Type A2

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

Last update: 9 March 2013

Type A2

The type A2 root system is

R= { ±α1, ±α2, ±(α1+α2) } ,

where α1,α2 =-1= α2,a1 . Then R is a root system as defined in (1.2.1), and the Weyl group W0S3. The simple roots are α1 and α2, and α1+α2 is the only other positive root.

α1 α2 α1+α2 ω2 ω1 The typeA2root system

The fundamental weights satisfy

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


P=-span {ω1,ω2}

be the weight lattice of R.

Hα1 Hα2 ω1 ω2 0 The weight latticeP

The affine Hecke algebra H (see 1.2.2) is generated as a -algebra by T1, T2, and X={XλλP}, with relations

XλXμ = Xλ+μ,for λ,μP (2.4) T1T2T1 = T2T1T2 (2.5) Ti2 = (q-q-1)Ti +1,fori=1,2 (2.6) Xω1T1 = T1 Xω2-ω1+ (q-q-1) Xω1 (2.7) Xω1T2 = T2 Xω1-ω2+ (q-q-1) Xω2 (2.8) Xω1T2 = T2Xω1 (2.9) Xω2T1 = T1Xω2 (2.10)


[X]= {XλλP} ,

a subalgebra of H, and let

T=Hom-alg ([X],).

Then W0 acts on X by

s1·Xω1 = Xω2-ω1, s1·Xω2 = Xω2, s2·Xω1 = Xω1, and s2·Xω2 = Xω1-ω2,

and W0 acts on T by

(w·t) (Xλ)=t (Xw-1λ) .



be the root lattice of R. Let

[Q]= {XλλQ}

and let

TQ= Hom-alg ([Q],).


tz,w: [Q] by Xα1 z Xα2 w.

We can also visualize TQ using the following picture. Here, the hyperplane Hα= { tTQ t(Xα)=1 } is drawn as a solid line. The hyperplanes Hα+δ= { tTQ t(Xα)=q2 } are drawn in this picture as dashed lines. The weight tqx,qy is the point x units away from Hα1 and y units away from Hα2. The action of W0 is also visible in this picture, as si is given by reflection in the hyperplane Hαi.

Hα1 Hα2 Hα1+α2 TQ

For each tz,wTQ, there are 3 elements tT with tQ=tz,w, determined by

t(Xω1)3= z2wandt (Xω2)=t (X-ω1) ·zw.

The dimension of the modules with central character t and the submodule structure of M(t) depends only on tQ. Thus we begin by examining the W0-orbits in TQ. The structure of the modules with weight t depends virtually exclusively on P(t)= { αR+ t(Xα)= q±2 } and Z(t)= { αR+ t(Xα)=1 } . For a generic weight t, P(t) and Z(t) are empty, so we examine only the non-generic orbits.

Proposition 2.3. If tTQ, and P(t)Z(t), then t is in the W0-orbit of one of the following weights:

t1,1, t1,q2, tq2,1, tq2,q2, { t1,z z×z1, q±2 } ,or { tq2,z z×z1, q±2,q-4 }


First, assume generic q.

Case 1: If Z(t) contains two positive roots, then it must contain the third. This implies t=t1,1.

Case 2: If Z(t) contains only one root, by applying an element of W0, assume that it is α1. Then t(Xα2)=t(Xα1+α2), so either P(t)= or P(t)={α2,α1+α2}. The first central character is t1,z for some z1 or q±2. (If z=1 or z=±2, either P(t) or Z(t) would be larger.) For the second case, there are two potential choices for the orbit, arising from choosing t(Xα2)=q2 or q-2. However, t1,q-2 is in the same orbit as tq2,1.

Case 3: Now assume that Z(t)=. If P(t) is not empty, assume that α1P(t) and t(Xα1)=q2. Then t(Xα2)q-2 by assumption on Z(t). Then it is possible that α2P(t), in which case t=tq2,q2. If α1+α2P(t) then t(Xα2)=q-4 and t=tq2,q-4= s2s1tq2q2. Otherwise, t=tq2,z for some z1,q±2,q-4.

Remark: There is some redundancy present in the list of central characters above for specific values of q. If q2=-1, then t1,q2, tq2,1, and tq2,q2 are all in the same W0-orbit. Also in this case, tq2,z= t-1,z= s1t-1,-z. If q2=1, then t1,1= tq2,1= t1,q2= tq2,q2, and t1,z=tq2,z. Also note that for every generic weight tz,w, there are six weights in its W0-orbits, all of which are of course generic.

