## Kyoto Journal of Mathematics

### Quantum continuous $\mathfrak{gl}_{\infty}$: Semiinfinite construction of representations

#### Abstract

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.

#### Article information

Source
Kyoto J. Math. Volume 51, Number 2 (2011), 337-364.

Dates
First available in Project Euclid: 22 April 2011

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1303494506

Digital Object Identifier
doi:10.1215/21562261-1214375

Mathematical Reviews number (MathSciNet)
MR2793271

Zentralblatt MATH identifier
1278.17012

#### Citation

Feigin, B.; Feigin, E.; Jimbo, M.; Miwa, T.; Mukhin, E. Quantum continuous $\mathfrak{gl}_{\infty}$ : Semiinfinite construction of representations. Kyoto J. Math. 51 (2011), no. 2, 337--364. doi:10.1215/21562261-1214375. https://projecteuclid.org/euclid.kjm/1303494506

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