We exhibit a strong link between the Hall algebra HX of an elliptic curve X defined over a finite field 𝔽l (or, more precisely, its spherical subalgebra U+X) and Cherednik’s double affine Hecke algebras of type GLn, for all n. This allows us to obtain a geometric construction of the Macdonald polynomials Pλ(q,t−1) in terms of certain functions (Eisenstein series) on the moduli space of semistable vector bundles on the elliptic curve X.

A new type of basic hypergeometric series based on Macdonald polynomials is introduced. Besides a pair of Macdonald polynomials attached to two different sets of variables, a key ingredient in the basic hypergeometric series is a bisymmetric function related to Macdonald’s commuting family of q-difference operators, to the Selberg integrals of Tarasov and Varchenko, and to alternating sign matrices. Our main result for series is a multivariable generalization of the celebrated q-binomial theorem. In the limit this q-binomial sum yields a new Selberg integral for Jack polynomials.

We present an explicit construction of the Macdonald polynomials and of the Heckman–Opdam generalized Jacobi polynomials associated with root systems.

We prove the existence of an involution on double affine Hecke algebras. This involution plays a central role in the theory of Macdonald polynomials, being responsible for a Fourier type duality and allowing the construction of affine intertwiners.

We construct certain raising operators of row type for Macdonald's symmetric polynomials by an interpolation method.

We give a short proof of the inner product conjecture for the symmetric Macdonald polynomials of type A$_n-1$. As a special case, the corresponding constant term conjecture is also proved.

We extend some results about shifted Schur functions to the general context of shifted Macdonald polynomials. We strengthen some theorems of F. Knop and S. Sahi and give two explicit formulas for these polynomials: a q-integral representation and a combinatorial formula. Our main tool is a q-integral representation for ordinary Macdonald polynomial. We also discuss duality for shifted Macdonald polynomials and Jack degeneration of these polynomials.