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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 $E_7$.
@article{10_4007_annals_2010_171_169,
author = {Eric M. Rains},
title = {Transformations of elliptic hypergeometric integrals},
journal = {Annals of mathematics},
pages = {169--243},
publisher = {mathdoc},
volume = {171},
number = {1},
year = {2010},
doi = {10.4007/annals.2010.171.169},
mrnumber = {2630055},
zbl = {1209.33014},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.171.169/}
}
TY - JOUR AU - Eric M. Rains TI - Transformations of elliptic hypergeometric integrals JO - Annals of mathematics PY - 2010 SP - 169 EP - 243 VL - 171 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.171.169/ DO - 10.4007/annals.2010.171.169 LA - en ID - 10_4007_annals_2010_171_169 ER -
Eric M. Rains. Transformations of elliptic hypergeometric integrals. Annals of mathematics, Tome 171 (2010) no. 1, pp. 169-243. doi: 10.4007/annals.2010.171.169
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