Classification of Calibrated Representations

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

Last update: 27 April 2013

Classification of Calibrated Representations

Structural results

We first examine some properties that hold for irreducible modules that are calibrated, i.e., can be decomposed into a direct sum of weight spaces (see (2.10)). This section follows closely similar results for affine Hecke algebras in [Ram1998].

Lemma 4.1. Let M be an irreducible calibrated module. Then, for all γ𝔥 such that Mγ0,

  1. γ(αi)0 for all 1in,
  2. dim(Mγ)=1.


The proof is by contradiction. Assume γ(αi)=0. Let A1 be the subalgebra of generated by tsi and all x𝔥*. Then the two-dimensional A1 principal series module M(γ) is irreducible and there is an A1-module homomorphism given by

M(γ) M vγ mγ

where mγ is a nonzero element of Mγ. Since M(γ) is simple, this is an injection and thus, M is not calibrated since M(γ) is not calibrated. Thus γ(αi)0.
(b) The proof is by contradiction. Assume γ𝔥 is such that dim(Mγ)>1. Let mγ be a nonzero element of Mγ. Since M is calibrated τi acts on mγ as a linear combination of the action of tsi and a multiple of the identity. Since M is irreducible, it follows from Proposition 2.5(b) that the action of the τ-operators must generate all of M. Thus, since dim(Mγ)>1, there is a sequence of τ-operators such that

nγ=τi1 τi2 τipmγ

is a nonzero vector in Mγ which is not a multiple of mγ.

Assume that the sequence τi1τi2τip is chosen so that p is minimal. Since the τ-operators in this sequence are all well defined, the elements siksipγ,1kp, in the orbit Wγ correspond (under the bijection in (2.7)) to a sequence of chambers in 𝔥* on the positive side of all Hα,αZ(γ). Each chamber in this sequence shares a face with the next chamber in the sequence. Since both nγ and mγ are in Mγ, this is a sequence which begins and ends at the chamber C. Since the chambers are in bijection with the elements of W, it follows that si1sip=1 in W.

This means that there is some 1<kp such that si1sik is not reduced and we can use the braid relations to rewrite this word as si1 sik-2 siksik. By Proposition 2.5(e) the τ-operators also satisfy the braid relations and so

nγ= τi1 τi2 τik-2 τik τik τipmγ.

By Proposition 2.5(c), the operator τikτik above will act (on τik+1τipmγ) by a constant ξ and so

nγ=ξ· τi1 τi2 τik-2 τik τik τipmγ,

where the constant ξ is nonzero since nγ is nonzero. But the expression

ξ-1nγ= τi1 τi2 τik-2 τik τik τipmγ,

is shorter than the original expression of nγ and this contradicts the minimality of p. It follows that dim(Mγ)1.

Lemma 4.2. Let M be an irreducible calibrated module. Suppose that Mγ and Msiγ are both nonzero. Then the map τi:MγMsiγ is a bijection.


By Lemma 4.1(b), dim(Mγ)=dim(Msiγ)=1, and thus it is sufficient to show that τi is not the zero map. Let vγ be a nonzero vector in Mγ. Since M is irreducible, there must be a sequence of τ-operators such that

vsiγ= τi1 τipvγ

is a nonzero element of Msiγ. Let p be minimal such that this is the case. Since τiτi1τipvγMγ, it follows, as in the second paragraph of the proof of Lemma 4.1(b), that sisi1sip=1 in W. For notational convenience let i0=i. Let 0k<p be maximal such that siksik+1sip is not reduced. If k0, then we can use the braid relations to get

vsiγ= τi1 τik τik τik+2 τipvγ.

Since τikτik acts on τik+2 τipvγ by a constant ξ,

vsiγ=ξ· τi1 τik-1 τik+2 τip vγ,

and ξ0 since vsiγ is not 0. But this contradicts the minimality of p. Thus we must have that k=0,p=1 and


Thus, since vsiγ0, τi0.

For simple roots αi and αj in R, let Rij be the rank two root subsystem of R generated by αi and αj. A weight μ𝔥 is skew if

  1. for all simple roots αi, 1in, μ(αi)0,
  2. for all pairs of simple roots αi, αj such that {αRij|μ(α)=0}, the set {αRij|μ(α)=±cα} contains more than two elements.

Condition (a) says that μ is regular for all rank 1 subsystems of R generated by simple roots. Condition (b) is an “almost regular” condition on μ with respect to rank 2 subsystems generated by simple roots. By the analysis in Section 3, the weights which appear in calibrated modules for graded Hecke algebras corresponding to rank two root systems are skew.

