Haar expectations of ratios of random characteristic polynomials

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 algebra in its canonical realization on the complex A V of holomorphic differential forms for a C-vector space V 0 . From it we construct the Fock representation of an orthosymplectic Lie superalgebra osp associated to V 0 . Particular attention is paid to defining Howe's oscillator semigroup and the representation that partially exponentiates the Lie algebra representation of sp ⊂ 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 0 = C n ⊗ C N where C N is equipped with the standard K-representation, and focus on the subspace A K V 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 ≥ 1. This completes earlier work of Conrey, Farmer, and Zirnbauer for the case of U N .

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