Transformations of elliptic hypergeometric integrals

We prove a pair of transformations relating elliptic hypergeometric integrals of different dimensions, corresponding to the root systems BC and An; as a special case, we recover some integral identities conjectured by van Diejen and Spiridonov. For BC, we also consider their "Type II" integral. Their proof of that integral, together with our transformation, gives rise to pairs of adjoint integral operators; a different proof gives rise to pairs of adjoint difference operators. These allow us to construct a family of biorthogonal abelian functions generalizing the Koornwinder polynomials, and satisfying the analogues of the Macdonald conjectures. Finally, we discuss some transformations of Type II-style integrals. In particular, we find that adding two parameters to the Type II integral gives an integral invariant under an appropriate action of the Weyl group E-7.