## $\stackrel{\sim }{H}\text{-modules}$

Last update: 13 November 2012

## $\stackrel{\sim }{H}\text{-modules}$

### 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 $\stackrel{\sim }{H},$ see (1.2), (1.9), and (1.14). The Weyl group acts on

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

Let $M$ be a finite dimensional $\stackrel{\sim }{H}\text{-module}$ and let $t\in T\text{.}$ The $t\text{-weight}$ space and the generalized t-weight space of $M$ are

$Mt = { m∈M∣Xλ m=t(Xλ)m for allXλ∈X } and Mtgen = { m∈M∣ for eachXλ∈X, ( Xλ-t(Xλ) ) k m=0for some k∈ℤ>0 } ,$

respectively. Then

$M=⨁t∈T Mtgen (2.2)$

is a decomposition of $M$ into Jordan blocks for the action of $ℂ\left[X\right],$ and we say that $t$ is a weight of $M$ if ${M}_{t}^{\text{gen}}\ne 0\text{.}$ Note that ${M}_{t}^{\text{gen}}\ne 0$ if and only if ${M}_{t}\ne 0\text{.}$ A finite-dimensional $\stackrel{\sim }{H}\text{-module}$

$Miscalibrated ifMtgen=Mt, for allt∈T.$

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 $\stackrel{\sim }{H}\text{-module.}$ As an $X\left(T\right)\text{-module,}$ $M$ 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 ${M}_{t}\ne 0$ for some $t\in T\text{.}$

### 2.3. Central characters

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

$H∼=H⊗ℂ[X]= H⊗ℂ[X]W⊗𝒦, where𝒦=ℂ-span { Xλw∣ w∈W } ,$

and $H$ is the Iwahori–Hecke algebra defined in (1.8). Thus $\stackrel{\sim }{H}$ is a free module over $Z\left(\stackrel{\sim }{H}\right)=ℂ{\left[X\right]}^{W}$ of rank $\text{dim}\left(H\right)·\text{dim}\left(𝒦\right)={\mid W\mid }^{2}\text{.}$ By Dixmier’s version of Schur’s lemma (see [44, Lemma 0.5.1]), $Z\left(\stackrel{\sim }{H}\right)$ acts on a simple $\stackrel{\sim }{H}\text{-module}$ by scalars and so it follows that every simple $\stackrel{\sim }{H}\text{-module}$ is finite dimensional of dimension $\le {\mid W\mid }^{2}\text{.}$ Theorem 2.12(d) below will show that, in fact, the dimension of a simple module is $\le \mid W\mid \text{.}$

Let $M$ be a simple $\stackrel{\sim }{H}\text{-module.}$ The central character of $M$ is an element $t\in T$ such that

$pm=t(p)m, for allm∈M,p∈ℂ [X]W=Z (H∼).$

The element $t$ is only determined up to the action of $W$ since $t\left(p\right)=wt\left(p\right)$ for all $w\in W\text{.}$ Because of this, any element of the orbit $Wt$ is referred to as the central character of $M\text{.}$

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 ${W}_{t}$ of a point $t\in T$ under the action of $W$ is the reflection group

$Wt= ⟨ sα∣α ∈Z(t) ⟩ ,whereZ(t)= { α∈R+∣ t(Xα)=1 } .$

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

$Wt↔W/Wt↔ { w∈W∣ R(w)∩Z(t) =∅ } ↔ { chambers on the positive side ofHα forα∈Z(t) } , (2.4)$

where the last bijection is the restriction of the map in (1.6). If the root system $Z\left(t\right)$ is generated by the simple roots ${\alpha }_{i}$ that it contains then $Wt$ is a parabolic subgroup of $W$ and $\left\{w\in W\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}R\left(w\right)\cap Z\left(t\right)\right\}$ is the set of minimal length coset representatives of the cosets in $W/{W}_{t}\text{.}$

### 2.5. Principal series modules

For $t\in T$ let $ℂ{v}_{t}$ be the one-dimensional $ℂ\left[X\right]\text{-module}$ given by

$Xλvt=t (Xλ)vt, forXλ∈X.$

The principal series representation $M\left(t\right)$ is the $\stackrel{\sim }{H}\text{-module}$ defined by

$M(t)=H∼ ⊗ℂ[X] ℂvt= Indℂ[X]H∼ (ℂvt). (2.6)$

The module $M\left(t\right)$ has basis $\left\{{T}_{w}\otimes {v}_{t}\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}w\in W\right\}$ with $H$ acting by left multiplication.