It is helpful to draw a picture of the weights { tqx,qy x,y } for various values of q. Solid lines in these pictures show sets of the form

Hα= { tTQ t(Xα)=1 } ,

for αR+, while dashed lines denote sets of the form

Hα±δ= { tTQ t(Xα)= q±2 } ,

for αR+.

Hα1 Hα2 Hα1+α2 Hα1+α2+δ Hα1+δ Hα2+δ t1,1 t1,q2 t1,z tz,w tq2,1 tq2,q2 tq2,z Figure 1: Representatives of some central characters of modules overH, with generalq. Hα1 Hα1+δ Hα1-δ Hα1 Hα2-δ Hα2+δ Hα2 Hα1+α2 Hα1+α2-δ Hα1+α2+δ t1,1 t1,q2 t1,z tz,w tq2,1 tq2,q2 tq2,z Figure 2:q2a primitive third root of unity. Hα1 Hα1±δ Hα2±δ Hα2 Hα1+α2 Hα1+α2±δ t1,1 t1,q2 t1,z tz,w tq2,z Figure 3:q2-1. Hα1 Hα2 Hα1+α2 t1,1 tz,w tq2,z Figure 4:q2=1.

Analysis of the characters

Proposition 2.4. Fix ε=e2πi/3. The 1-dimensional H-modules are

Lq2,q2: H T1q T2q Xω1q2 Xω2q2 Lεq2,ε2q2: H T1q T2q Xω1εq2 Xω2ε2q2 Lε2q2,εq2: H T1q T2q Xω1ε2q2 Xω2εq2 Lq-2,q-2: H T1-q-1 T2-q-1 Xω1q-2 Xω2q-2 Lεq-2,ε2q-2: H T1-q-1 T2-q-1 Xω1εq-2 Xω2ε2q-2 Lε2q-2,εq-2: H T1-q-1 T2-q-1 Xω1ε2q-2 Xω2εq-2


A straightforward check shows that the maps above respect the defining relations for H (2.4) - (2.10), so that the maps are homomorphisms.

Let v be any 1-dimensional H-module. By (2.6) and (2.5),

Tiv=qvor Tiv=-q-1v andT1v= T2v.

Case 1: T1v=T2v=qv. By (2.7) and (2.8),

X2ω1v=q2 Xω2v,and X2ω2v=q2 Xω1v.


Xα1v= X2ω1-ω2v= q2v,and Xα2v= X2ω2-ω1= q2v,

so that

X3ω1v= X2α1+α2v =q6v,and X3ω2v= Xα1+2α2 v=q6v.

Hence, Xω1v=εiq2v, where i=0,1, or 2, and Xω2v=ε2iq2v.

Case 2: T1v=T2v=-q-1v. Then

X2ω1v=q-2 Xω2vand X2ω2v=q-2 Xω1v.

This implies that

Xα1v= X2ω1-ω2v= q-2v,and Xα2v= X2ω2-ω1v =q-2v,

so that

X3ω1v= X2α1+α2 v=q-6v, andX3ω2v= Xα1+2α2v= q-6v.

Hence, Xω1v=εiq-2v, where i=0,1, or 2, and Xω2v=ε2iq-2v.

Principal Series Modules and Local Regions

Let tT. The principal series module is

M(t)= Ind [X] H t=H [X] t,

where t is the one-dimensional [X]-module given by

t=span{vt} andXλ vt=t(Xλ) vt.

By (1.6), every irreducible H module is a quotient of some principal series module M(t). Thus, finding all the composition factors of M(t) for all central characters t will find all the irreducible H-modules.

Assume for now that q21.

Case 1: P(t) empty. By 1.8, if P(t)= { αR+ t(Xα1)=1 } is empty, then M(t) is irreducible and is the only irreducible module with central character t. This case includes the central characters t1,1, t1,z, and tz,w for generic z,w.

Since P(t)=, there is one local region

t,=W0 /Wt,

the set of minimal length coset representatives of Wt cosets in W0, where Wt is the stabilizer of t in W0. If w and siw are both in (t,) then τi: M(t)wtgen M(t)siwtgen is a bijection. The following pictures show (t,) with one dot in the chamber w-1C for each basis element of M(t)wtgen.

