Michal, Thank you. This is what I was looking for, with the added bonus of seeing the relevance of diffeomorphism groups to hydrodynamics.

Jeffrey -- thanks for the reference -- and for your previous answer to what is known about the unitary dual of $O(p,q)$.

Emerton - thank you for pointing me to the Vogan paper. This has opened up a useful search direction. --Mark

Although the question is motivated by physics considerations, I believe the question is mathematically well-posed: is there a classification of the irreducible unitary representations of the Lie groups O(4,1) and O(3,2), and more generally for O(p,q) for all p and q. Hope this clarifies the question a bit.

Thanks for the responses so far. To try to sharpen my question a bit, let me give a bit more context. In 1939 Wigner classified all of the irreducible unitary representations of the Poincare group (semi-direct product of the Lorentz group and the group of translations in 3+1 dimensional Minkowski space). Minkowski space is flat, but deSitter and anti-deSitter space have constant curvature, and are also maximally symmetric, with isometry groups O(4,1) and O(3,2). I would be quite happy to understand the classification of their (infinite-dimensional) irreducible unitary representations.