# Particles in curved space-time and group representation

It is well-known, that particles in Minkowski space can be constructed as unitary projective representations of the Poincaré group, i.e. isometry group of Minkowski space: $$M_d= \frac{Poincare_d}{SO(1,d-1)}$$

Is such construction for another maximal-symmetric space-times:

$$AdS_d = \frac{SO(2, d-1)}{SO(1, d-1)}\;\;\;\;\;\; and \;\;\;\;\;\; dS_d =\frac{SO(1,d-1)}{SO(1, d-2)}$$

For example, in AdS there is special representation -- singeleton. It's field representation are pure gauge degrees of freedom and do not propogate. However any massless field in AdS can be constructed by taking the product of two of them.