  We investigate refined algebraic quantisation with group averaging in a
finite-dimensional constrained Hamiltonian system that provides a simplified
model of general relativity. The classical theory has gauge group SL(2,R) and a
distinguished o(p,q) observable algebra. The gauge group of the quantum theory
is the double cover of SL(2,R), and its representation on the auxiliary Hilbert
space is isomorphic to the (p,q) oscillator representation. When p>1, q>1 and
p+q == 0 (mod 2), we obtain a physical Hilbert space with a nontrivial
representation of the o(p,q) quantum observable algebra. For p=q=1, the system
provides the first example known to us where group averaging converges to an
indefinite sesquilinear form.
