Classification of calibrated representations

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

Last update: 13 November 2012

Classification of calibrated representations

For simple roots αi and αj in R and let Rij be the rank-two root subsystem of R generated by αi and αj. A weight tT is calibratable if, for every pair i,j,ij,t is a weight of a calibrated representation of the rank-two affine Hecke (sub)algebra generated by Ti, Tj and [X]. A local region

(t,J)is skewifwt is calibratable for allw (t,J).

The classification of irreducible representations of rank-two affine Hecke algebras given in [Ram2002] can be used to state this condition combinatorially. Specifically, a weight tT is calibratable if

  1. for all simple αi, 1in, t(Xαi)1, and
  2. for all pairs of simple roots αi and αj such that { αRij t(Xα)=1 } , the set { αRij t(Xα)=q±2 } contains more than two elements.

Condition (a) says that t is regular for all rank-1 subsystems of R generated by simple roots. This condition guarantees that the weight is “calibratable” (i.e., appears as a weight of some calibrated representation) for all rank-1 affine Hecke subalgebras of H. Condition (b) is an “almost regular” condition on t with respect to rank-2 subsystems generated by simple roots.

Remark. The conversion between the definition of calibratable weight and the combinatorial condition given in (a) and (b) is as follows. Consider a rank-two affine Hecke algebra H.

  1. By Theorem 2.12, (a) and (d), local regions (t,J) with t regular satisfy (a) and (b) and always contribute calibrated representations of H.
  2. Using the notation of [Ram2002], the local regions (t,J) with t nonregular and which satisfy both conditions (a) and (b) are:
    type A2: none,
    type C2: (tb,{α1}) and (tb,{α1,α1+α2}) (for each of these P(t) contains 3 elements),
    type G2: (te,J) with J and JP(te) (for each of these P(te) contains 4 elements).

From (A) and (B) it follows that the local regions which satisfy (a) and (b) do contribute calibrated weights. The following shows that the other local regions do not contribute calibratable weights.

  1. By Lemma 3.1(a) local regions (t,J) with a weight ξ=wt, wF(t,J) such that ξ(Xαi)=1 do not satisfy (a) and, by inspection of the tables in [Ram2002], they never contribute a calibrated representation.
  2. Using the notation of [Ram2002], the local regions which satisfy condition (a) but not condition (b) are
    type A2: (tc,{α2}) and (td,{α1}),
    type B2: (td,{α2}),
    type G2: (ti,{α2}), (tf,{α1}).

(Note that to satisfy (b) Z(t) must be nonempty.) From the tables in [Ram2002] we see that none of these local regions supports a calibrated representation.

Remark. The paper [Ram2002] does not treat roots of unity. However, it is interesting to note that, provided q2±1, the methods of [Ram2002] go through without change to classify all representations of rank-two affine Hecke algebras even when q2 is a root of unity. This classification can be used (as in the previous remark) to show that (a) and (b) above still characterize calibratable weights when q2 is a root of unity such that q2±1. The key point is that Lemma 1.19 of [Ram2002] still holds. If q2=-1 then Lemma 1.19 of [Ram2002] breaks down at the next to last line of the proof in the statement “... forces ϕ(wt(Tj)), to have Jordan blocks of size 1 ....” When q2=-1 it is possible that ϕ(wt(Tj)) has a Jordan block of size 2. If q2=1 then one can change the definition of the τ-operators and use similar methods to produce a complete analysis of simple H-modules, but we shall not do this here, choosing instead to exclude the case q2=1 for simplicity of exposition.

The following lemma provides fundamental results about the structure of irreducible calibrated H-modules. We omit the proof since it is accomplished in exactly the same way as in [KRa2002, Lemmas 4.1 and 4.2].

Lemma 3.1 Let M be an irreducible calibrated module. Then, for all tT such that Mt0,

  1. If tT such that Mt0 then t(Xαi)1 for all 1in.
  2. If tT such that Mt0 then dim(Mt)=1.
  3. If tT such that Mt and Msit are both nonzero then the map τi:MtMsit is a bijection.

This lemma together with the classification of irreducible modules for rank-two affine Hecke algebras gives the following fundamental structural result for irreducible calibrated H-modules. The proof is essentially the same as the proof of Proposition 4.3 in [KRa2002]. We repeat the proof here for continuity.

Theorem 3.2 If M is an irreducible calibrated H-module with central character tT then there is a unique skew local region (t,J) such that

dim(Mwt)= { 1 for allw F(t,J), 0 otherwise.


By Lemma 3.1(b) all nonzero generalized weight spaces of M have dimension 1 and by Lemma 3.1(c) all τ-operators between these weight spaces are bijections. This already guarantees that there is a unique local region (t,J) which satisfies the condition. It only remains to show that this local region is skew.

Let Hij be the subalgebra generated by Ti, Tj and [X]. Since M is calibrated as an H-module it is calibrated as a Hij-module and so all factors of a composition series of M as an Hij-module are calibrated. Thus the weights of M are calibratable. So (t,J) is a skew local region.

