Residue construction of Hecke algebras

Abstract

Hecke algebras are usually defined algebraically, via generators and relations. We give a new algebra-geometric construction of affine and double-affine Hecke algebras (the former is known as the Iwahori-Hecke algebra, and the latter was introduced by Cherednik) in terms of residues. More generally, to any generalized Cartan matrixAand a pointqin a one-dimensional complex algebraic group C we associate an associate algebraHq. IfAis of finite type and C=C×, the algebraHqis theaffineHecke algebra of the corresponding finite root system. IfAis of affine type and C=C×, thenHqis, essentially, the Cherednik algebra. The case C=C+corresponds to "degenerate" counterparts of the above objects considered by Drinfeld and Lusztig. Finally, taking C to be an elliptic curve, one gets some new elliptic analogues of the affine Hecke algebra. © 1997 Academic Press.