Extended double Riemann–Hilbert problem and multiple representations of the infinite-dimensional symmetric group for the stationary axisymmetric vacuum field equations

Abstract

The double-complex function method suggested in previous papers is extended and the extended method is used to establish two mutually dual extended double Hauser–Ernst equations and related Riemann–Hilbert problems, then quadruple representations of the semidirect product of the Kac–Moody and Virasoro groups are given. These show that, for the stationary axisymmetric vacuum space–times, there are more and richer symmetry structures than previously expected. The multiple forms of some well-known transformations can be obtained as some special cases.