  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).
