The many faces of the elliptic beta integral

The elliptic beta integral, a contour integral of a particular product of elliptic gamma functions, admits an explicit evaluation. This formula represents (i) an elliptic binomial theorem, (ii) top known univariate extension of the Euler beta integral, (iii) a germ for building an elliptic analogue of the Euler-Gauss hypergeometric function and of very many elliptic hypergeometric integrals on root systems, (iv) a normalization of the measure for biorthogonal functions comprising all classical systems of orthogonal functions, (v) an integral operator realization of the Coxeter relations of a permutation group, (vi) a confinement criterion in a four-dimensional supersymmetric quantum field theory. After a brief explanation of the above points, I'll discuss a recent proposal by Kels of an extension of this identity by addition of discrete parameters related to the lens spaces.