Elliptic Littlewood identities

We prove analogues for elliptic interpolation functions of Macdonaldʼs version of the Littlewood identity for (skew) Macdonald polynomials, in the process developing an interpretation of general elliptic “hypergeometric” sums as skew interpolation functions. One such analogue has an interpretation as a “vanishing integral”, generalizing a result of Rains and Vazirani (2007) [17]; the structure of this analogue gives sufficient insight to enable us to conjecture elliptic versions of most of the other vanishing integrals of Rains and Vazirani (2007) [17] as well. We are thus led to formulate ten conjectures, each of which can be viewed as a multivariate quadratic transformation, and can be proved in a number of special cases.