Last update: 31 March 2013
The root system for has one simple root and fundamental weight
Irreducible representations. Table 2.1 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 2.1. 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 2.1 we have replaced the representation by the weight The nilpotent element is a representative of the root space for 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.
Figure 2.1 displays the real parts of the central characters in Table 2.1. If then the polar decomposition determines an element such that (see (1.3)). For each central character the point labeled by in Figure 2.1 is the graph of the corresponding Assume (for pictorial convenience) that is a positive real number and let
The marks indicate the (affine) hyperplanes
Tempered and square integrable representations. The tempered (resp. square integrable) are the ones which have all their weight spaces in the closure (resp. interior) of the dotted region of Figure 2.2.
The irreducible tempered representations with real central character can be indexed by the irreducible representations of the symmetric group (see [BMo1989]). These representations are indexed by the partitions (2), of 2. Let be an element of the root space for the Lie algebra The two nilpotent orbits in and the corresponding tempered representations of are as in Table 2.2.
Central character There are two one-dimensional representations and with central character These representations are given explicitly by
respectively. One uses Theorem 1.15 and the fact that the principal series module is two dimensional to conclude that and are the only irreducible representations of with central character
Central character By Theorem 1.15 and Kato’s irreducibility criterion, Theorem 1.16, the only irreducible representation with central character is the principal series module Alternatively, this module can be constructed by applying Theorem 1.11 to the placed skew shape
Central character The weights given by are the two central characters which satisfy In either case Kato’s irreducibility criterion (Theorem 1.16) tells us that the principal series module is irreducible. This module has basis and action given by
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.