Group averaging and BRST quantization in de Sitter space

In this note we give a BRST interpretation of inner product of group averaging on de Sitter group SO(2,1). Quantization approaches can be divided in two classes, without introducing additional degrees of freedom and with introducing ghosts, antighosts, Lagrange multipliers and canonically conjugated momenta. The first class of quantization methods includes Dirac method and group averaging method. The Dirac method is based on imposing the constraint conditions on physical states, while in group averaging method one modifies the physical inner product due to constrains. One should distinguish the abelian case for which you can have continuous spectrum and discrete spectrum for which there are internal topological problems in BRST-BFV approach, and the non-abelian case.