If $w\in W$ and ${X}^{\lambda }\in X$ then the defining relation (1.10) for $\stackrel{\sim }{H}$ implies that

$Xλ(Tw⊗vt) =t(Xwλ) (Tw⊗vt)+ ∑u

where the sum is over $u in the Bruhat–Chevalley order and ${a}_{u}\in ℂ\text{.}$ Let ${W}_{t}=\text{Stab}\left(t\right)$ be the stabilizer of $t$ under the $W\text{-action.}$ It follows from (2.7) that the eigenvalues of $X$ on $M\left(t\right)$ are of the form $wt,$ $w\in W,$ and by counting the multiplicity of each eigenvalue we have

$M(t)= ⨁wt∈Wt M(t)wtgen where dim (M(t)wtgen) =∣Wt∣, for allw∈W. (2.8)$

In particular, if $t$ is regular (i.e., when ${W}_{t}$ is trivial), there is a unique basis $\left\{{v}_{wt}\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}w\in W\right\}$ of $M\left(t\right)$ determined by

$Xλvwt= (wt)(Xλ) vwt,for all w∈Wandλ∈P, vwt=Tw⊗vt +∑u

Let $t\in T\text{.}$ The spherical vector in $M\left(t\right)$ is

$1t=∑w∈W qℓ(w)Tw ⊗vt. (2.10)$

Up to multiplication by constants this is the unique vector in $M\left(t\right)$ such that ${T}_{w}{1}_{t}={q}^{\ell \left(w\right)}{1}_{t}$ for all $w\in W\text{.}$ The following is due to Kato [Kat1981, Proposition 1.20 and Lemma 2.3],

Proposition 2.11. Let $t\in T$ and let ${W}_{t}$ be the stabilizer of $t$ under the $W\text{-action.}$

1. If ${W}_{t}=\left\{1\right\}$ and ${v}_{wt},$ $w\in W$ is the basis of $M\left(t\right)$ defined in (2.9) then $1t=∑z∈W t(cz), wherecz= ∏α∈R(w0z) q-q-1Xα 1-Xα .$
2. The spherical vector ${1}_{t}$ generated $M\left(t\right)$ if and only if $t\left(\prod _{\alpha \in {R}^{+}}\left({q}^{-1}-q{X}^{\alpha }\right)\right)\ne 0\text{.}$
3. The module $M\left(t\right)$ is irreducible if and only if ${1}_{wt}$ generates $M\left(wt\right)$ for all $w\in W\text{.}$

 Proof. 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 Use ${T}_{i}\left(\sum _{w\in W}{q}^{\ell \left(w\right)}{T}_{w}\right)=q\left(\sum _{w\in W}{q}^{\ell \left(w\right)}{T}_{w}\right)$ and ${1}_{t}=\left(\sum _{w\in W}{q}^{\ell \left(w\right)}{T}_{w}\right){v}_{t}$ and the $\tau \text{-operators}$ 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 $\tau \text{-operators,}$ which we will define in the next section, there is no logical gap here.) For the proof of (b) use the Steinberg basis $\left\{{X}^{{\lambda }_{y}}\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}y\in W\right\}$ and the determinant $\text{det}\left({X}^{{z}^{-1}{\lambda }_{y}}\right)$ from Theorem 1.17(b) in place of the basis $\left\{{b}_{y}\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}w\in W\right\}$ and the determinant used in [KRa2002]. $\square$

