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

Last update: 13 November 2012


2.1. Weights

In view of the results in Section 1.18 we shall (for the remainder of this paper, except Sections 5–7 where we use the data in (1.1)) assume that L=P in the definition of the group X and H, see (1.2), (1.9), and (1.14). The Weyl group acts on

T=Hom(X,*)= {group homomorphismst:X*} by(wt) (Xλ)=t (Xw-1λ).

Let M be a finite dimensional H-module and let tT. The t-weight space and the generalized t-weight space of M are

Mt = { mMXλ m=t(Xλ)m for allXλX } and Mtgen = { mM for eachXλX, ( Xλ-t(Xλ) ) k m=0for some k>0 } ,

respectively. Then

M=tT Mtgen (2.2)

is a decomposition of M into Jordan blocks for the action of [X], and we say that t is a weight of M if Mtgen0. Note that Mtgen0 if and only if Mt0. A finite-dimensional H-module

Miscalibrated ifMtgen=Mt, for alltT.

Remark. The term tame is sometimes used in place of the term calibrated particularly in the context of representations of Yangians, see [NTa1998]. The word calibrated is preferable since tame also has many other meanings in different parts of mathematics.

Let M be a simple H-module. As an X(T)-module, M contains a simple submodule and this submodule must be one-dimensional since all irreducible representations of a commutative algebra are one dimensional. Thus, a simple module always has Mt0 for some tT.

2.3. Central characters

The Pittie–Steinberg theorem, Theorem 1.17, shows that, as vector spaces,

H=H[X]= H[X]W𝒦, where𝒦=-span { Xλw wW } ,

and H is the Iwahori–Hecke algebra defined in (1.8). Thus H is a free module over Z(H)=[X]W of rank dim(H)· dim(𝒦)= W2. By Dixmier’s version of Schur’s lemma (see [44, Lemma 0.5.1]), Z(H) acts on a simple H-module by scalars and so it follows that every simple H-module is finite dimensional of dimension W2. Theorem 2.12(d) below will show that, in fact, the dimension of a simple module is W.

Let M be a simple H-module. The central character of M is an element tT such that

pm=t(p)m, for allmM,p [X]W=Z (H).

The element t is only determined up to the action of W since t(p)=wt(p) for all wW. Because of this, any element of the orbit Wt is referred to as the central character of M.

Because P=L in the construction of X, a theorem of Steinberg [Ste1968-2, 3.15, 4.2, 5.3] tells us that the stabilizer Wt of a point tT under the action of W is the reflection group

Wt= sαα Z(t) ,whereZ(t)= { αR+ t(Xα)=1 } .

Thus the orbit Wt can be viewed in several different ways via the bijections

WtW/Wt { wW R(w)Z(t) = } { chambers on the positive side ofHα forαZ(t) } , (2.4)

where the last bijection is the restriction of the map in (1.6). If the root system Z(t) is generated by the simple roots αi that it contains then Wt is a parabolic subgroup of W and {wWR(w)Z(t)} is the set of minimal length coset representatives of the cosets in W/Wt.

2.5. Principal series modules

For tT let vt be the one-dimensional [X]-module given by

Xλvt=t (Xλ)vt, forXλX.

The principal series representation M(t) is the H-module defined by

M(t)=H [X] vt= Ind[X]H (vt). (2.6)

The module M(t) has basis { Twvt wW } with H acting by left multiplication.

If wW and XλX then the defining relation (1.10) for H implies that

Xλ(Twvt) =t(Xwλ) (Twvt)+ u<wau (Tuvt), (2.7)

where the sum is over u<w in the Bruhat–Chevalley order and au. Let Wt=Stab(t) be the stabilizer of t under the W-action. It follows from (2.7) that the eigenvalues of X on M(t) are of the form wt, wW, and by counting the multiplicity of each eigenvalue we have

M(t)= wtWt M(t)wtgen where dim (M(t)wtgen) =Wt, for allwW. (2.8)

In particular, if t is regular (i.e., when Wt is trivial), there is a unique basis {vwtwW} of M(t) determined by

Xλvwt= (wt)(Xλ) vwt,for all wWandλP, vwt=Twvt +u<wawu (t)(Tuvt) ,whereawu (t). (2.9)

Let tT. The spherical vector in M(t) is

1t=wW q(w)Tw vt. (2.10)

Up to multiplication by constants this is the unique vector in M(t) such that Tw1t= q(w) 1t for all wW. The following is due to Kato [Kat1981, Proposition 1.20 and Lemma 2.3],

Proposition 2.11. Let tT and let Wt be the stabilizer of t under the W-action.