Recall from Section 2.3 that a pair (γ,J) is a local region if the set

(γ,J)= { wW|R(w) Z(γ)= andR(w) P(γ)=J }

is nonempty. A local region (γ,J) is skew if, for all w(γ,J), the weight wγ is skew for all pairs αi,αj of simple roots in R.

The following theorem specifies the weight space structure of an irreducible calibrated -module.

Theorem 4.3. If M is an irreducible calibrated -module with central character γ𝔥, then there is a unique skew local region (γ,J) such that

dim(Mwγ)= { 1 for allw (γ,J), 0 otherwise.


By Lemma 4.1 all nonzero generalized weight spaces of M have dimension 1 and by Lemma 4.2 all τ-operators between these weight spaces are bijections. This already guarantees that there is a unique local region (γ,J) which satisfies the condition. It only remains to show that this local region is skew.

Let ij be the subalgebra of generated by tsi,tsj and S(𝔥*). Since M is calibrated as an -module it is calibrated as an ij-module and so all factors of a composition series of M as an ij-module are calibrated. Thus, by the classification in Section 3, the weights of M are skew. So (γ,J) is a skew local region.


The following Proposition shows that the weight structure of calibrated representations as determined in Theorem 4.3 essentially forces the -action on a weight basis.

Proposition 4.4. Let M be a calibrated -module and for all γ𝔥 such that Mγ0, assume that

γ(αi)0 for all1in, (A1)


dim(Mγ)=1. (A2)

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

tsivb=c (tsi)cb vcandx vb=b(x)vb forx𝔥*.


  1. (tsi)bb=cαib(αi) for all vb in the basis,
  2. if (tsi)cb0, then c=sib,
  3. (tsi)b,sib (tsi)sib,b =1-(tsi)bb2 = (1+(tsi)bb) (1+(tsi)sib,sib) .


The relation

xtsi-tsisi (x)=cαi x-si(x)αi


c ( c(x) (tsi)cb- (tsi)cb b(six) ) vc=cαi b(x)-b(six) b(αi) vb.

Comparing coefficients yields

c(x)(tsi)cb -(tsi)cb b(six)=0, ifbc,


b(x)(tsi)bb -(tsi)bbb (six)=cαi b(x)-b(six) b(αi) .

These equations imply that

if(tsi)cb 0,thenb (six)=c(x) for allx𝔥*,


(tsi)bb= cαib(αi) ifb(αi)0 andb(x)b (six)for somex 𝔥*.


tsivb= (tsi)bbvb +(tsi)sib,b vsibwith (tsi)bb= cαib(αi) .

This completes the proof of (a) and (b). The relation tsi2=1 in implies that

vb = tsi2vb = [ (tsi)bb2+ (tsi)b,sib (tsi)sib,b ] vb+ [ (tsi)bb+ (tsi)sib,sib ] (tsi)sib,b vsib = [ (tsi)bb2+ (tsi)b,sib (tsi)sib,b ] vb,

since (tsi)bb+ (tsi)sib,sib =0. Thus

(tsi)b,sib (tsi)sib,b =1-(tsi)bb2= (1+(tsi)bb) (1+(tsi)sib,sib) .

Theorem 4.5. Let (γ,J) be skew and let (γ,J) index the chambers in the local region (γ,J). Define

H(γ,J)= -span { vw|w (γ,J) } ,

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

xvw=(wγ) (x)vwfor x𝔥*,


tsivw= cαiwγ(αi) vw+ ( 1+cαiwγ(αi) ) vsiwfor 1in,

where we set vsiw=0 if siw(γ,J).


Since (γ,J) is skew, (wγ)(αi)0 for all w(γ,J) and all simple roots αi. This implies that the coefficients in tsivw are well defined for all i and w(γ,J).

By construction, the nonzero weight spaces of (γ,J) are ((γ,J))wγgen =((γ,J))wγ where w(γ,J). Since dim(((γ,J))uγ)=1 for u(γ,J), any proper submodule N of (γ,J) must have Nwγ0 and Nwγ=0 for some ww, with w,w(γ,J). This is a contradiction to Corollary 2.6. So (γ,J) is irreducible if it is an -module.

It remains to show that the defining relations for are satisfied. Let w(γ,J). Then

( si(x)tsi +cαi x-sixαi ) vw = six [ cαiwγ(αi) vw+ (1+cαiwγ(αi)) vsiw ] +cαi wγ(x)-wγ(six) wγ(αi) vw = cαiwγ(αi) wγ(x)vw+ (1+cαiwγ(αi)) siwγ(six) vsix = tsixvw.

