Quantum continuous $\mathfrak{gl}_\infty$: Semi-infinite construction of representations

TL;DR

This paper explores the representation theory of quantum continuous rak{gl}_\u221e, introducing new constructions of modules, homomorphisms to double affine Hecke algebras, and connections to Macdonald polynomials and Hilbert schemes.

Contribution

It provides a semi-infinite construction of representations of quantum continuous rak{gl}_ and links these to double affine Hecke algebras and geometric models.

Findings

Constructed surjective homomorphisms to spherical double affine Hecke algebras.

Identified bases of tensor products with Macdonald polynomials.

Connected Fock representations to K-theory of Hilbert schemes.

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, semi-infinite 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 bases 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.