Type A1

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

Last update: 9 March 2013

Type A1

The Affine Hecke Algebra

The type A1 affine Hecke algebra is built on the root data of SL2. This is the only reduced rank one root system (up to isometry.) Though our main goal is the rank two affine Hecke algebras, the rank one case is fundamental in the classifcation of the higher-rank cases.


P=ω1andX= {Xkω1k}

so that X is the group generated by Xω1 and is isomorphic to P. The Weyl group is

W0={1,s1} withs12=1 ,ands1 Xω1= X-ω1

defines an action of W0 on X.

Let q×. The affine Hecke algebra of type A1 is

H=-span { Xkω1, T1Xkω1 k } ,

with relations

T12 = (q-q-1) T1+1 (2.1) Xkω1 Xmω1 = X(k+m)ω1 (2.2) Xω1 T1 = T1X-ω1+ (q-q-1) Xω1. (2.3)


[X]=span {Xkω1k} ,with Xkω1 Xω1= X(k+)ω1

is a subalgebra of H. A weight t is an element of

T=Hom-alg ([X],)= HomGp(X,×).

A weight t:[X] is completely determined by the value t(Xω1), which must be invertible, so that T×.

This definition is a special case of the definitions given in sections 1.2.1 and 1.2.2, using the root system


with α1=2ω1.

Proposition 2.1. There are four 1-dimensional H representations

Lq: H Xω1 q T1 q L-q: H Xω1 -q T1 q L-q-1: H Xω1 -q-1 T1 -q-1 Lq-1: H Xω1 q-1 T1 -q-1

If q2=-1, then q=-q-1, so that Lq=L-q-1 and L-q=Lq-1.


A straightforward check of the relations (2.1) − (2.3) shows that the maps L±q±1 are homomorphisms.

Assume M is a 1-dimensional H-module, with M=span{v}. By (2.1),

T1v=qvorT1 v=-q-1v.

By (2.3), if T1v=qv, then

X2ω1v=q2v and eitherMLqor ML-q.

Similarly, if T1v=-q-1v, then

X2ω1v=q-2v and eitherMLq-1 orML-q-1.

Let tT and let t=span{vt} be the one-dimensional [X]-module given by

Xλvt=t (Xλ)vt.

Then the principal series module is

M(T)= Ind[X]H t=H [X] t=span {vt,T1vt},

so that {vt,T1vt} is a basis for M(t).

Proposition 2.2.

  1. If t(Xω1) ±q±1, then M(t) is irreducible.
  2. If t(Xω1)=±q±1, M(t) has composition series M(t)M10.
    If t(Xω1)=±q, then
    M1 = span{T1vt-qvt} L±q-1 and M(t)/M1 L±q,and
    If t(Xω1)=±q-1, M1 = span{T1vt-q-1vt} L±q and M(t)/M1 L±q-1.
  3. If q2=1 and t(Xω1)=±q, then M(t)=M+M-, where M+ = span{T1vt-qvt}, and M- = span{T1vt+q-1vt} .

Note that if q is a primitive fourth root of unity, the two cases in (b) coincide.


(a) Any non-trivial proper submodule of M(t) must be 1-dimensional. Such a submodule can exist only if t(Xω1)=±q±1, by proposition 2.1.

(b) If t(Xω1)=±q, let n=T1vt-qvt. Then

T1n=T12vt-q T1vt=-q-1T1 vt+vt=-q-1n


Xω1n = T1(X-ω1) vt+(q-q-1) Xω1vt-q Xω1vt = t(X-ω1) ( T1vt-q-1t (X2ω1)vt ) = t(X-ω1) (T1vt-qvt) = t(X-ω1)n,

so that N=n is a 1-dimensional submodule of M(t). Since n=T1vt-qvt, if vt is the image of vt in M(t)/N, then T1vt=qvt and Xω1vt=t(Xω1)vt.

If t(Xω1)=±q-1, then n=T1vt+q-1vt spans a 1-dimensional submodule NM(t), with T1vt=-q-1vt and Xω1vt= t(Xω1)vt in M(t)/N.

If t(Xω1)t(X-ω1), then M(t) has two distinct weight spaces: M(t)t=vt and M(t)s1t=n. If M(t) were the direct sum of two 1-dimensional submodules, it would have to be the direct sum of the two weight spaces. However, Hvt is all of M(t), so that M(t) is indecomposable.

