Howe pairs, supersymmetry, and ratios of random characteristic polynomials for the unitary groups U(N)

Abstract

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.