  1. If Wt={1} and vwt, wW is the basis of M(t) defined in (2.9) then 1t=zW t(cz), wherecz= αR(w0z) q-q-1Xα 1-Xα .
  2. The spherical vector 1t generated M(t) if and only if t ( αR+ (q-1-qXα) ) 0.
  3. The module M(t) is irreducible if and only if 1wt generates M(wt) for all wW.


The proof is accomplished in exactly the same way as done for the graded Hecke algebra in [KRa2002, Proposition 2.8]. The only changes which need to be made to [KRa2002] are

  1. Use Ti ( wW q(w) Tw ) =q ( wW q(w) Tw ) and 1t= ( wW q(w)Tw ) vt and the τ-operators defined in Proposition 2.14 for the proof of (a). (We have included this result in this section since it is really a result about the structure of principal series modules. Though the proof uses the τ-operators, which we will define in the next section, there is no logical gap here.)
  2. For the proof of (b) use the Steinberg basis {XλyyW} and the determinant det(Xz-1λy) from Theorem 1.17(b) in place of the basis {bywW} and the determinant used in [KRa2002].

Part (b) of the following theorem is due to Rogawski [Rog1985, Proposition 2.3] and part (c) is due to Kato [Kat1981, Theorem 2.1]. Parts (a) and (d) are classical.

Theorem 2.12. Let tT and wW and define P(t)= { αR+ t(Xα)= q±2 } .

  1. If Wt={1} then M(t) is calibrated.
  2. M(t) and M(wt) have the same composition factors.
  3. M(t) is irreducible if and only if P(t)=.
  4. If M is a simple H-module with Mt0 then M is a quotient of M(t).


(a) follows from (2.8) and the definition of calibrated. Part (b) accomplished exactly as in [KRa2002, Proposition 2.8] and (c) is a direct consequence of the definition of P(t) and Proposition 2.11.

(d) Let mt be a nonzero vector in Mt. If vt is as in the construction of M(t) in (2.6) then, as [X]-modules, mtvt. Thus, since induction is the adjoint functor to restriction there is a unique H-module homomorphism given by

ϕ: M(t) M, vt mt.

This map is surjective since M is irreducible and so M is a quotient of M(t).

2.13. The τ operators

The following proposition defines maps τi: Mtgen Msitgen on generalized weight spaces of finite-dimensional H-modules M. These are “local operators” and are only defined on weight spaces Mtgen such that t(Xαi) 1. In general, τi does not extend to an operator on all of M.

Proposition 2.14. Fix i, let tT be such that t(Xαi)1 and let M be a finite-dimensional H-module. Define

τi: Mtgen Msitgen, m ( Ti- q-q-1 1-X-αi ) m.
  1. The map τi:MtgenMsitgen is well defined.
  2. As operators on Mtgen, Xλτi=τi Xsiλ, for all XλX.
  3. As operators on Mtgen, τiτi= (q-q-1Xαi) (q-q-1X-αi)/ ( (1-Xαi) (1-X-αi) ) .
  4. Both maths τi:Mtgen Msitgen and τi: Msitgen Mtgen are invertible if and only if t(Xαi) q±2.
  5. Let 1ijn and let mij be as in (1.7). Then τiτjτi mijfactors = τjτiτj mijfactors , whenever both sides are well defined operators on Mtgen.


(a) The element Xαi acts on Mtgen by t(Xαi) times a unipotent transformation. As an operator on Mtgen, 1-X-αi is invertible since it has determinant (1-t(X-αi))d where d=dim(Mtgen). Since this determinant is nonzero (q-q-1)/ (1-X-αi)= (q-q-1)× (1-X-αi)-1 is a well defined operator on Mtgen. Thus the definition of τi makes sense.

