Quantum continuous gl∞: Semiinfinite construction of representations

We begin a study of the representation theory of quantum continuous gl(infinity), winch we denote by E. This algebra depends on two parameters and is a deformed version of the enveloping algebra of the Lie algebra of difference operators acting on the space of Laurent polynomials in one variable. Fundamental representations of epsilon are labeled by a continuous parameter u is an element of C. The representation theory of epsilon has many properties familiar from the representation theory of gl(infinity) : vector representations, Fock modules, and semiinfinite constructions of modules. Using tensor products of vector representations, we construct surjective homomorphisms from epsilon to spherical double affine Hecke algebras S(H) over dot(N) for all N. A key step in this construction is an identification of a natural basis of the tensor products of vector representations with Macdonald polynomials. We also show that one of the Fock representations is isomorphic to the module constructed earlier by means of the K-theory of Hilbert schemes.