Open Access
Summer 2011 Quantum continuous gl: Semiinfinite construction of representations
B. Feigin, E. Feigin, M. Jimbo, T. Miwa, E. Mukhin
Kyoto J. Math. 51(2): 337-364 (Summer 2011). DOI: 10.1215/21562261-1214375


We begin a study of the representation theory of quantum continuous gl, which 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 E are labeled by a continuous parameter uC. The representation theory of E has many properties familiar from the representation theory of gl: vector representations, Fock modules, and semiinfinite constructions of modules. Using tensor products of vector representations, we construct surjective homomorphisms from E to spherical double affine Hecke algebras SN 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.


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B. Feigin. E. Feigin. M. Jimbo. T. Miwa. E. Mukhin. "Quantum continuous gl: Semiinfinite construction of representations." Kyoto J. Math. 51 (2) 337 - 364, Summer 2011.


Published: Summer 2011
First available in Project Euclid: 22 April 2011

zbMATH: 1278.17012
MathSciNet: MR2793271
Digital Object Identifier: 10.1215/21562261-1214375

Primary: 05E10 , 17B37 , 81R10

Rights: Copyright © 2011 Kyoto University

Vol.51 • No. 2 • Summer 2011
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