Affine Hecke algebras (Ram-Ramagge)
Last update: 25 June 2012
Deducing the representation theory from
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 [KL] or a adic Chevalley group [IM]. 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 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 modules from the simple modules.
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 of acting by automorphisms, such that
is the subalgebra of fixed points under the action of the group
There are two cases to consider, depending of 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 [RR, Eq. 4.6] and [RR, Eq. 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
and define automorphisms of by
where and are the two intermediate lattices strictly between and
A. Ram and J. Ramagge,
Affine Hecke Algebras,