The following proposition shows that the weight space structure of calibrated representations, as determined in Theorem 3.2, essentially forces the H-action on a weight basis. The proof is quite similar to the proof of Proposition 4.4 in [KRa2002]. However, we include the details since there is a technicality here; to make the conclusion in (3.4) we use the fact that the group X corresponds to the weight lattice L=P.

Proposition 3.3. Let M be a calibrated H-module and assume that for all tT such that Mt0,

(A1)t(Xiα) 1for all1i n,and(A2) dim(Mt)=1.

For each bT such that Mb0 let vb be a nonzero vector in Mb. The vectors {vb} form a basis of M. Let (Ti)cb and b(Xλ) given by

Tivb=c (Ti)cbvc andXλ vb=b(Xλ) vb.


  1. (Ti)bb= (q-q-1)/ (1-b(X-αi)) , for all vb in the basis,
  2. if (Ti)cb0 then c=sib,
  3. (Ti)b,sib (Ti)sib,b= ( q-1+ (Ti)bb ) ( q-1+ (Ti)sib,sib ) .


The definition equation for H,

XλTi-Ti Xsiλ= (q-q-1) Xλ-Xsiλ 1-X-αi ,


c ( c(Xλ) (Ti)cb- (Ti)cbb (Xsiλ) ) vc= (q-q-1) b(Xλ)- b(Xsiλ) 1-b(X-αi) vb.

Comparing coefficients gives

c(Xλ) (Ti)cb- (Ti)cbb (Xsiλ) =0,ifbc ,and b(Xλ) (Ti)bb- (Ti)bbb (Xsiλ) =(q-q-1) b(Xλ)- b(Xsiλ) 1-b(X-αi) .

These relations give:

if (Ti)cb0 thenb(Xsiλ) =c(Xλ)for all XλX,and (Ti)bb= q-q-1 1-b(X-αi) ifb(X-αi) 1andb(Xλ) b(Xsiλ) for someXλX.

By assumption (A1), b(Xαi)1 for all i. For each fundamental weight ωi, XωiX and b(Xsiωi)= b(Xωi-αi) b(Xωi) since b(Xαi)1. Thus we conclude that

Tivb= (Ti)bbvb+ (Ti)sib,b vsibwith (Ti)bb= q-q-1 1-b(X-αi) . (3.4)

This completes the proof of (a) and (b). By the definition of H, the vector

Ti2vb= ( (Ti)bb2+ (Ti)b,sib (Ti)sib,b ) vb+ ( (Ti)bb+ (Ti)sib,sib ) (Ti)sib,b vsib

must equal

( (q-q-1) Ti+1 ) vb= ( (q-q-1) (Ti)bb+1 ) vb+(q-q-1) (Ti)sib,b vsib.

Using the formula for (Ti)bb and (Ti)sib,sib, we find (Ti)bb+ (Ti)sib,sib =(q-q-1). So, by comparing coefficients of vb, we obtain the equation

(Ti)b,sib (Ti)sib,b = (q-(Ti)bb) ((Ti)bb+q-1) = (q-1+(Ti)bb) ( q-1+ (Ti)sib,sib ) .

Theorem 3.5. Let (t,J) be a skew local region and let (t,J) index the chambers in the local region (t,J). Define

H(t,J)=-span { vww (t,J) } ,

so that the symbols vw are a labeled basis of the vector space H(t,J) Then the following formulas make H(t,J) into an irreducible H-module: For each w(t,J),

Xλvw = (wt) (Xλ)vw, forXλX, and Tivw = (Ti)wwvw + ( q-1+ (Ti)ww ) vsiw, for1in,

where (Ti)ww= (q-q-1)/ (1-(wt)(X-αi)) , and we set vsiw=0 if siw(t,J).


Since (t,J) is a skew local region (wt)(X-αi)1 for all w(t,J) and all simple roots αi. This implies that the coefficient (Ti)ww is well defined for all i and w(t,J).

By construction, the nonzero weight spaces of H(t,J) are (H(t,J)) wt gen = (H(t,J)) wt where w(t,J). Since dim (H(t,J)) =1 for u(t,J) any proper submodule N of H(t,J) must have Nwt0 and Nwt=0 for some ww with w,w(t,J). This is a contradiction to Corollary 2.19. So H(t,J) is irreducible if it is an H-module.

It remains to show that the defining relations for H are satisfied. This is accomplished as in the proof of [KRa2002, Theorem 4.5]. The only relation which is tricky to check is the braid relation. This can be verified as in [KRa2002] or it can be checked by case by case arguments (as in [Ram1998]).

We summarize the results of this section with the following corollary of Theorem 3.2 and the construction in Theorem 3.5.

Theorem 3.6. Let M be an irreducible calibrated H-module. Let tT be (a fixed choice of) the central character of M and let J=R(w)P(t) for any wW such that Mwt0. Then (t,J) is a skew local region and MH(t,J) where H(t,J) is the module defined in Theorem 3.5.

Notes and References

This is an excerpt of the paper entitled Affine Hecke algebras and generalized standard Young tableaux written by Arun Ram in 2002, published in the Academic Press Journal of Algebra 260 (2003) 367-415. The paper was dedicated to Robert Steinberg.

Research supported in part by the National Science Foundation (DMS-0097977) and the National Security Agency (MDA904-01-1-0032).

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