Classification for A2

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

Last update: 31 March 2013

Classification for A2

The root system R for A2 has simple roots α1 and α2, fundamental weights ω1 and ω2, and

α1,α2 =-1 α2,α1 =-1, ω1=13 (2α1+α2) ω2=13 (α1+2α2), and α1=2ω1-ω2 α2=-ω1+2ω2.

Irreducible representations. Table 4.1 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 4.1. The Langlands parameters usually consist of a pair (𝒯,I) where I is a subset of {1,2} 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 4.1 we have replaced the representation 𝒯 by the weight t. The nilpotent elements eα1 and eα2 are representatives of the root spaces 𝔤α1 and 𝔤α2, respectively, where 𝔤 is the Lie algebra 𝔤=𝔰𝔩3. 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,α2} 1 (ta,) (ta,0,1) 2 (s1ta,{2}) (ta,eα2,1) {α2} 2 (s2ta,{1}) (ta,eα1,1) {α1} 1 tempered (ta,eα1+eα2,1) {α1,α2} tb {α2} 3 (tb,) (tb,0,1) 3 (s2tb,{2}) (tb,eα2,1) {α2} tc {α2,α1+α2} {α1} 3 (tc,{1}) (tc,0,1) nc 3 (s2tc,{2}) (tc,eα2,1) nc td {α1,α1+α2} {α2} 3 (td,{2}) (td,0,1) nc 3 (s1td,{1}) (td,eα1,1) nc te {α1} 6 (te,{1}) (te,0,1) nc tf {α2} 6 (tf,{2}) (tf,0,1) nc tg 6 (tg,) (tg,0,1) to {α1+α2} 6 tempered (to,0,1) nc Table 4.1.Irreducible representations

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 tT then the polar decomposition t=trtc determines an element νn such that tr(Xλ)=eλ,ν (see (1.3)). For each central character tp the point labeled by p in Figure 4.1 is the graph of the corresponding νpn. Assume (for pictorial convenience) that q is a positive real number and let

Hβ= {xnβ,x=0} ,andHβ±δ± { xn β,x= ln(q±2) } ,

for each positive root β. The dotted lines display the (affine) hyperplanes Hβ±δ.

Hα1 Hα2 Hα1+α2 a b c d e f g o Figure 4.1. Real parts of central characters in Table 4.1

Tempered and square integrable representations. The tempered (resp. square integrable) H-modules 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 tT be given by t(X-α1)=±q, t(X-α2)=±q. This is a special case of the central character tb in Table 4.1. For this particular special case there is one tempered representation with central character t.

Hα1 Hα2 Hα1+α2 to s1t t s2t s2s1s2ta Figure 4.2. Real parts of weights of tempered representations

The irreducible tempered representations with real central character are in one-to-one cor- respondence with the irreducible representations of the symmetric group S3 (see [BMo1989]). These representations are indexed by the partitions (3), (21), (13) of 3. Equivalently, they can be indexed by the pairs (n,ρ) which appear in the Springer correspondence. The n and ρ will also be elements of the indexing triple for the corresponding tempered representation of H. Here n is a nilpotent element of the Lie algebra 𝔤=𝔰𝔩3 and ρ is an irreducible representation of the component group ZG(n)/ZG(n). In type A the component group is always trivial. For each root βR let eβ be an element of the root space 𝔤β. The three nilpotent orbits in 𝔤 and the corresponding tempered representations of H are as in Table 4.2.

Nilpotent orbit ZG(n)/ ZG(n) Indexing triple Square integrable Wrepresentation regular 1 (ta,eα1+eα2,1) yes (3) subregular 1 (s2s1t,eα2,1) no (21) 0 1 (to,0,1) no (13) Table 4.2. Tempered representations and the Springer correspondence s1ta ta s2ta s1s2ta s2s1s2ta s2s1ta s1tb tb s2tb s1s2tb s2s1s2tb s2s1tb s1tc tc s2tc s1s2tc s2s1s2tc s2s1tc s1td td s2td s1s2td s2s1s2td s2s1td s1te te s2te s1s2te s2s1s2te s2s1te s1tf tf s2tf s1s2tf s2s1s2tf s2s1tf s1tg tg s2tg s1s2tg s2s1s2tg s2s1tg s1to to s2to s1s2to s2s1s2to s2s1to Figure 4.3. Calibration graphs for central characters in Table 4.1

The analysis.

Central characters ta, tb, tg: Since Z(t)= 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 Γ(t) and can be constructed explicitly with the use of Theorem 1.11. Up to isomorphism the principal series module M(tg) is the only irreducible H-module with central character tg. 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 tb when the irreducible module constructed by applying Theorem 1.11 to the placed skew shape (tb,) is tempered. This happens when tb=s2s1t for the weight tT given by t(X-α1)=±q, t(X-α2)=±q. The indexing triple and the calibration set for this case are still given by (tb,eα2,1) and J={α2}, respectively.

Central characters tc and td: One can use the defining relations of H to check that the only 1-dimensional representations of H are the ones with central character ta. Construct two 3- dimensional representations of H by

IndH{2}H (vc)and IndH{2}H (vs2c),

where vc and vs2c are the two one-dimensional representations of H{2} given by

T2vc=qvc, Xα1vc=vc, Xα2vc=q2vc, T2vs2c=- q-1vs2c, Xα1vs2c= q2vs2c, Xα2vs2c= q-2vs2c.

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 tc.

The central characters tc and td are taken into each other under the automorphism of the Dynkin diagram of A2 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 α1 and α2). Thus the representations with central character td can be obtained from the ones with central character tc by switching all 1’s and 2’s and changing all c’s to d’s.

Central characters te and tf: Since P(te)=, Kato’s irreducibility criterion (Theorem 1.16) implies that the principal series module M(te) is irreducible. By Theorem 1.15 this is the only irreducible with central character te. As for the case of tc and td, the central characters te and tf are taken into each other under the automorphism of the Dynkin diagram of A2. Thus the irreducible representations with central character tf can be obtained from the one with central character te by switching all 1’s and 2’s and changing all e’s to f’s.

Central characters to, Since P(to)=, Kato’s irreducibility criterion (Theorem 1.16) implies that the principal series module M(to) is irreducible. By Theorem 1.15 this is the only irreducible with central character to.

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.

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