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By identifying U(N) as one member of a dual pair acting in the spinor representation of Spin(4N), the expansion of \u03c9U(s) in powers of s2 is shown to be a decomposition into irreducible characters of U(N). In close analogy with the ballistic nonlinear sigma model, a coherent-state integral representation of \u03c9U(s) is developed. For generic U this integral has (N2N) saddle points and the leading-order saddle-point approximation turns out to reproduce \u03c9U(s)\u2008exactly, up to a constant factor. This miracle is explained by interpreting \u03c9U(s) as a character of U(2N), and arguing that the leading-order saddle-point result corresponds to the Weyl character formula. Unfortunately, the Weyl decomposition behaves nonsmoothly in the semiclassical limit N\u2192\u221e, and to make further progress some additional averaging needs to be introduced. Several schemes are investigated, including averaging over basis states and an \u201cisotropic\u201d average. 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