  For the classical compact Lie groups K = U(N) the autocorrelation functions
of ratios of random characteristic polynomials are studied. Basic to our
treatment is a property shared by the spinor representation of the spin group
with the Shale-Weil representation of the metaplectic group: in both cases the
character is the analytic square root of a determinant or the reciprocal
thereof. By combining this fact with Howe's theory of supersymmetric dual pairs
(g,K), we express the K-Haar average product of p ratios of characteristic
polynomials and q conjugate ratios as a character which is associated with an
irreducible representation of the Lie superalgebra g = gl(n|n) for n = p+q.
This primitive character is shown to extend to an analytic radial section of a
real supermanifold related to gl(n|n), and is computed by invoking Berezin's
description of the radial parts of Laplace-Casimir operators for gl(n|n). The
final result for the character looks like a natural transcription of the Weyl
character formula to the context of highest-weight representations of Lie
supergroups.
  While several other works have recently reproduced our results in the stable
range where N is no less than max(p,q), the present approach covers the full
range of matrix dimensions N.
  To make this paper accessible to the non-expert reader, we have included a
chapter containing the required background material from superanalysis.
