Transformations of elliptic hypergometric integrals

TL;DR

This paper establishes new transformations between elliptic hypergeometric integrals related to root systems BC_n and A_n, leading to novel biorthogonal functions and symmetries connected to Weyl groups.

Contribution

It introduces new transformations of elliptic hypergeometric integrals, constructs biorthogonal functions, and uncovers symmetries involving Weyl groups, advancing the understanding of these integrals.

Findings

Derived transformations relating BC_n and A_n integrals.

Constructed biorthogonal functions generalizing Koornwinder polynomials.

Identified Weyl group symmetries in Type II integrals.

Abstract

We prove a pair of transformations relating elliptic hypergeometric integrals of different dimensions, corresponding to the root systems BC_n and A_n; as a special case, we recover some integral identities conjectured by van Diejen and Spiridonov. For BC_n, 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.