Classification for A1

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 31 March 2013

Classification for A1

The root system R for A1 has one simple root α1 and fundamental weight ω1=12α1.

Irreducible representations. Table 2.1 lists the irreducible H-modules by their central characters. The sets P(t) and Z(t) 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 (𝒯,I) where I is a subset of {1} and 𝒯 is a tempered representation for the parabolic subalgebra HI. In our cases the tempered representation 𝒯 of HI is completely determined by a character tT. Specifically, 𝒯 is the only tempered representation of HI which has t as a weight. In the labeling in Table 2.1 we have replaced the representation 𝒯 by the weight t. The nilpotent element eα1 is a representative of the root space 𝔤α1 for the Lie algebra 𝔤=𝔰𝔩2. For each calibrated module with central character t we have listed the subset JP(t) such that (t,J) is the corresponding placed skew shape (see Theorem 1.11). The abbreviation ‘nc’ indicates modules that are not calibrated.

Central P(t) Z(t) Dimension Langlands Indexing Calibration character parameters triple setJ ta {α1} 1 (ta,) (ta,0,1) 1 tempered (ta,eα1,1) {α1} tb 2 (tb,) (tb,0,1) to {α1} 2 tempered (to,0,1) nc Table 2.1. Irreducible representations

Figure 2.1 displays the real parts of the central characters in Table 2.1. If tT then the polar decomposition t=trtc determines an element μn such that tr(Xλ)=eλ,μ (see (1.3)). For each central character tp the point labeled by p in Figure 2.1 is the graph of the corresponding μpn. Assume (for pictorial convenience) that q is a positive real number and let

Hα1= {xα1,x=0} ,andHα1±δ= {xα1,x=ln(q±2)}.

The marks indicate the (affine) hyperplanes Hα1±δ.

to ta tb Hα1-δ Hα1 Hα1+δ Figure 2.1. Real parts of central characters in Table 2.1

Tempered and square integrable representations. The tempered (resp. square integrable) H-modules are the ones which have all their weight spaces in the closure (resp. interior) of the dotted region of Figure 2.2.

s1ta to Figure 2.2. Real parts of weights of tempered representations

The irreducible tempered representations with real central character can be indexed by the irreducible representations of the symmetric group S2 (see [BMo1989]). These representations are indexed by the partitions (2), (12) of 2. Let eα1 be an element of the root space 𝔤α1 for the Lie algebra 𝔤=𝔰𝔩2. The two nilpotent orbits in 𝔤 and the corresponding tempered representations of H are as in Table 2.2.

Nilpotent orbit Indexing triple Square integrable Wrepresentation regular (ta,eα1,1) yes (12) 0 (to,0,1) no (2) Table 2.2. Tempered representations and the Springer correspondence s1ta ta s1tb tb to Figure 2.3. Calibration graphs for central characters in Table 2.1

The analysis.

Central character ta: There are two one-dimensional representations va and vs1a with central character ta. These representations are given explicitly by

Xλva = ta(Xλ) va, T1va = qva, and Xλvs1a = (s1ta) (Xλ) vs1a, T1vs1a = -q-1 vs1a,

respectively. One uses Theorem 1.15 and the fact that the principal series module M(ta) is two dimensional to conclude that va and vs1a are the only irreducible representations of H with central character ta.

Central character tb: By Theorem 1.15 and Kato’s irreducibility criterion, Theorem 1.16, the only irreducible representation with central character tb is the principal series module M(tb). Alternatively, this module can be constructed by applying Theorem 1.11 to the placed skew shape (tb,).

Central character to: The weights given by to(Xω1)=±1 are the two central characters toT which satisfy P(t)=, Z(t)={α1}. In either case Kato’s irreducibility criterion (Theorem 1.16) tells us that the principal series module M(to) is irreducible. This module has basis {vt,T1vt} and action given by

ϕ(Xλ)=t (Xλ) ( 1 (q-q-1) λ,α1 0 1 ) andϕ(T1)= ( 01 1q-q-1 ) .

Notes and References

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.

page history