These pictures also show the weight space structure of M(t). For t=t1,1, all of M(t) is in the t weight space, so all the dots lie in the same chamber. For t=t1,z, s1t=t, so that the weights of M(t) are t, s2t, and s1s2t, and each weight space is 2-dimensional. The weight t=tz,w is regular, so there are six different weights. In each case, all the dots lie in the same local region, a region bounded by solid and/or dashed lines.

(M(t)) tgen =M(t) (M(t)) tgen (M(t)) s2tgen (M(t)) s1s2tgen (M(t))t (M(t))s2t (M(t))s1s2t (M(t))s1s2s1t (M(t))s2s1t (M(t))s1t t1,1,q21 t1,z,q21 tz,w

Case 2: Z(t)=, P(t). This case includes the central characters tq2,q2, and tq2,z. If Z(t) is empty, then M(t) is calibrated and the irreducible modules with central character t are in one-to-one correspondence with the local regions and the components of the calibration graph. Each local region in the following pictures is a set of chambers between two dashed lines.

For the weight tq2,q2 for generic q, there are four local regions. If q6=1, each weight is in a local region by itself and thus there are six different representations in that case. For the weight tq2,z, there are two local regions. These local regions are bounded by dashed lines, which also serve as barriers of sorts between the composition factors. This is a combinatorial reflection of the fact that if the τ operator between two weight spaces of M(t) is not invertible, those weight spaces will be in different composition factors. Weight spaces with invertible τ operators between them are separated by dotted lines since they are in the same composition factor and local region.

(M(t))t (M(t))s2t (M(t))s1s2t (M(t))s1s2s1t (M(t))s2s1t (M(t))s1t (M(t))t (M(t))s2t (M(t))s1s2t (M(t))s1s2s1t (M(t))s2s1t (M(t))s1t (M(t))t (M(t))s2t (M(t))s1s2t (M(t))s1s2s1t (M(t))s2s1t (M(t))s1t tq2,q2, q41,q61 tq2,q2,q2 a primitive third root of unity tq2,z,q2 1

Case 3: P(t), Z(t). The only central characters with both Z(t) and P(t) nonempty are t1,z=tq2,z when q2=1, and tq2,1 and t1,q2 in all cases. If q4=1, then tq2,1 and t1,q2 are in the same orbit, and are in the same orbit as tq2,q2. If q2=1, then tq2,1= t1,q2= t1,1. In each of these pictures, there are three local regions, but the structure of M(t) is slightly more complicated in this case.

The composition factors of M(t) each have some weight wt with t(Xαi)=1 for some simple root αi. Then 1.11 shows that when q2±1, this composition factor must be at least 3-dimensional. In this case, these composition factors are shown with the two dots in that weight space M(t)wt connected to each other, as well as to a dot in the next chamber, since the basis vectors correpsonding to these dots must lie in the same irreducible. When q2=-1, the two dimensional weight space M(t)wt makes up an entire composition factor.

(M(t))t (M(t))s2t (M(t))s1s2t (M(t))t (M(t))s2s1t (M(t))s1t (M(t))t (M(t))s2t (M(t))s1s2t t1,q2, q41 tq2,1, q41 t1,q2, q2=-1

To prove that these pictures do reflect the structure of M(t), rather than analyzing M(t) directly, it is easier to construct several irreducible modules with central character t and show that they must account for all the composition factors of M(t). Let q2,1 be the 1-dimensional H{1}-module spanned by vt and let 1,q-2 be the 1-dimensional H{2}-module spanned by vs2s1t, given by

Xλvt=t (Xλ)vt, andT1vt= qvt,and Xλvs2s1t =(s2s1t) (Xλ) vs2s1t andT1 vs2s1t=- q-1vs2s1t .


M=HH{1} q2,1and N=HH{2} 1,q-2

are 3-dimensional H-modules with central character tq2,1.

Proposition 2.5. Let M=HH{1}q2,1 and N=H H{2} 1,q-2.

  1. If q41 then M and N are irreducible.
  2. If q2=-1 then Ms1t is an irreducible submodule of M and Ns1t is an irreducible submodule of N. The quotients N/Ns1t and M/Ms1t are irreducible.


(a) Assume q41. If either M or N were reducible, it would have a 1-dimensional submodule or quotient, which cannot happen since the 1-dimensional modules have central character tq2,q2. Thus both M and N are reducible.

