Last update: 28 February 2013
Let be a reduced root system and let be the root system formed by the coroots for Let be the Weyl group of and fix a system of positive roots in Let be the corresponding simple roots and let be the corresponding simple reflections in The fundamental weights are defined by the equations and the lattices
are the weight lattice and the root lattice, respectively. The Dynkin diagrams and the corresponding extended Dynkin diagrams are given in Figure 3. If is a Dynkin diagram or extended Dynkin diagram define
The extended affine Weyl group is the group
for and where corresponds to translation by Define by the equation
see [Bou1968 Ch. IV §1 no. 2.1]. The subgroup of is presented by generators and relations
where the are determined from the extended Dynkin diagram of the root system Define
is the longest element of and is the longest element of the group see [Bou1968 Ch. IV §2 Prop. 6]. Then and each element corresponds to an automorphism of the extended Dynkin diagram of in the sense that
where is the permutation of the nodes determined by the automorphism. Equation (4.4) means that The usual length function on the Coxeter group is extended to the group by
Let be a lattice such that View as a subgroup of and let
Then and is a subgroup of
Let be a lattice such that The affine braid group is the group given by generators and relations
Let View as a subgroup of Then is presented by generators and relations
where is as in (4.4), and the are specified by the extended Dynkin diagram of
Let be the dominant weights in Define elements by
By [Mac1423624, 3.4] and [Lus1989], the are well defined and do not depend on the choice and
Then if and only if
Fix The affine Hecke algebra is the quotient of the group algebra by the relations
In (see [Mac1995, 4.2]),
The Iwahori-Hecke algebra is the subalgebra of generated by
To our knowledge, the following theorem is due to Bernstein and Zelevinsky in type A, and to Bernstein in general type (unpublished). Lusztig has given an exposition in [Lus1989]. We give a new proof which we believe is more elementary and more direct.
Theorem 4.12. (Bernstein, Zelevinsky, Lusztig [Lus1989]) Let be a lattice such that where is the root lattice and is the weight lattice of the root system Let be the affine Hecke algebra corresponding to and let Let be the Weyl group of Then the center of is
Let be maximal in Bruhat order subject to for some If there exists a dominant such that (otherwise for every dominant which is impossible since is a finite linear combination of Since we have
Repeated use of the relation (4.11) yields
where are constants such that for and unless So
and comparing the coefficients of gives Since it follows that which is a contradiction. Hence
The relation (4.11) gives
where Comparing coefficients of on both sides yields Hence and therefore for So
Although we have not taken this point of view in our presentation, the affine Hecke algebras defined above are naturally associated to a reductive algebraic group over [KLu0862716] or a Chevalley group [IMa1965]. In this formulation, the lattice is determined by the group of characters of the maximal torus of It is often convenient to work only with the adjoint version or only with the simply connected version of the group and therefore it seems desirable to be able to derive the representation theory of the affine Hecke algebras from the representation theory of the affine Hecke algebra The following theorem shows that this can be done in a simple way by using the extension of Clifford theory in the Appendix. In particular, Theorem A.13, can be used to construct all of the simple from the simple
Theorem 4.14. Let be a lattice such that where is the root lattice and is the weight lattice of the root system Let be the affine Hecke algebra corresponding to Then there is an action of a finite group on acting by automorphisms, such that
is the subalgebra of fixed points under the action of the group
There are two cases to consider, depending on whether the group is cyclic or not.
In each case we construct the group and its action on explicitly. This is necessary for the effective application of Theorem A.13 on examples.
Case 1. If is a cyclic group then the subgroup is a cyclic subgroup. Suppose
where Let be a primitive root of unity and define an automorphism
The map is an algebra isomorphism since it preserves the relations in (4.6) and (4.10). Furthermore, gives rise to a action on and
Case 2. If the root system is of type even, then and the subgroups of correspond to the intermediate lattices Suppose
and define automorphisms of by
where and are the two intermediate lattices strictly between and
This is an excerpt of the paper entitled Affine Hecke Algebras, Cyclotomic Hecke algebras and Clifford Theory authored by Arun Ram and Jacqui Ramagge. It was dedicated to Professor C.S. Seshadri on the occasion of his 70th birthday.
Research partially supported by the Natioanl Science Foundation (DMS-0097977) and the National Security Agency (MDA904-01-1-0032).