Since (q-q-1)/ (1-X-αi) is not an element of H or [X] it should be viewed only as an operator on Mtgen in calculations. With this in mind it is straightforward to use the defining relation (1.10) to check that

Xλτim=Xλ ( Ti- q-q-1 1-X-αi ) m= ( Ti- q-q-1 1-X-αi ) Xsiλm= τiXsiλm and τiτim= ( Ti- q-q-1 1-X-αi ) ( Ti- q-q-1 1-X-αi ) m= (q-q-1Xαi) (q-q-1X-αi) (1-Xαi) (1-X-αi) m,

for all mMtgen and lX. This proves (a)-(c).

(d) The operator Xαi acts on Mtgen as t(Xαi) times a unipotent transformation. Similarly for X-αi. Thus, as an operator on Mtgen det ( (q-q-1Xαi) (q-q-1X-αi) ) =0 if and only if t(Xαi)=q±2. Thus part (c) implies that τiτi, and each factor in this composition, is invertible if and only if t(Xαi) q±2.

(e) Let tT be regular. By part (a), the definition of the τi, and the uniqueness in (2.9), the basis {vwt} wW of M(t) in (2.9) is given by

vwt=τw vt, (2.15)

where τw=τi1τip for a reduced word w=si1sip of w. Use the defining relation (1.10) for H to expand the product of τi and compute

vw0t = τiτjτi mijfactors vt = TiTjTi mijfactors vt +w<w0Tw Pwvt=Tw0 vt+w<w0 t(Pw)Twvt = τjτiτj mijfactors vt = TjTiTj mijfactors vt +w<w0Tw Qwvt=Tw0 vt+w<w0 t(Qw)Twvt

where Pw and Qw are rational functions in the Xλ. By the uniqueness in (2.9), t(Pw)= aw0w(t)= t(Qw) for all wW, ww0. Since the values of Pw and Qw coincide on all generic points tT it follows that

Pw=Qw for allwW,w w0. (2.16)


τiτjτi mijfactors =Tw0+w<w0 TwPw=Tw0+ w<w0Tw Qw= τjτiτj mijfactors ,

whenever both sides are well defined operators on Mtgen.

Let tT and recall that

Z(t)= { αR+t t(Xα)=1 } andP(t)= { αR+ t(Xα)= q±2. } (2.17)

If JP(t) define

(t,J)= { wW R(w)Z(t) =,R(w) P(t)=J } . (2.18)

We say that the pair (t,J) is a local region if (t,J). Under the bijection (2.4) the set (t,J) maps to the set of chambers whose union is the set of points x𝔥* which are

  1. on the positive side of the hyperplanes Hα for αZ(t),
  2. on the positive side of the hyperplanes Hα for αP(t)\J,
  3. on the negative side of the hyperplanes Hα for αJ.

See the picture in Example 4.11(d). In this way the local region (t,J) really does correspond to a region in 𝔥*. This is a connected convex region in 𝔥* since it is cut out by half spaces in 𝔥*n. The elements w(t,J) index the chambers w-1C in the local region and, as J runs over the subsets of P(t), the sets (t,J) form a partition of the set { wWR (w)Z(t)= } (which, by (2.4), indexes the cosets in W/Wt).

Corollary 2.19. Let M be a finite dimensional H-module. Let tT and let JP(t). Then

dim(Mwtgen)= dim(Mwtgen), forw,w (t,J),


Suppose w,siw(t,J). We may assume that siw>w. Then α=w-1αi>0, αR(w) and αR(siw). Now, R(w)Z(t)= R(siw)Z(t) implies t(Xα)1, and R(w)P(t) implies t(Xα)q±2. Since wt(Xαi)=t (Xw-1αi) =t(Xα)1 and wt(Xαi) q±2, it follows from Proposition 2.14(d) that the map τi:MwtgenMsiwtgen is well defined and invertible. It remains to note that if w,w(t,J), then w=si1siw where sik siw (t,J) for all 1k. This follows from the fact that (t,J) corresponds to a connected convex region in h*.

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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