(b) If q2=-1, then note that the map

θ: H H T1T2 T2T1 Xω1Xω2 Xω2Xω1

is an automorphism of H. If ϕM:HGL3 and ϕN:HGL3 are the maps describing the action of H on M and N respectively, then ϕMθ=ϕN and ϕNθ=ϕM. Thus if one of M and N is irreducible, the other is as well. However, this would account for all the composition factors of M(t), a contradiction since there is a 1-dimensional module with central character t. Thus M and N must both be reducible, so M has a 1-dimensional submodule or quotient. It must have weight s1t since that is the only weight of M that supports a 1-dimensional module. Then the 2-dimensional composition factor is Mtgen, by Lemma 1.11. Then since

HomH (M,Mtgen)= HomH{1} ( vt, Mtgen ) 0,

Ms1t is a submodule of M. Similarly, Ns1t is a submodule of N since Ns2s1tgen must be a quotient of N.

(M(t))t (M(t))s2t (M(t))s1s2t (M(t))t (M(t))s2s1t (M(t))s1t t1,q2, q41 tq2,1, q41

To construct the irreducibles with tQ=t1,q2, composing with

ϕ: H H T1T2 T2T1 Xω1Xω2 Xω2Xω1,

an automorphism of H, gives a bijection between representations with central character tq2,1 and those with central character t1,q2.

(M(t))t (M(t))s2t (M(t))s1s2t t1,q2,q2= -1

If q2=1, then the results of section 1.2.9 suffice to classify the representations of H with central characters t1,1 and t1,z for zq±2. Specifically, if tQ=t1,1, then Wt=W0 and a H-module is merely a W0-module (via the isomorphism H[W0]) on which Xλ[X] acts by the scalar t(Xλ). In fact, M(t) considered as a W0 module is the regular W0 module. If tQ=t1,z for z1, then Wt={1,s1}. Since Wt has two 1-dimensional irreducible representations, H has two irreducible 3-dimensional representations obtained by inducing up from H{1}.

(M(t))t=M(t) (M(t)) t (M(t)) s2t (M(t)) s1s2t (M(t))t (M(t))s2t (M(t))s1s2t (M(t))s2s1s2t (M(t))s2s1t (M(t))s1t t1,1,q2=1 t1,z,q2=1 tz,w,q2=1

Note that in all cases, the local region picture is a picture of a small neighborhood around the point corresponding to t in the picture of TQ. The necessary information to understand the composition factors of M(t) is contained in the local region picture around t.

The following tables summarize the classification. It should be noted that for any value of q with q2±1 and q2 not a primitive third root of unity, the representation theory of H can be described in terms of q only. If q2 is a primitive root of unity of order 3 or less, then the representation theory of H does not fit that same description. This fact can be seen through a number of different lenses. It is a reflection of the fact that the sets P(t) and Z(t) for all possible central characters t can be described solely in terms of q. In the local region pictures, this is reflected in the fact that the hyperplanes Hα and Hα±δ are distinct unless q2 is a root of unity of order 3 or less. When these hyperplanes coincide, the sets P(t) and Z(t) change for characters on those hyperplanes.

tQ Z(t) P(t) Dimensions of Irreds. t1,1 R+ 6 t1,z {α1} 6 t1,q2 {α1} {α2,α1+α2} 3,3 tq2,1 {α2} {α1,α1+α2} 3,3 tq2,q2 {α1,α2} 1,1,2,2 tq2,z {α1} 3,3 tz,w 6 Table 4: Table of possible central characters in TypeA2, withqgeneric.

tQ Z(t) P(t) Dimensions of Irreds. t1,1 R+ 6 t1,z {α1} 6 t1,q2 {α1} {α2,α1+α2} 3,3 tq2,1 {α2} {α1,α1+α2} 3,3 tq2,q2 R+ 1,1,1,1,1,1 tq2,z {α1} 3,3 tz,w 6 Table 5: Table of possible central characters in TypeA2, withq6=1.

tQ Z(t) P(t) Dimensions of Irreds. t1,1 R+ 6 t1,z {α1} 6 t1,q2 {α1} {α2,α1+α2} 1,2,2 tq2,z {α1} 3,3 tz,w 6 Table 6: Table of possible central characters in TypeA2, withq2=-1.

tQ Z(t) P(t) Dimensions of Irreds. t1,1 R+ R+ 1,1,2 t1,z {α1} {α1} 3,3 tz,w 6 Table 7: Table of possible central characters in TypeA2, withq=-1.

Notes and References

This is an excerpt from Matt Davis' Ph.D Thesis entitled Representations of Rank Two Affine Hecke Algebras at Roots of Unity, University of Wisconsin, 2010.

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