Last update: 9 March 2013
The type root system is
where Then is a root system as defined in (1.2.1), and the Weyl group The simple roots are and and is the only other positive root.
The fundamental weights satisfy
be the weight lattice of
The affine Hecke algebra (see 1.2.2) is generated as a by and with relations
a subalgebra of and let
Then acts on by
and acts on by
be the root lattice of Let
We can also visualize using the following picture. Here, the hyperplane is drawn as a solid line. The hyperplanes are drawn in this picture as dashed lines. The weight is the point units away from and units away from The action of is also visible in this picture, as is given by reflection in the hyperplane
For each there are 3 elements with determined by
The dimension of the modules with central character and the submodule structure of depends only on Thus we begin by examining the in The structure of the modules with weight depends virtually exclusively on and For a generic weight and are empty, so we examine only the non-generic orbits.
Proposition 2.3. If and then is in the of one of the following weights:
First, assume generic
Case 1: If contains two positive roots, then it must contain the third. This implies
Case 2: If contains only one root, by applying an element of assume that it is Then so either or The first central character is for some or (If or either or would be larger.) For the second case, there are two potential choices for the orbit, arising from choosing or However, is in the same orbit as
Case 3: Now assume that If is not empty, assume that and Then by assumption on Then it is possible that in which case If then and Otherwise, for some
Remark: There is some redundancy present in the list of central characters above for specific values of If then and are all in the same Also in this case, If then and Also note that for every generic weight there are six weights in its all of which are of course generic.
It is helpful to draw a picture of the weights for various values of Solid lines in these pictures show sets of the form
for while dashed lines denote sets of the form
Proposition 2.4. Fix The 1-dimensional are
A straightforward check shows that the maps above respect the defining relations for (2.4) - (2.10), so that the maps are homomorphisms.
Let be any 1-dimensional By (2.6) and (2.5),
Case 1: By (2.7) and (2.8),
Hence, where or 2, and
Case 2: Then
This implies that
Hence, where or 2, and
Principal Series Modules and Local Regions
Let The principal series module is
where is the one-dimensional given by
By (1.6), every irreducible module is a quotient of some principal series module Thus, finding all the composition factors of for all central characters will find all the irreducible
Assume for now that
Case 1: empty. By 1.8, if is empty, then is irreducible and is the only irreducible module with central character This case includes the central characters and for generic
Since there is one local region
the set of minimal length coset representatives of cosets in where is the stabilizer of in If and are both in then is a bijection. The following pictures show with one dot in the chamber for each basis element of
These pictures also show the weight space structure of For all of is in the weight space, so all the dots lie in the same chamber. For so that the weights of are and and each weight space is 2-dimensional. The weight is regular, so there are six different weights. In each case, all the dots lie in the same local region, a region bounded by solid and/or dashed lines.
Case 2: This case includes the central characters and If is empty, then is calibrated and the irreducible modules with central character are in one-to-one correspondence with the local regions and the components of the calibration graph. Each local region in the following pictures is a set of chambers between two dashed lines.
For the weight for generic there are four local regions. If each weight is in a local region by itself and thus there are six different representations in that case. For the weight there are two local regions. These local regions are bounded by dashed lines, which also serve as barriers of sorts between the composition factors. This is a combinatorial reflection of the fact that if the operator between two weight spaces of is not invertible, those weight spaces will be in different composition factors. Weight spaces with invertible operators between them are separated by dotted lines since they are in the same composition factor and local region.
Case 3: The only central characters with both and nonempty are when and and in all cases. If then and are in the same orbit, and are in the same orbit as If then In each of these pictures, there are three local regions, but the structure of is slightly more complicated in this case.
The composition factors of each have some weight with for some simple root Then 1.11 shows that when this composition factor must be at least 3-dimensional. In this case, these composition factors are shown with the two dots in that weight space connected to each other, as well as to a dot in the next chamber, since the basis vectors correpsonding to these dots must lie in the same irreducible. When the two dimensional weight space makes up an entire composition factor.
To prove that these pictures do reflect the structure of rather than analyzing directly, it is easier to construct several irreducible modules with central character and show that they must account for all the composition factors of Let be the 1-dimensional spanned by and let be the 1-dimensional spanned by given by
are 3-dimensional with central character
Proposition 2.5. Let and
(a) Assume If either or were reducible, it would have a 1-dimensional submodule or quotient, which cannot happen since the 1-dimensional modules have central character Thus both and are reducible.
(b) If then note that the map
is an automorphism of If and are the maps describing the action of on and respectively, then and Thus if one of and is irreducible, the other is as well. However, this would account for all the composition factors of a contradiction since there is a 1-dimensional module with central character Thus and must both be reducible, so has a 1-dimensional submodule or quotient. It must have weight since that is the only weight of that supports a 1-dimensional module. Then the 2-dimensional composition factor is by Lemma 1.11. Then since
is a submodule of Similarly, is a submodule of since must be a quotient of
To construct the irreducibles with composing with
an automorphism of gives a bijection between representations with central character and those with central character
If then the results of section 1.2.9 suffice to classify the representations of with central characters and for Specifically, if then and a is merely a (via the isomorphism on which acts by the scalar In fact, considered as a module is the regular module. If for then Since has two 1-dimensional irreducible representations, has two irreducible 3-dimensional representations obtained by inducing up from
Note that in all cases, the local region picture is a picture of a small neighborhood around the point corresponding to in the picture of The necessary information to understand the composition factors of is contained in the local region picture around
The following tables summarize the classification. It should be noted that for any value of with and not a primitive third root of unity, the representation theory of can be described in terms of only. If is a primitive root of unity of order 3 or less, 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 is a root of unity of order 3 or less. 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.