Haar expectations of ratios of random characteristic polynomials Article Swipe
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· 2016
· Open Access
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· DOI: https://doi.org/10.1186/s40627-015-0005-3
· OA: W1712443865
We compute Haar ensemble averages of ratios of random characteristic\npolynomials for the classical Lie groups K = O(N), SO(N), and USp(N). To that\nend, we start from the Clifford-Weyl algebera in its canonical realization on\nthe complex of holomorphic differential forms for a C-vector space V. From it\nwe construct the Fock representation of an orthosymplectic Lie superalgebra osp\nassociated to V. Particular attention is paid to defining Howe's oscillator\nsemigroup and the representation that partially exponentiates the Lie algebra\nrepresentation of sp in osp. In the process, by pushing the semigroup\nrepresentation to its boundary and arguing by continuity, we provide a\nconstruction of the Shale-Weil-Segal representation of the metaplectic group.\n To deal with a product of n ratios of characteristic polynomials, we let V =\nC^n \\otimes C^N where C^N is equipped with its standard K-representation, and\nfocus on the subspace of K-equivariant forms. By Howe duality, this is a\nhighest-weight irreducible representation of the centralizer g of Lie(K) in\nosp. We identify the K-Haar expectation of n ratios with the character of this\ng-representation, which we show to be uniquely determined by analyticity, Weyl\ngroup invariance, certain weight constraints and a system of differential\nequations coming from the Laplace-Casimir invariants of g. We find an explicit\nsolution to the problem posed by all these conditions. In this way we prove\nthat the said Haar expectations are expressed by a Weyl-type character formula\nfor all integers N \\ge 1. This completes earlier work by Conrey, Farmer, and\nZirnbauer for the case of U(N).\n
