Last update: 27 April 2013
In this section we analyze the structure of all simple for rank 2 graded Hecke algebras The results, the classification of simple modules and various other data (central character dimension, calibrated or not calibrated, Langlands parameters), are listed in Table 1. An irreducible representation that is calibrated (see (2.10)) has all its weights of the form with for a unique and this is the set that is displayed in the fourth column of Table 1. The notation ‘nc’ indicates that the representation is not calibrated. The Langlands parameters of a simple of central character consists of a pair where is a subset of and is a tempered representation of (see Theorem 2.4). If is empty there is a unique tempered representation of of central character so we place the pair in the corresponding entry of column 5 of Table 1. If consists of one element, then and each representation is naturally indexed by its maximal weight so we place in column 5 of Table 1. If then the corresponding simple is tempered.
The classification of the simple is accomplished in three steps:
The reflection group is the dihedral group of order Let be an orthonormal basis of and define
Fix the roots, positive roots and simple roots for the reflection group by
For and If is even there are two orbits of roots, and Let be a choice of parameters for the graded Hecke algebra When is odd all of the are equal and, when is even, there are two, possibly unequal, parameters and Figure 1 displays the roots and hyperplanes for and When is even each root lies on the hyperplane and this is why, in the picture of hyperplanes and roots for there are multiple labels on each line.
Figure 2 displays, using thin and thick lines, the hyperplanes
for and (and a particular choice of the parameters
Using the orthonormal basis we can identify with and with If then
In terms of the pictures in Figure 2, if is a point in then the elements of label the (thin lines) that is on and the elements of label the set of (thick lines) that is on.
Let us analyze the possibilities for and as runs over representatives of in
If then and so we may, without loss of generality, assume that for all when is odd, and and when is even.
|a.||If contains 2 roots or more, then since any two distinct positive roots are linearly independent. This is the central character in Table 1.|
|b.||If contains one root, then, by choosing our representative of the Wγ appropriately, we may arrange that when is odd, and or when is even. When is even there is an automorphism of the root system (and of the graded Hecke algebra) which switches and The automorphism extends linearly to and if then and Thus, even when is even, it will be sufficient to analyze the case|
|b'.||If and then the equations and uniquely determine Since must also be in This happens for the central characters and in Table 1.|
|b''.||If and then equation (3.1) for and forces which forces even and and to be of different parity. Furthermore, the parameters must satisfy and, when this happens, it happens for a unique choice of the 4-tuple Thus, the only possible option is (if then This is the central character in Table 1.|
|c.||If and such that then is uniquely determined by the equations These equations force if is even (the easiest way to see this is to look at the pictures in Figure 2). Since we assumed it follows that is odd. If contains 3 elements, then at least two of them would satisfy even, and so it follows that contains a maximum of two elements. By appropriately choosing our representative of the orbit we can assume that for some This case corresponds to the central character in Table 1.|
This analysis shows that Table 1 covers all possibilities.
The following analysis determines the structure of each of the irreducible the dimensions of each generalized weight space and the Langlands parameters. The derivation of the irreducible representations proceeds by considering, separately, each central character In each case we have included a picture showing the local regions In these pictures the solid lines correspond to hyperplanes for and the dotted lines correspond to hyperplanes for Each local region is labeled by the corresponding set of roots which determines its location in the picture (see the discussion before Corollary 2.6).
The Langlands parameters of an irreducible are determined by the real parts of weights of This means that, according to the labeling of the simple modules as in Table 1, the Langlands parameters can depend on the choice of the parameters In our calculations of Langlands parameters, and in the Langlands data displayed in Table 1, we assume that all (this assumption is used only in the analysis of Langlands parameters).
In the case when is even not all roots are in the orbit of and one should really consider central characters which have (see the remark in Section 3.2(b)). These central characters are the images of the central characters and under the automorphism of the root system which switches and This automorphism extends to an automorphism of and thus it follows that the modules with central characters have exactly the same structures as the modules with central characters and respectively.
By Theorem 2.10 the principal series module is irreducible and, by Proposition 2.8(a), this is the unique irreducible module with central character Since is regular, is calibrated.
The weight is uniquely determined by the fact that and where
Suppose is an irreducible module with central character and Then by Lemma 2.7(a), for all
Now apply operators of the form to If but then and Therefore, by Lemma 2.7(b),
Thus, by Corollary 2.6,
By applying more operators, if but then Thus, by Corollary 2.6,
The module is
Similar reasoning applied to an irreducible module with central character and for some yields the dimensions of the generalized weight spaces of which sum to Thus the decomposition of the principal series module consists of two irreducible modules and with central character and
and all other weight spaces of and are 0. Neither of the two irreducible modules and with central character are calibrated.
