Eisenstein series and quantum affine algebras

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  • 27 April 1996
Abstract
Let X be a smooth projective curve over a finite field. We consider the Hall algebra H whose basis is formed by the isomorphism classes of all coherent sheaves on X and whose typical structure constant is the number of subsheaves in a given sheaf of given isomorphism class and with given isomorphism class of quotient. We study this algebra in detail. It turns out that its structure is strikingly similar to that of quantized affine algebras in their loop realization of Drinfeld. In this analogy the set of simple roots of a semisimple Lie algebra g corresponds to the set of cusp forms on X, the entries of the Cartan matrix of g to the values of Rankin-Selberg L-functions associated to two cusp forms. For the simplest instances of both theories (the curve P^1, the Lie algebra g=sl_2) the analogy is in fact the identity.

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