Quantum continuous \(\mathfrak{gl}_{\infty}\): semiinfinite construction of representations. (English) Zbl 1278.17012

Summary: We begin a study of the representation theory of quantum continuous \(\mathfrak{gl}_{\infty}\), which we denote by \(\mathcal{E}\). 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 \(\mathcal{E}\) are labeled by a continuous parameter \(u\in \mathbb{C}\). The representation theory of \(\mathcal{E}\) has many properties familiar from the representation theory of \(\mathfrak{gl}_{\infty}\): vector representations, Fock modules, and semiinfinite constructions of modules. Using tensor products of vector representations, we construct surjective homomorphisms from \(\mathcal{E}\) to spherical double affine Hecke algebras \(S\ddot{H}_{N}\) 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.
See also the authors’ follow-up paper [Kyoto J. Math. 51, No. 2, 365–392 (2011; Zbl 1278.17013)].


17B37 Quantum groups (quantized enveloping algebras) and related deformations
05E10 Combinatorial aspects of representation theory
20C08 Hecke algebras and their representations
33D52 Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)


Zbl 1278.17013
Full Text: DOI arXiv


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