Last update: 9 March 2013
The type affine Hecke algebra is built on the root data of This is the only reduced rank one root system (up to isometry.) Though our main goal is the rank two affine Hecke algebras, the rank one case is fundamental in the classifcation of the higher-rank cases.
so that is the group generated by and is isomorphic to The Weyl group is
defines an action of on
Let The affine Hecke algebra of type is
is a subalgebra of A weight is an element of
A weight is completely determined by the value which must be invertible, so that
This definition is a special case of the definitions given in sections 1.2.1 and 1.2.2, using the root system
Proposition 2.1. There are four 1-dimensional representations
If then so that and
A straightforward check of the relations (2.1) − (2.3) shows that the maps are homomorphisms.
Assume is a 1-dimensional with By (2.1),
By (2.3), if then
Similarly, if then
Let and let be the one-dimensional given by
Then the principal series module is
so that is a basis for
Note that if is a primitive fourth root of unity, the two cases in (b) coincide.
(a) Any non-trivial proper submodule of must be 1-dimensional. Such a submodule can exist only if by proposition 2.1.
(b) If let Then
so that is a 1-dimensional submodule of Since if is the image of in then and
If then spans a 1-dimensional submodule with and in
If then has two distinct weight spaces: and If were the direct sum of two 1-dimensional submodules, it would have to be the direct sum of the two weight spaces. However, is all of so that is indecomposable.
(c) If and then the weight space is all of Both and span 1-dimensional submodules of which are disjoint.
Remark: The machinery of section 1.2 also suffices to obtain the information in this theorem. Part a is exactly Kato’s criterion (Theorem 1.8). Part b follows from the properties of the operators, since but exactly when In that case, is a multiple of so spans a 1-dimensional submodule of We have given an explicit proof to keep the type example as clear as possible.
For let be the weight given by
Pictorially, identify with the real line. In this picture, the set
is marked with a solid line, while
are denoted by dashed lines.
If is a primitve root of unity then is identified with and
The following is the specific case so that The periodicity is evident in the picture.
The structure of can be seen using the τ operator described above (1.2.4). If then is not defined. If then is non-zero on but is zero on This accounts for the structure of described above (Proposition 2.2 (b)). If or 1, then is invertible on The following pictures show the “local regions” as described in (1.2.6), which describe
Each region is divided into two chambers, representing the two weights in the of the central character. Each chamber contains a number of large dots equal to the dimension of the generalized weight space with weight and lines connecting the dots show that the corresponding basis vectors are in the same composition factor. A solid line is drawn where is not defined, as in the following picture.
These two pictures represent weights with so that Hence the two dots representing the weight spaces are drawn in the same chamber. If is irreducible, so the weight spaces are connected by arcs. The picture reflects the fact that there is only one local region, since
These pictures show for the other possible weights A dashed line denotes a weight with so that the operator on the weight spaces of is well-defined, but not invertible in both directions. This means that the corresponding weight spaces will be in different composition factors of A dotted line denotes a weight with or 1. In this case is invertible, forcing the weight spaces to be in the same composition factor of Accordingly, the dots representing them are connected.
Notice also the connection to the drawings of central characters above. The pictures of the local regions at a weight are a picture of a small open neighborhood around the point in the picture of the characters. This correspondence is not as clear in this case as in others, since it is the smallest example of the affine Hecke algebra, but the essential ingredients are present. The weights of the are all displayed, as are the actions of the operators that determine the composition factors of
The complete classification of modules is summarized in the following tables.
It should be noted that for any value of with the representation theory of can be described in terms of only. If then the representation theory of does not fit that same description. This fact can be seen through a number of different lenses. It is a reflection of the fact that the sets and for all possible central characters can be described solely in terms of In the local region pictures, this is reflected in the fact that the hyperplanes and are distinct unless When these hyperplanes coincide, the sets and change for characters on those hyperplanes.
This is an excerpt from Matt Davis' Ph.D Thesis entitled Representations of Rank Two Affine Hecke Algebras at Roots of Unity, University of Wisconsin, 2010.