(c) If q2=1 and t(Xω1)=±q=±q-1, then the weight space M(t)t is all of M. Both T1vt-qvt and T1vt+q-1vt span 1-dimensional submodules of M(t) which are disjoint.

Remark: The machinery of section 1.2 also suffices to obtain the information in this theorem. Part a is exactly Kato’s criterion (Theorem 1.8). Part b follows from the properties of the τ operators, since τ1(vt)0, but τ1τ1(vt)=0 exactly when t(Xα1)=q±2. In that case, T1·τ1(m) is a multiple of τ1(m), so τ1(m) spans a 1-dimensional submodule of M(t). We have given an explicit proof to keep the type A1 example as clear as possible.

For z×, let tzT be the weight given by


Pictorially, identify {tqxx} with the real line. In this picture, the set

Hα= { xtqx (Xα1)=1 }

is marked with a solid line, while

Hα+δ= { xtqx (Xα1)=q2 } and Hα-δ= { xtqx (Xα1)=q-2 }

are denoted by dashed lines.

Hα-δ Hα Hα+δ tq-1 tq0 tq tq2 tq3 Characterstqx, genericq.

If q is a primitve 2th root of unity then {tqxx} is identified with /2 and Hα= {kk} ={0,} .

The following is the specific case =2, so that Hα= {0,±2,±4,} ={0,2} The periodicity is evident in the picture.

Hα Hα±δ Hα Hα±δ Hα Hα±δ Hα Hα±δ Hα tq0 tq1 tq2 tq3 tq0 tq tq2 tq3 tq0 Characterstqx,q4=1.

If q2=1, then Hα±δ=Hα= {0,1} .

Hα Hα Hα Hα Hα Hα Hα Hα Hα tq0 tq tq0 tq tq0 tq tq0 tq tq0 Characterstqx,q2=1.

The structure of M(t) can be seen using the τ operator described above (1.2.4). If t(Xα1)=1, then τ1 is not defined. If t(Xα1)=q±21, then τ1 is non-zero on M(t)t, but is zero on M(t)s1t. This accounts for the structure of M(t) described above (Proposition 2.2 (b)). If t(Xω1)q2 or 1, then τ1 is invertible on M(t). The following pictures show the “local regions” as described in (1.2.6), which describe M(t).

Each region is divided into two chambers, representing the two weights in the W0-orbit of the central character. Each chamber wC contains a number of large dots equal to the dimension of M(t)wtgen, the generalized weight space with weight wt, and lines connecting the dots show that the corresponding basis vectors are in the same composition factor. A solid line is drawn where τ1 is not defined, as in the following picture.

t1, t-1, q21 t1, t-1, q2=1

These two pictures represent weights with t(Xα1)=1, so that s1t=t. Hence the two dots representing the weight spaces are drawn in the same chamber. If q21, M(t) is irreducible, so the weight spaces are connected by arcs. The picture reflects the fact that there is only one local region, since P(t)=.

tq, t-q, q21 tz,z q2,1

These pictures show M(t) for the other possible weights t. A dashed line denotes a weight t with t(Xα1)=q±2, so that the operator τ1 on the weight spaces of M(t) is well-defined, but not invertible in both directions. This means that the corresponding weight spaces will be in different composition factors of M(t). A dotted line denotes a weight t with t(Xα1)q2 or 1. In this case τ1 is invertible, forcing the weight spaces to be in the same composition factor of M(t). Accordingly, the dots representing them are connected.

Notice also the connection to the drawings of central characters above. The pictures of the local regions at a weight tqx are a picture of a small open neighborhood around the point tqx in the picture of the characters. This correspondence is not as clear in this case as in others, since it is the smallest example of the affine Hecke algebra, but the essential ingredients are present. The weights of the M(t) are all displayed, as are the actions of the τ operators that determine the composition factors of M(t).

The complete classification of H modules is summarized in the following tables.

tDimensions of Irreds. t12 t-12 tq1,1 t-q1,1 tz,z±1or±q2 Table 1: Table of possible central characters in TypeA1, withqgeneric.

tDimensions of Irreds. t12 t-12 tq1,1 tz,z±1or±q2 Table 2: Table of possible central characters in TypeA1, withq2=-1.

tDimensions of Irreds. t11,1 t-11,1 tz,z±1or±q2 Table 3: Table of possible central characters in TypeA1, withq=-1.

It should be noted that for any value of q with q2±1, the representation theory of H can be described in terms of q only. If q2=±1, 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=±1. When these hyperplanes coincide, the sets P(t) and Z(t) change for characters on those hyperplanes.

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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