Louisiana State University
The Huyghens' Principle and Symmetric Space Duality (joint work with T. Branson)
It has been know for some time that the wave equation on a Lorentzian space of the form $G/K \times R$, $G$ non compact and semisimple, satisfy the Huyghens' principle (HP) if and only if the group $G$ has one conjugacy class of Cartan subalgebras. On the other hand the answer to the same problem for $G$ compact was much harder. On a Lorentzian space the strong HP for a shifted Laplace-d'Alambert operator $\square +b$ is equivalent to the property of vanishing logarithmic terms (VLT) for the fundamental solution of the operator. Our main result is that the property VLT is invariant under symmetric space duality. This, in particular, implies that a compact symmetric space $U/K$ satisfy the VLT (and hence the local HP holds) if and only if the corresponding non-compact dual space $G/K$ satisfy the HP.