Let w(γ,J). Then

tsi2vw = tsi [ cαiwγ(αi) vw+ (1+cαiwγ(αi)) vsiw ] = cαiwγ(αi) [ cαiwγ(αi) vw+ (1+cαiwγ(αi)) vsiw ] +(1+cαiwγ(αi)) [ cαisiwγ(αi) vsiw+ (1+cαisiwγ(αi)) vw ] = (cαiwγ(αi))2 vw+(1+cαiwγ(αi)) (1-cαiwγ(αi)) vw+0 = vw.

Now let us check the braid relations. Write tsi=τi+di where

τivw= (1+cαi(wg)(αi)) vsiwand divw= cαi(wg)(αi) vw,

for w(γ,J). Then di is a diagonal matrix and τi is a pseudo-permutation matrix, in the sense that each row and each column contains at most one nonzero entry. For a sequence j1,,jp define a diagonal matrix dij1,,jp by the relation

diτj1 τjp=τj1 τjp dij1,,jp . (4.1)

If γ is generic, then, for all wW,

dij1,,jp vw= (cαi(sjpsj1wγ)(αi)) vw,

and all diagonal entries are nonzero, but, in general, some diagonal entries of dij1,,jp may be 0. Use this method to expand the expression

tsitsjtsi mijfactors = (τi+di) (τi+dj) (τi+di) mijfactors =zWτz pz,

and move all the diagonal operators di to the right of the τi and obtain diagonal operators pz. The operators τw are pseudo-permutation operators that may have some rows and columns without a nonzero entry. By replacing some diagonal entries of the pz operators by 0, we may "fix the τz" and replace the τz with operators τz which have exactly one nonzero entry in each row and each column. This yields the expression

tsitsjtsi mijfactors =zW τz pz. (4.2)

If γ is generic, then the diagonal entries (pz)ww of pz are nonzero and have the form (pz)ww=wγ(Pz), wW, where Pz is a rational function in the αi. A similar expansion gives

tsitsjtsi mijfactors =zW τz qz, (4.3)

where the qz are diagonal operators which, for generic γ, have diagonal entries (qz)ww=wγ(Qz), where Qz is a rational function of the αi. As in the proof of Proposition 2.5(e), γ(Pz)=γ(Qz) for all generic γ, and so it follows that Pz=Qz as rational functions.

When γ is not generic, the operators pz and qz may have some diagonal entries equal to zero. From the classification of representations of rank two graded Hecke algebras we know that there exists a calibrated representation of ij when (γ,J) is skew. This representation has a unique, up to constant multiples, basis of simultaneous eigenvectors for the action of λ𝔥*, and Proposition 4.4 shows that the action on this basis is forced except for the values of the off diagonal elements of the tsi. These values depend on the normalization of the basis. Because we know that this representation exists we know that there are choices of the nonzero entries in the τz such that (4.2) and (4.3) are equal. If a diagonal entry (pz)ww of pz is nonzero, then it is equal to (wγ)(Pz) and (pz)ww= (wγ)(Pz)= (wγ)(Qz)= (qz)ww, since (as shown above) Pz=Qz. Thus it follows that nonzero contributions from the terms τzpz and τzqz are equal and that tsitsjtsivw is equal to tsjtsitsjvw.

Remark 4.6. The action of on a weight basis of (γ,J) is forced up to the freedom in Proposition 4.4(c). Our choice (tsi)sib,b =1+(tsi)bb in Theorem 4.5 and the alternative choice (tsi)sib,b =1+(tsi)sib,sib yield isomorphic modules. The change of basis vb=1(1+(tsi)bb)vb provides the isomorphism.

We summarize the results of this section with the following corollary of Theorem 4.3 and the construction in Theorem 4.5.

Theorem 4.7. Let M be an irreducible calibrated -module. Let γ𝔥 be (a fixed choice of) the central character of M and let J=R(w)P(γ) for any wW such that Mwγ0. Then (γ,J) is skew and M(γ,J), where (γ,J) is the module defined in Theorem 4.5.

Notes and References

This is an excerpt of a paper entitled Representations of graded Hecke algebras, written by Cathy Kriloff (Department of Mathematics, Idaho State University, Pocatello, Idaho 83209-8085) and Arun Ram.

Research of the first author supported in part by an NSF-AWM Mentoring Travel Grant. Research of the second author supported in part by National Security Agency grant MDA904-01-1-0032 and EPSRC Grant GR K99015.

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