Last update: 31 March 2013
The root system for has simple roots and fundamental weights and and
Irreducible representations. Table 4.1 lists the irreducible by their central characters. The sets and are as given in (1.7) and correspond to the choice of representative for the central character displayed in Figure 4.1. 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 4.1 we have replaced the representation by the weight The nilpotent elements and are representatives of the root spaces and respectively, where is the Lie algebra For each calibrated module with central character we have listed the subset such that is the corresponding placed skew shape (see Theorem 1.11). The abbreviation ‘nc’ indicates modules that are not calibrated.
There is one case when this representation is tempered, see Table 4.2.
Figure 4.1 displays the real parts of the central characters in Table 4.1. If then the polar decomposition determines an element such that (see (1.3)). For each central character the point labeled by in Figure 4.1 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 tempered (resp. square integrable) are the ones which have the real parts of all their weights in the closure (resp. interior) of the shaded region of Figure 4.2. Let be given by This is a special case of the central character in Table 4.1. For this particular special case there is one tempered representation with central character
The irreducible tempered representations with real central character are in one-to-one cor- respondence with the irreducible representations of the symmetric group (see [BMo1989]). These representations are indexed by the partitions (3), (21), of 3. Equivalently, they 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 and is an irreducible representation of the component group In type A the component group is always trivial. For each root let be an element of the root space The three nilpotent orbits in and the corresponding tempered representations of are as in Table 4.2.
Central characters Since these weights are regular. Thus the representations corresponding to these central characters are in one to one correspondence with the connected components of the calibration graph and can be constructed explicitly with the use of Theorem 1.11. Up to isomorphism the principal series module is the only irreducible with central character The Langlands parameters for each module can be determined from its weight structure and the indexing triple is then determined from the Langlands data by using the induction theorem of Kazhdan and Lusztig (see the discusssion in [BMo1989, p.34]).
There is one special case of the central character when the irreducible module constructed by applying Theorem 1.11 to the placed skew shape is tempered. This happens when for the weight given by The indexing triple and the calibration set for this case are still given by and respectively.
Central characters and One can use the defining relations of to check that the only 1-dimensional representations of are the ones with central character Construct two 3- dimensional representations of by
where and are the two one-dimensional representations of given by
These representations must be irreducible since, if not, they would either have a 1-dimensional submodule or a one dimensional quotient. But there are no 1-dimensional modules with central character
The central characters and are taken into each other under the automorphism of the Dynkin diagram of which switches the two nodes and thus these two central characters will produce modules which have the same structure (up to twisting by the automorphism which switches and Thus the representations with central character can be obtained from the ones with central character by switching all 1’s and 2’s and changing all to
Central characters and Since Kato’s irreducibility criterion (Theorem 1.16) implies that the principal series module is irreducible. By Theorem 1.15 this is the only irreducible with central character As for the case of and the central characters and are taken into each other under the automorphism of the Dynkin diagram of Thus the irreducible representations with central character can be obtained from the one with central character by switching all 1’s and 2’s and changing all to
Central characters Since Kato’s irreducibility criterion (Theorem 1.16) implies that the principal series module is irreducible. By Theorem 1.15 this is the only irreducible with central character
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.