Left-invariant metrics with hidden symmetries and integrating the Laplace-Beltrami equation on Lie groups

In this talk, we consider left-invariant pseudo-Riemannian metrics on Lie groups and the corresponding Laplace-Beltrami equations. We discuss an integration method of these equations, which is based on a “non-commutative” version of the harmonic analysis on Lie groups. The main steps of this method are: 1) to construct the special family of irreducible representations of the respective Lie algebra; 2) to construct the generalized Fourier transform on the Lie group; 3) to solve the equation that is the Fourier-image of the original Laplace-Beltrami equation. In the talk, we also consider a special class of left-invariant metrics, whose Laplace-Beltrami operators become, after the generalized Fourier transform, first-order differential operators. It turns out that, in general, these Laplace-Beltrami operators admit additional hidden symmetries belonging to the class of integro-differential operators. The talk concludes a few examples.