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 $t\in T$ and $w\in W$ and define $P\left(t\right)=\left\{\alpha \in {R}^{+}\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}t\left({X}^{\alpha }\right)={q}^{±2}\right\}\text{.}$

1. If ${W}_{t}=\left\{1\right\}$ then $M\left(t\right)$ is calibrated.
2. $M\left(t\right)$ and $M\left(wt\right)$ have the same composition factors.
3. $M\left(t\right)$ is irreducible if and only if $P\left(t\right)=\varnothing \text{.}$
4. If $M$ is a simple $\stackrel{\sim }{H}\text{-module}$ with ${M}_{t}\ne 0$ then $M$ is a quotient of $M\left(t\right)\text{.}$

 Proof. (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 $P\left(t\right)$ and Proposition 2.11. (d) Let ${m}_{t}$ be a nonzero vector in ${M}_{t}\text{.}$ If ${v}_{t}$ is as in the construction of $M\left(t\right)$ in (2.6) then, as $ℂ\left[X\right]\text{-modules,}$ $ℂ{m}_{t}\cong ℂ{v}_{t}\text{.}$ Thus, since induction is the adjoint functor to restriction there is a unique $\stackrel{\sim }{H}\text{-module}$ homomorphism given by $ϕ: M(t) ⟶ M, vt ⟼ mt.$ This map is surjective since $M$ is irreducible and so $M$ is a quotient of $M\left(t\right)\text{.}$ $\square$

### 2.13. The $\tau$ operators

The following proposition defines maps ${\tau }_{i}:\phantom{\rule{0.2em}{0ex}}{M}_{t}^{\text{gen}}\to {M}_{{s}_{i}t}^{\text{gen}}$ on generalized weight spaces of finite-dimensional $\stackrel{\sim }{H}\text{-modules}$ $M\text{.}$ These are “local operators” and are only defined on weight spaces ${M}_{t}^{\text{gen}}$ such that $t\left({X}^{{\alpha }_{i}}\right)\ne 1\text{.}$ In general, ${\tau }_{i}$ does not extend to an operator on all of $M\text{.}$

Proposition 2.14. Fix $i,$ let $t\in T$ be such that $t\left({X}^{{\alpha }_{i}}\right)\ne 1$ and let $M$ be a finite-dimensional $\stackrel{\sim }{H}\text{-module.}$ Define

$τi: Mtgen ⟶ Msitgen, m ⟼ ( Ti- q-q-1 1-X-αi ) m.$
1. The map ${\tau }_{i}:\phantom{\rule{0.2em}{0ex}}{M}_{t}^{\text{gen}}\to {M}_{{s}_{i}t}^{\text{gen}}$ is well defined.
2. As operators on ${M}_{t}^{\text{gen}},$ ${X}^{\lambda }{\tau }_{i}={\tau }_{i}{X}^{{s}_{i}\lambda },$ for all ${X}^{\lambda }\in X\text{.}$
3. As operators on ${M}_{t}^{\text{gen}},$ ${\tau }_{i}{\tau }_{i}=\left(q-{q}^{-1}{X}^{{\alpha }_{i}}\right)\left(q-{q}^{-1}{X}^{-{\alpha }_{i}}\right)/\left(\left(1-{X}^{{\alpha }_{i}}\right)\left(1-{X}^{-{\alpha }_{i}}\right)\right)\text{.}$
4. Both maths ${\tau }_{i}:\phantom{\rule{0.2em}{0ex}}{M}_{t}^{\text{gen}}\to {M}_{{s}_{i}t}^{\text{gen}}$ and ${\tau }_{i}:\phantom{\rule{0.2em}{0ex}}{M}_{{s}_{i}t}^{\text{gen}}\to {M}_{t}^{\text{gen}}$ are invertible if and only if $t\left({X}^{{\alpha }_{i}}\right)\ne {q}^{±2}\text{.}$
5. Let $1\le i\ne j\le n$ and let ${m}_{ij}$ be as in (1.7). Then $τiτjτi… ⏟mijfactors = τjτiτj… ⏟mijfactors ,$ whenever both sides are well defined operators on ${M}_{t}^{\text{gen}}\text{.}$

