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 .

We consider free affine actions of unipotent complex algebraic groups on C n and prove that such actions admit an analytic geometric quotient if their degree is at most 2. Moreover, we classify free affine C 2 -actions on C n of degree n − 1 and n − 2. For every n > 4, an action of degree n − 2 appears in the classification whose quotient topology is not Hausdorff.

Abstract. Let X be a holomorphically separable irreducible reduced complex space, K a connected compact Lie group acting on X by holomorphic transformations, θ : K → K a Weyl involution, and µ : X → X an antiholomorphic map satisfying µ 2 = Id and µ(kx) = θ(k)µ(x) for x ∈ X, k ∈ K. We show that if O(X) is a multiplicity free K-module, then µ maps every K-orbit onto itself. For a spherical affine homogeneous space X = G/H of the reductive group G = K C we construct an antiholomorphic map µ with these properties.

We consider pairs (V, H) of subgroups of a connected unipotent complex Lie group G for which the induced V ×H-action on G by multiplication from the left and from the right is free. We prove that this action is proper if the Lie algebra g of G is 3-step nilpotent. If g is 2-step nilpotent, then there is a global slice of the action that is isomorphic to C n . Furthermore, a global slice isomorphic to C n exists if dim V = 1 = dim H or dim V = 1 and g is 3-step nilpotent. We give an explicit example of a 3-step nilpotent Lie group and a pair of 2-dimensional subgroups such that the induced action is proper but the corresponding geometric quotient is not affine.Research supported by SFB/TR12 "Symmetrien und Universalität in mesoskopischen Systemen" of the Deutsche Forschungsgemeinschaft.

We consider free algebraic actions of the additive group of complex numbers on a complex vector space X embedded in the complex projective space. We find an explicit formula for the map p that assigns to a generic point x ∈ X the Chow point of the closure of the orbit through x. The properties Hausdorff quotient topology and proper action are equivalently characterized by the closure of the image of p in the closed Chow variety.