Last update: 28 February 2013
The homomorphism of Proposition 2.5 is a powerful tool for transporting results about the affine Hecke algebra of type to the cyclotomic Hecke algebras. In this section we show how this homomorphism arises naturally, from a folding of the Dynkin diagram of and we give some generalizations of the homomorphism to other types.
Example 1. Type The root system of type can be realized by
where are an orthonormal basis of The simple roots and the fundamental weights are given by
If is the highest root of then and
Then is the only miniscule weight,
Thus, from (4.2), (4.3) and (4.6),
The braid group is generated by and which satisfy relations in (4.6), where the are given by the extended Dynkin diagram see Figure 3. The braid group is the subgroup generated by These elements satisfy the relations in (4.6), where the are given by the Dynkin diagram A straightforward check verifies that the map defined by
extends to a well defined surjective group homomorphism. From the identity (4.3),
By inductively applying the relation we get
Example 2. Type Since the weight lattice for the root system of type is the same as the lattice defined in (1.5) we have an injective homomorphism
The composition of and the map from (5.5) is the surjective homomorphism defined by
In fact, it follows from the defining relations of and that the map is an isomorphism!
The cyclotomic Hecke algebras are quotients of and in this way the group homomorphism is the source of the algebra homomorphism
which was used extensively in Section 3 to relate the representation theory of the cyclotomic Hecke algebras to the affine Hecke algebra of type A.
Example 3. Type Let be the root system of type Then is also of type and inspection of the Dynkin diagrams of types and yields a surjective algebra homomorphism defined by
Examples 1, 2, and 3 show that, for types and there exist surjective homomorphisms from the affine Hecke algebra to the corresponding Iwahori-Hecke subalgebra. The following example shows that this is not a general phenomenon: there does not exist a surjective algebra homomorphism from the affine Hecke algebra of type to the corresponding Iwahori-Hecke subalgebra of type
Example 4. Type If is the root system of type then and
Proposition 5.6. Let be the affine Hecke algebra of type as given by (4.6) and (4.10) and let be the Iwahori-Hecke subalgebra of type generated by and There does not exist an algebra homomorphism such that for
There is an irreducible representation of given by
(see [Ram1997, Theorem 6.11]). We show that there does not exist a matrix which satisfies
If exists then must be diagonal since commutes with and is a diagonal matrix with distinct eigenvalues. The first equation shows that is invertible and the second equation shows that is conjugate to It follows that must have one eigenvalue and one eigenvalue Thus, either
However, neither of these matrices satisfies the relation This contradiction shows that the representation cannot be extended to be a representation of
In spite of the fact, demonstrated by the previous example, that there does not always exist a surjective algebra homomorphism from the affine Hecke algebra onto its Iwahori-Hecke subalgebra, there are interesting surjective homomorphisms from affine Hecke algebras of exceptional type.
Example 5. Type For the root system of type Let where is as given by (4.3) for the minuscule weight (see [Bou1968, p. 261]). There are surjective algebra homomorphisms
Example 6. Type For the root system of type we have and There is a surjective homomorphism
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).