  Static black holes contain regions of spacetime which not even light can
escape from. In the centre of mass frame, these blocks are separated from each
other by event horizons. Unlike pointlike particles, fields can spread and
interact non-causally across the horizons. The microlocal theory of this is
somewhat incomplete however. For instance, the theory of real principal type
operators does not apply on the horizon. In this article, we address this issue
for the extremal rotating black hole. Namely, we show that null covectors on
the horizon of the extremal Kerr spacetime form an involutive double
characteristic manifold and then extend the construction of parametrix across
the event horizon. This provides a mathematical basis for the asymptotic
oscillatory solutions in the region. Such approximations are central in quantum
mechanics. In contrast to the real principal case, solutions near the horizon
admit two channels for propagation of singularities. Indeed, there is
additional propagation along the double characteristic variety.