 Proof. (a) The element ${X}^{{\alpha }_{i}}$ acts on ${M}_{t}^{\text{gen}}$ by $t\left({X}^{{\alpha }_{i}}\right)$ times a unipotent transformation. As an operator on ${M}_{t}^{\text{gen}},$ $1-{X}^{-{\alpha }_{i}}$ is invertible since it has determinant ${\left(1-t\left({X}^{-{\alpha }_{i}}\right)\right)}^{d}$ where $d=\text{dim}\left({M}_{t}^{\text{gen}}\right)\text{.}$ Since this determinant is nonzero $\left(q-{q}^{-1}\right)/\left(1-{X}^{-{\alpha }_{i}}\right)=\left(q-{q}^{-1}\right)×{\left(1-{X}^{-{\alpha }_{i}}\right)}^{-1}$ is a well defined operator on ${M}_{t}^{\text{gen}}\text{.}$ Thus the definition of ${\tau }_{i}$ makes sense. Since $\left(q-{q}^{-1}\right)/\left(1-{X}^{-{\alpha }_{i}}\right)$ is not an element of $\stackrel{\sim }{H}$ or $ℂ\left[X\right]$ it should be viewed only as an operator on ${M}_{t}^{\text{gen}}$ 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 $m\in {M}_{t}^{\text{gen}}$ and $l\in X\text{.}$ This proves (a)-(c). (d) The operator ${X}^{{\alpha }_{i}}$ acts on ${M}_{t}^{\text{gen}}$ as $t\left({X}^{{\alpha }_{i}}\right)$ times a unipotent transformation. Similarly for ${X}^{-{\alpha }_{i}}\text{.}$ Thus, as an operator on ${M}_{t}^{\text{gen}}$ $\text{det}\left(\left(q-{q}^{-1}{X}^{{\alpha }_{i}}\right)\left(q-{q}^{-1}{X}^{-{\alpha }_{i}}\right)\right)=0$ if and only if $t\left({X}^{{\alpha }_{i}}\right)={q}^{±2}\text{.}$ Thus part (c) implies that ${\tau }_{i}{\tau }_{i},$ and each factor in this composition, is invertible if and only if $t\left({X}^{{\alpha }_{i}}\right)\ne {q}^{±2}\text{.}$ (e) Let $t\in T$ be regular. By part (a), the definition of the ${\tau }_{i},$ and the uniqueness in (2.9), the basis ${\left\{{v}_{wt}\right\}}_{w\in W}$ of $M\left(t\right)$ in (2.9) is given by $vwt=τw vt, (2.15)$ where ${\tau }_{w}={\tau }_{{i}_{1}}\dots {\tau }_{{i}_{p}}$ for a reduced word $w={s}_{{i}_{1}}\dots {s}_{{i}_{p}}$ of $w\text{.}$ Use the defining relation (1.10) for $\stackrel{\sim }{H}$ to expand the product of ${\tau }_{i}$ and compute $vw0t = …τiτjτi ⏟mijfactors vt = …TiTjTi ⏟mijfactors vt +∑w where ${P}_{w}$ and ${Q}_{w}$ are rational functions in the ${X}^{\lambda }\text{.}$ By the uniqueness in (2.9), $t\left({P}_{w}\right)={a}_{{w}_{0}w}\left(t\right)=t\left({Q}_{w}\right)$ for all $w\in W,$ $w\ne {w}_{0}\text{.}$ Since the values of ${P}_{w}$ and ${Q}_{w}$ coincide on all generic points $t\in T$ it follows that $Pw=Qw for allw∈W,w ≠w0. (2.16)$ Thus, $…τiτjτi ⏟mijfactors =Tw0+∑w whenever both sides are well defined operators on ${M}_{t}^{\text{gen}}\text{.}$ $\square$

