Residue Construction of Hecke Algebras

✍ Scribed by Victor Ginzburg; Mikhail Kapranov; Eric Vasserot

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 393 KB
• Volume: 128
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1997.1620

✦ Synopsis

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 matrix A and a point q in a one-dimensional complex algebraic group C we associate an associate algebra H q . If A is of finite type and C=C _ , the algebra H q is the affine Hecke algebra of the corresponding finite root system. If A is of affine type and C=C _ , then H q is, 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

Let W be the Weyl group and T the maximal torus of a Kac Moody group associated to the Cartan matrix A. Write X * (T ) for the lattice of one-parameter subgroups in T and C(T ) for the field of rational functions on T with the natural W-action. There is a well-known description of affine Hecke algebra as the subalgebra of End C C(T) generated by the so-called Demazure Lusztig operators, cf. [Lu]. We observe that all the operators obtained in this way have poles of a very special kind. Thus, for any 1-dimensional algebraic group C, we can give an ``external'' definition of an article no. AI971620 1 0001-8708Â97 25.00