Last update: 1 April 2013
The root system for has simple roots and fundamental weights and and
Irreducible representations.
Table 6.1 lists the irreducible by their central characters. We have listed only those central characters for which the principal series module is not irreducible (see Theorem 1.16). The sets and are as given in (1.7) and correspond to the choice of representative for the central character displayed in Figure 6.1. The Langlands parameters usually consist of a pair where is a subset of and is a tempered representation for the parabolic subalgebra In our cases the tempered representation of is completely determined by a character Specifically, is the only tempered representation of which has as a weight. In the labeling in Table 6.1 we have replaced the representation by the weight The notation for the nilpotent elements in the indexing triples is as in Table 6.3.
Table 6.2 lists the irreducible calibrated For each module with central character we have listed the subset such that is the corresponding placed skew shape (see Theorem 1.11). We have listed only those central characters for which the principal series module is not irreducible (see Theorem 1.16).
Figure 6.1 displays the real parts of the central characters in Table 6.1. If then the polar decomposition determines an element such that (see (1.3)). For each central character the point labeled by in Figure 6.1 is the graph of the corresponding Assume (for pictorial convenience) that is a positive real number and let
for each positive root The dotted lines display the (affine) hyperplanes
Tempered and square integrable representations. The irreducible tempered representations with real central character can be indexed by the irreducible representations of the Weyl group of type (see [BMo1989, p. 34]). Equivalently, these representations can be indexed by the pairs which appear in the Springer correspondence. The and will also be elements of the indexing triple for the corresponding tempered representation of Here is a nilpotent element of the Lie algebra is the complex simple group over of type and is an irreducible representation of the component group (see [Car1985]). For each root let be an element of the root space The five nilpotent orbits in g and the corresponding tempered representations of are as in Table 6.3. The notation denotes the symmetric group on three elements, which has irreducible representations indexed by the partitions (3),(21), of 3. We have used the notation of Carter [Car1985, p.427] to label the irreducible representations of the Weyl group of type
The only other tempered representations are the representations labeled by the triples and These representations are square integrable but do not have real central character.
The modules labeled by are the ones constructed by Lusztig in [Lus1983-2] 4.20, 4.19, 4.7 and 4.22 respectively. In Lusztig’s notation these are the stars (see [Lus1983-2, 4.23]) of the modules labeled by the graphs and respectively.
The analysis.
Central characters The central characters and are the only ones which have both and nonempty. The other cases are handled by Theorem 1.16 and Theorem 1.11 as in the cases of central characters and for type
The Langlands parameters for each module can be determined from its weight structure. The indexing triple is determined from the Langlands data by using the induction theorem of Kazhdan and Lusztig (see the discussion in [BMo1989, p.34]). Let us give an example to illustrate the procedure. The Langlands parameters for the 4 dimensional representation with central character correspond to the indexing triple which is conjugate to the triple
The indexing triples for the tempered representations cannot be determined with the use of the Kazhdan-Lusztig induction theorem. The indexing triples for the tempered representations with real central character are determined from the Springer correspondence, see Table 6.3 and [BMo1989, p.34]. The two tempered representations with central characters and do not have real central character. By the last two sentences of [Lus1983-2, 2.10] we know that the indexing triples for these representations contain the subregular nilpotent and that the component groups are isomorphic to and respectively. In both cases the component group acts trivially on and so The fact that the elements and are representatives of the subregular nilpotent orbit can be derived from the analysis in [Jac1997, Theorem 4.40] or [Stu1971]. This determines the triples and
Central character Theorem 1.11 applied to the placed skew shapes and produces, respectively, a two dimensional irreducible module with a one dimensional irreducible module with and a one dimensional irreducible module with Lusztig [Lus1983-2] Theorem 4.7 constructs a 3-dimensional irreducible with and In Lusztig’s notation this is the module labeled by the graph for
As described in [Lus1983-2, 4.23] we can twist the module by an involutive automorphism of to obtain another 3-dimensional irreducible module which has and
All of the modules must appear as composition factors of the principal series module By comparing dimensions of weight spaces, any other module which appears in a composition series of must have Theorem 1.11(b) then implies that must be isomorphic to Thus Theorem 1.15 implies that and are (up to isomorphism) all the irreducible modules with central character
The Langlands parameters for each module are determined from its weight structure. The Kazhdan-Lusztig induction theorem allows us to use the Langlands
parameters to determine the indexing triples for the modules which are not tempered. Since is a real central character
the indexing triples for the tempered representations can be determined from the Springer correspondence, see Table 6.3. Alternatively, one can get these triples from
[Lus1983-2, 2.10] where it is explained that the nilpotent in the indexing triple is subregular, the variety
(where and is subregular) consists of three disjoint points and a projective
line, and the component group is isomorphic to the symmetric group The symmetric group
acts trivially on the line and permutes the three points, which implies that the line corresponds to
(trivial representation of
and the three points are split between the isotypic components and
In this case the standard module
The projective line in corresponds to the two dimensional weight space
Central character Let and be the one dimensional representations of given by
Let
By Lemma 1.17 these modules have weights and respectively. Both and are 6 dimensional.
Let be any such that By Lemma 1.19 and Proposition 1.6, Since it follows from Lemma 1.19 (b) that Then, by Proposition 1.6, Adding these numbers up we see that It follows that is irreducible. An analogous argument can be applied to conclude that is irreducible.
Central character Let and be the one dimensional representations of given by
Let
By Lemma 1.17 these modules have weights and respectively. Both and are 6 dimensional.
Let be any such that By Lemma 1.19 and Proposition 1.6, Thus It follows that is irreducible. An analogous argument can be applied to conclude that is irreducible.
Central character Let and be the one dimensional representations of given by
Let
An argument similar to that for the central character shows that and are irreducible.
Central character Let and be the one dimensional representations of given by
Let
An argument similar to that for the central character shows that and are irreducible.
This is an excerpt of a preprint entitled Representations of rank two affine Hecke Algebras, written by Arun Ram, Department of Mathematics, Princeton University, August 5, 1989.
Research supported in part by National Science Foundation grant DMS-9622985, and a Postdoctoral Fellowship at Mathematical Sciences Research Institute.