The maximal weight of is which is dominant and on the hyperplane The Langlands set for this weight is The maximal weight of is on the hyperplane if is even, and on the hyperplane if is odd. This observation determines the set in the Langlands decomposition of the (real part) of the maximal weight of (equation (2.11)).
Central character even,
We use Lemma 2.7 and an argument similar to that for central character to decompose the principal series module and conclude that there are two irreducible modules and with central character with
All other weight spaces of and are 0. Neither of the two irreducible modules and with central character are calibrated.
The maximal weight of is which is dominant and on the hyperplane The Langlands set for this weight is The module is tempered with maximal weight
It may be that so that the hyperplanes and are the same and contains only 3 roots. We do not have to consider this situation separately.
In some sense, the special central character occurs when the parameters are exactly right so that the central characters and “coalesce”. This occurs only if is even, and are of different parity, and the parameters satisfy For a fixed choice of parameters, there is at most one choice of the quadruple
We use Lemma 2.7 and Corollary 2.6 in an argument similar to that for central character to see that there are five nonisomorphic irreducible and with central character unless in which case there are only four has dimension 0).
and all other weight spaces of these modules are 0.
Both modules and are tempered and have the same maximal weight
The weight is uniquely determined by and
The dashed line in this picture is for reference only, it does not correspond to a root in or
Since is regular, the irreducible with central character are calibrated and can be indexed by the sets The irreducible calibrated module indexed by the set has
and all other weight spaces 0. A construction of is given in Theorem 4.5.
To compute the Langlands parameters of these modules we first assume that is odd and If the maximal weight of the module is in the same chamber as if is even, and in the same chamber as if is odd. If the maximal weight of is in the same chamber as if is odd, and in the same chamber as if is even. In each case this information determines the set in the Langlands parameters. If the module is tempered with maximal weights
If is even and all parameters are equal, then the Langlands parameters are as in the previous paragraph. In the case that is even and then it may happen that is not in the dominant chamber. The structure of the modules with central character does not change but the Langlands parameters of the representations may change significantly. One of the four irreducibles with central character will always be tempered, but which one (and thus the dimension of the tempered module with this central character) depends on the values of the parameters and
Since is regular the irreducible modules with central character are calibrated and can be indexed by the sets The module has
and all other weight spaces 0. A construction of is given in Theorem 4.5.
The Langlands parameters given in Table 1 for irreducible representations with central character assume that where is odd and In the particular case where is odd and the irreducible module indexed by the set is tempered.
The Springer correspondence for Weyl groups (see [BMo1989, p. 34]) associates to each tempered representation of with real central character, the unique “maximal” irreducible which is contained in For Weyl groups (crystallographic reflection groups) this is a one-to-one correspondence between tempered representations of and irreducible representations of Using our classification of in Table 1, we can establish a similar correspondence for the noncrystallographic groups
If is odd, then the group has 2 one-dimensional irreducible representations and two-dimensional irreducible representations. The trivial (resp. sign) representation of corresponds to the tempered irreducible with central character (resp. The two-dimensional representations of correspond to the tempered with central characters and Note that and can all be taken to be multiples of the root and in the dominant chamber. In this normalization the 1-dimensional representations correspond to the two extreme elements of this chain of weights.
If is even and the parameters are all equal, the trivial (resp. sign) representation of corresponds to the tempered irreducible with central character (resp. and the other two 1-dimensional representations of correspond to the tempered with central characters and where is the involution that switches and The 2-dimensional correspond to the tempered with central characters As in the case where is odd, the central characters and can be taken to be in the dominant chamber and on the line through the origin and the point In this normalization the trivial and the sign representations correspond to the two extreme elements of this chain of weights. In the case when the parameters are unequal, two of the points on this chain may coalesce in the weight and “become” the two tempered representations of with central character The case where contains only 3 roots comes from one of the central characters or coalescing with one of the
This analysis establishes the “Springer correspondence” for all dihedral groups and all choices of the parameters of with
This is an excerpt of a paper entitled Representations of graded Hecke algebras, written by Cathy Kriloff (Department of Mathematics, Idaho State University, Pocatello, Idaho 83209-8085) and Arun Ram.
Research of the first author supported in part by an NSF-AWM Mentoring Travel Grant. Research of the second author supported in part by National Security Agency grant MDA904-01-1-0032 and EPSRC Grant GR K99015.