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The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials

Published online by Cambridge University Press:  07 July 2010

O. Schiffmann
Affiliation:
Institut Mathématique de Jussieu, Université de Paris 6, 175 rue du Chevaleret, 75013 Paris, France (email: [email protected])
E. Vasserot
Affiliation:
Institut Mathématique de Jussieu, Université de Paris 7, 175 rue du Chevaleret, 75013 Paris, France (email: [email protected])
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Abstract

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We exhibit a strong link between the Hall algebra HX of an elliptic curve X defined over a finite field 𝔽l (or, more precisely, its spherical subalgebra U+X) and Cherednik’s double affine Hecke algebras of type GLn, for all n. This allows us to obtain a geometric construction of the Macdonald polynomials Pλ(q,t−1) in terms of certain functions (Eisenstein series) on the moduli space of semistable vector bundles on the elliptic curve X.

Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2010

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