  Perturbative gravity in global de Sitter space is subject to so-called
linearization stability constraints: If they are to couple consistently to the
gravitational field, quantum states must be invariant under the de Sitter
isometries. While standard Fock spaces contain no de Sitter-invariant states
apart from (possibly) the vacuum, a full Hilbert space of de Sitter-invariant
quantum states can be constructed via group averaging techiniques. We
re-examine the simple toy model of de Sitter group averaging given by the free
1+1 scalar field, expanding on an earlier analysis by Higuchi. Our purpose is
twofold: to include the scalar zero-mode, and to explicitly count the number of
de Sitter-invariant states as a function of an appropriately defined energy.