Let $t\in T$ and recall that

$Z(t)= { α∈R+t t(Xα)=1 } andP(t)= { α∈R+∣ t(Xα)= q±2. } (2.17)$

If $J\subseteq P\left(t\right)$ define

$ℱ(t,J)= { w∈W∣ R(w)∩Z(t) =∅,R(w) ∩P(t)=J } . (2.18)$

We say that the pair $\left(t,J\right)$ is a local region if ${ℱ}^{\left(t,J\right)}\ne \varnothing \text{.}$ Under the bijection (2.4) the set ${ℱ}^{\left(t,J\right)}$ maps to the set of chambers whose union is the set of points $x\in {𝔥}_{ℝ}^{*}$ which are

1. on the positive side of the hyperplanes ${H}_{\alpha }$ for $\alpha \in Z\left(t\right),$
2. on the positive side of the hyperplanes ${H}_{\alpha }$ for $\alpha \in P\left(t\right)\J,$
3. on the negative side of the hyperplanes ${H}_{\alpha }$ for $\alpha \in J\text{.}$

See the picture in Example 4.11(d). In this way the local region $\left(t,J\right)$ really does correspond to a region in ${𝔥}_{ℝ}^{*}\text{.}$ This is a connected convex region in ${𝔥}_{ℝ}^{*}$ since it is cut out by half spaces in ${𝔥}_{ℝ}^{*}\cong {ℝ}^{n}\text{.}$ The elements $w\in {ℱ}^{\left(t,J\right)}$ index the chambers ${w}^{-1}C$ in the local region and, as $J$ runs over the subsets of $P\left(t\right),$ the sets ${ℱ}^{\left(t,J\right)}$ form a partition of the set $\left\{w\in W\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}R\left(w\right)\cap Z\left(t\right)=\varnothing \right\}$ (which, by (2.4), indexes the cosets in $W/{W}_{t}\text{).}$

Corollary 2.19. Let $M$ be a finite dimensional $\stackrel{\sim }{H}\text{-module.}$ Let $t\in T$ and let $J\subseteq P\left(t\right)\text{.}$ Then

$dim(Mwtgen)= dim(Mw′tgen), forw,w′∈ ℱ(t,J),$

 Proof. Suppose $w,{s}_{i}w\in {ℱ}^{\left(t,J\right)}\text{.}$ We may assume that ${s}_{i}w>w\text{.}$ Then $\alpha ={w}^{-1}{\alpha }_{i}>0,$ $\alpha \notin R\left(w\right)$ and $\alpha \in R\left({s}_{i}w\right)\text{.}$ Now, $R\left(w\right)\cap Z\left(t\right)=R\left({s}_{i}w\right)\cap Z\left(t\right)$ implies $t\left({X}^{\alpha }\right)\ne 1,$ and $R\left(w\right)\cap P\left(t\right)$ implies $t\left({X}^{\alpha }\right)\ne {q}^{±2}\text{.}$ Since $wt\left({X}^{{\alpha }_{i}}\right)=t\left({X}^{{w}^{-1}{\alpha }_{i}}\right)=t\left({X}^{\alpha }\right)\ne 1$ and $wt\left({X}^{{\alpha }_{i}}\right)\ne {q}^{±2},$ it follows from Proposition 2.14(d) that the map ${\tau }_{i}:\phantom{\rule{0.2em}{0ex}}{M}_{wt}^{\text{gen}}\to {M}_{{s}_{i}wt}^{\text{gen}}$ is well defined and invertible. It remains to note that if $w,{w}^{\prime }\in {ℱ}^{\left(t,J\right)},$ then ${w}^{\prime }={s}_{{i}_{1}}\dots {s}_{{i}_{\ell }}w$ where ${s}_{{i}_{k}}\dots {s}_{{i}_{\ell }}w\in {ℱ}^{\left(t,J\right)}$ for all $1\le k\le \ell \text{.}$ This follows from the fact that ${ℱ}^{\left(t,J\right)}$ corresponds to a connected convex region in ${h}_{ℝ}^{*}\text{.}$ $\square$

## 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).