Haar expectations of ratios of random characteristic polynomials

TL;DR

This paper computes exact Haar ensemble averages of ratios of characteristic polynomials for classical Lie groups, expressing them through Weyl-type character formulas, extending previous results for unitary groups to orthogonal and symplectic groups.

Contribution

It provides explicit formulas for Haar averages of ratios of characteristic polynomials for O(N), SO(N), and USp(N), using Lie superalgebra and duality techniques, generalizing prior U(N) results.

Findings

Explicit Haar expectation formulas for classical groups.

Connection of expectations to Weyl-type character formulas.

Extension of previous U(N) results to orthogonal and symplectic groups.

Abstract

We compute Haar ensemble averages of ratios of random characteristic polynomials for the classical Lie groups K = O(N), SO(N), and USp(N). To that end, we start from the Clifford-Weyl algebera in its canonical realization on the complex of holomorphic differential forms for a C-vector space V. From it we construct the Fock representation of an orthosymplectic Lie superalgebra osp associated to V. Particular attention is paid to defining Howe's oscillator semigroup and the representation that partially exponentiates the Lie algebra representation of sp in osp. In the process, by pushing the semigroup representation to its boundary and arguing by continuity, we provide a construction of the Shale-Weil-Segal representation of the metaplectic group. To deal with a product of n ratios of characteristic polynomials, we let V = C^n \otimes C^N where C^N is equipped with its standard K-representation, and focus on the subspace of K-equivariant forms. By Howe duality, this is a highest-weight irreducible representation of the centralizer g of Lie(K) in osp. We identify the K-Haar expectation of n ratios with the character of this g-representation, which we show to be uniquely determined by analyticity, Weyl group invariance, certain weight constraints and a system of differential equations coming from the Laplace-Casimir invariants of g. We find an explicit solution to the problem posed by all these conditions. In this way we prove that the said Haar expectations are expressed by a Weyl-type character formula for all integers N \ge 1. This completes earlier work by Conrey, Farmer, and Zirnbauer for the case of U(N).