  An attempt is made to clarify the ballistic non-linear sigma model formalism
recently proposed for quantum chaotic systems, by the spectral determinant
Z(s)=Det(1-sU) of a quantized map U element of U(N). More precisely, we study
the correlator omega_U(s)=<|Z(st)|^2> (averaging t over the unit circle).
Identifying the group U(N) as one member of a dual pair acting in the spinor
representation of Spin(4N), omega_U(s) is expanded in terms of irreducible
characters of U(N). In close analogy with the ballistic non-linear sigma model,
a coherent-state integral representation of omega_U(s) is developed. We show
that the leading-order saddle-point approximation reproduces omega_U(s)
exactly, up to a constant factor; this miracle can be explained by interpreting
omega_U(s) as a character of U(2N), for which the saddle-point expansion yields
the Weyl character formula. Unfortunately, this decomposition behaves
non-smoothly in the semiclassical limit, and to make further progress some
averaging over U needs to be introduced. Several averaging schemes are
investigated. In general, a direct application of the saddle-point
approximation to these schemes is demonstrated to give incorrect results; this
is not the case for a `semiclassical averaging scheme', for which all loop
corrections vanish identically. As a side product of the dual pair
decomposition, we compute a crossover between the Poisson and CUE ensembles for
omega_U(s).
