Moment map and matrix integrals

We discuss the geometry behind some integrals related to structure constants of the Liouville conformal field theory. §0. Introduction. Three incarnations of a quadratic map This note is a followup of [BS]; it is mostly a review of known results. We discuss some geometry lying behind the computations from [ZZ] and [BR], and their p-adic and adelic analogs. 0.1. Moment ternoon. Let K be a field of characteristic 6= 2. We will discuss certain quadratic map between two affine spaces μ : K −→ K (0.1.1) It may be introduced in three ways. (a) As an exterior multiplication μ : K ×K −→ Λ(K), (x, y) 7→ x ∧ y. (0.1a) (b) As a moment map. Regard X(K) = K as the cotangent space to Y (K) = K; the group H = SO3(K) acts on Y (K) in an obvious way; this action is Hamiltonian, and μ is the momentum map, K on the right being identified with h := Lie(H): μ : T Y −→ h (0.1b) 1 This is the archetypical moment map, wherefrom its very name has appeared. Its three components are "angular momenta" . (c) As a quotient map. Identify X(K) with the space of 2×3 matrices; the groupG = SL2(K) acts upon X(K) from the left, and μ may be identified (at least birationally) with the quotient map μ : X(K) −→ G(K)\X(K) = Y (K), (0.1c) Y being identified with a categorical quotient of X, by the Igusa criterion, cf. [I]. 0.2. In [BR] the map μ (for K = R) has been used for a computation of certain triple integral I(a, b, c;R), a, b, c ∈ C over Y (R), see (1.1.7) below. A similar integral for K = C has appeared previously in [ZZ] (cf. also [Z]). We can take K to be a nonarchimedian local field; the integral I(a, b, c;Qp) has been introduced and computed in [BS]. In op. cit. a q-deformation of I(a, b, c) is discussed as well. The upshot of the trick from [BR] is that an integral I(a, b, c;K) over Y (K) is represented as a ratio of two Gaussian integrals over X(K) and Y (K). From our viewpoint it might be considered as an integral over a fiber Xt := μ(t), t ∈ Y (K), and indeed, in the original definition in [ZZ] I(a, b, c;C) has appeared as an integral over G(C), see §1 below. 0.3. I am much obliged to M.Finkelberg for consultations; among others things he explained to me that 0.1 is a particular case (and a part) of a general superalgebra construction described e.g. in [BFT], see §4 below. §1. Some geometry behind an integral 1.1. Complex and real integrals. The following integral appears in [ZZ] (4.17) I(σ1, σ2, σ3;C) = ∫