Quantum continuous gl_∞: Semiinfinite construction of representations

We begin a study of the representation theory of quantum continuous gl _α, which we denote by ε. 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 ε are labeled by a continuous parameter u ∈ ℂ. The representation theory of ε has many properties familiar from the representation theory of gl_α: vector representations, Fock modules, and semiinfmite constructions of modules. Using tensor products of vector representations, we construct surjective homomorphisms from ε to spherical double affine Hecke algebras SḦ_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.