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).
Highlights
1.1 Comparison with results of other approaches 1.2 Howe duality and weight expansion 1.3 Group representation and differential equations 2 Howe dual pairs in the orthosymplectic Lie superalgebra 2.1 Notion of Lie superalgebra 2.2 Structure of osp(W) 2.3 Howe pairs in osp(W) 2.4 Clifford–Weyl algebra q(W ) 2.5 osp(W) inside q(W) 2.6 Spinor–oscillator representation 2.7 Real structuresHuckleberry et al Complex Analysis and its Synergies (2016) 2:1
3.1 The oscillator semigroup Here we introduce the basic semigroup in the complex symplectic group
Recall that we are concerned with the Lie algebra representation of spR ⊂ sp which is defined by the identification of sp with the set of symmetrized elements of degree two in the Weyl algebra w(W ) and the representation of w(W ) on a(V )
Summary
K where K is one of the classical compact Lie groups ON, SON, or USpN equipped with Haar measure dk of unit mass K dk = 1 and. Our explicit formula for I(t) looks exactly like a classical Weyl formula and is derived in terms of the roots of the Lie superalgebra g and the Weyl group W Let us state this formula for K = ON, USpN without going into the details of the -positive even and odd roots. To prove this formula we establish certain properties of I(t) which uniquely characterize it and are satisfied by the right-hand side.
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