Level-0 structure of level-1 $U_q({\widehat{sl}}_2)$-modules and Macdonald polynomials

Abstract

The level-$1$ integrable highest weight modules of $U_q({\widehat{sl}}_2)$ admit a level-$0$ action of the same algebra. This action is defined using the affine Hecke algebra and the basis of the level-$1$ module generated by components of vertex operators. Each level-$1$ module is a direct sum of finite-dimensional irreducible level-$0$ modules, whose highest weight vector is expressed in terms of Macdonald polynomials. This decomposition leads to the fermionic character formula for the level-$1$ modules.

22 pages, LaTeX 2.09 (and amsfonts preferably). (Minor corrections)

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• Anisotropic spin generalization of elliptic Macdonald-Ruijsenaars operators and R-matrix identities
• Representations of the Quantum Toroidal Algebra on highest weight modules of the Quantum Affine Algebra of type gl(N)
• The decomposition of level-1 irreducible highest weight modules with respect to the level-0 actions of the quantum affine algebra $U_q^{\prime }({\widehat{sl}}_n)$
• An integrable $U_q(\widehat{gl}(2|2){)}_1$-Model: Corner Transfer Matrices and Young Skew Diagrams
• Shuffle approach to wreath Pieri operators
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