in terms of invariants under diagonal action of isometries (Lorentz transformations) (Q917721)

Let H be a hyperboloid in the real m-dimensional space \(R^ m\), and let G be the isometry group of H. Let \(H^ n\) be the n-th power of H, i.e. the set of points \((p_ 1,p_ 2,...,p_ n)\), \(p_ j\in H\). There is a natural action of the group G in \(H^ n\). This action divides \(H^ n\) into orbits \(O_ y\). Let A be the set of orbits \(O_ y\). The main result of the paper is factorization of the measure on \(H^ n\) into the product of invariant measures on the orbits \(O_ y\) and the measure on the set A of orbits. It is shown that the Jacobian, which appears under this factorization, is expressed in terms of certain determinants. The case, when the hyperboloid is replaced by the sphere is also considered.