## Preliminaries

Last update: 25 April 2013

## Preliminaries

Let $W$ be a finite reflection group, defined by its action on its reflection representation ${𝔥}_{ℝ}^{*}\text{.}$ For each reflection ${s}_{\alpha }\in W$ fix a root $\alpha$ in the $-1$ eigenspace of ${s}_{\alpha }\text{.}$ The roots $\alpha$ are chosen so that the set $R$ of roots is $W\text{-invariant.}$ Then ${s}_{\alpha }$ fixes a hyperplane

$Hα= (+1 eigenspace of sα)= {x∈𝔥ℝ* | α∨(x)=0},$

where we fix the linear function ${\alpha }^{\vee }\in {𝔥}_{ℝ}={\text{Hom}}_{ℝ}\left({𝔥}_{ℝ}^{*},ℝ\right)$ so that ${\alpha }^{\vee }\left(\alpha \right)=2\text{.}$ By fixing a nondegenerate symmetric $W-invariant$ bilinear form on ${𝔥}_{ℝ}^{*}$ we may identify ${𝔥}_{ℝ}$ and ${𝔥}_{ℝ}^{*},$ which will be used at times to view the ${H}_{\alpha }$ as lying in ${𝔥}_{ℝ}\text{.}$ Then

$sαx=x- ⟨x,α∨⟩α ,for all x∈𝔥ℝ* , where ⟨x,α∨⟩= α∨(x). (2.1)$

Fix simple roots ${\alpha }_{1},\dots ,{\alpha }_{n}$ in the root system for $W$ and let ${s}_{i}={s}_{{\alpha }_{i}}$ be the corresponding reflections.

By extension of scalars, $W$ acts on the complexification ${𝔥}_{ℂ}^{*}=ℂ{\otimes }_{ℝ}{𝔥}_{ℝ}^{*}$ and, in terms of its action on ${𝔥}_{ℂ}^{*},$ $W$ is a complex reflection group. Then $W$ acts on the symmetric algebra $S\left({𝔥}_{ℂ}^{*}\right)$ which is naturally identified with the algebra of polynomial functions on the vector space ${𝔥}_{ℂ}={\text{Hom}}_{ℂ}\left({𝔥}_{ℂ}^{*},ℂ\right)\text{.}$

Fix parameters ${c}_{\alpha }\in ℂ,$ ${c}_{\alpha }\ne 0,$ labeled by the roots, such that

$cα=cwα, for w∈W.$

If $W$ acts irreducibly on ${𝔥}_{ℝ}^{*},$ this amounts to the choice of one or two values, depending on whether there are one or two orbits of roots under the action of $W\text{.}$ The group algebra of $W$ is

$ℂW= ℂ-span{tw | w∈W} with multiplication twtw′= tww′.$

$ℍ=ℂW⊗S (𝔥ℂ*)$

with multiplication determined by the multiplication in $S\left({𝔥}_{ℂ}^{*}\right)$ and the multiplication in $ℂW$ and the relations

$xtsi=tsisi (x)+cαi ⟨x,αi∨⟩, for x∈𝔥ℂ*, (2.2)$

where ${\alpha }_{1}^{\vee },\dots ,{\alpha }_{n}^{\vee }\in {𝔥}_{ℝ}$ are the simple co-roots. More generally, it follows that for any $p\in S\left({𝔥}_{ℂ}^{*}\right),$

$ptsi=tsi si(p)+cαi Δi(p)and tsip=si (p)tsi+cαi Δi(p),$

where ${\Delta }_{i}:S\left({𝔥}_{ℂ}^{*}\right)\to S\left({𝔥}_{ℂ}^{*}\right)$ is the BGG-operator given by

$Δi(p)= p-si(p)αi for p∈S(𝔥ℂ*) .$

Proposition 2.1 ([Lus1988, Theorem 6.5]). The center of the graded Hecke algebra $ℍ$ is $Z\left(ℍ\right)=S{\left({𝔥}_{ℂ}^{*}\right)}^{W},$ the ring of $W\text{-invariant}$ polynomials on ${𝔥}_{ℂ}\text{.}$

 Proof. If $p\in S{\left({𝔥}_{ℂ}^{*}\right)}^{W},$ then $ptsi=tsisi (p)+cαi p-si(p)αi =tsip+0= tsip,$ and so $p$ commutes with ${t}_{{s}_{i}}\text{.}$ Therefore, $S{\left({𝔥}_{ℂ}^{*}\right)}^{W}\subseteq Z\left(ℍ\right)\text{.}$ Let $p\in Z\left(ℍ\right)$ and write $p=\sum _{w\in W}{p}_{w}{t}_{w}\text{.}$ Fix $v$ of maximal length such that ${p}_{v}$ has maximal degree. Let $\mu \in {𝔥}_{ℂ}^{*}$ be regular, meaning that the stabilizer ${W}_{\mu }$ is trivial. Then $μp= ∑w∈Wμpw tw equalspμ= ∑w∈Wpwtwμ= ∑w∈Wpw ( (wμ)tw+ ∑u where ${c}_{u,w}^{\mu }\in ℂ\text{.}$ Comparing coefficients of ${t}_{v}$ yields $μpv=pv· (vμ).$ So $\mu =\left(v\mu \right)$ and thus $v=1$ since $\mu$ is regular. So $p\in S\left({𝔥}_{ℂ}^{*}\right)\text{.}$ Comparing coefficients of ${t}_{{s}_{i}}$ in $ptsi=si(p) tsi+cαi p-si(p)αi$ shows that $p={s}_{i}\left(p\right)$ for all $1\le i\le n\text{.}$ So $p\in S{\left({𝔥}_{ℂ}^{*}\right)}^{W}\text{.}$ Thus $Z\left(ℍ\right)=S{\left({𝔥}_{ℂ}^{*}\right)}^{W}\text{.}$ $\square$

### Harmonic polynomials

Let us briefly review the relationship between $S\left({𝔥}_{ℂ}^{*}\right),$ $S{\left({𝔥}_{ℂ}^{*}\right)}^{W},$ and harmonic polynomials [CGi1433132, §6.3]. Let ${x}_{1},{x}_{2},\dots ,{x}_{n}$ be an orthonormal basis of ${𝔥}_{ℂ}$ and define a symmetric bilinear form $⟨,⟩$ on $S\left({𝔥}_{ℂ}^{*}\right)$ by

$⟨P,Q⟩= (P(∂)Q) |xi=0, for P,Q∈S (𝔥ℂ*),$

where $P\left(\partial \right)=P\left(\frac{\partial }{\partial {x}_{1}},\dots ,\frac{\partial }{\partial {x}_{n}}\right)$ and ${|}_{{x}_{i}=0}$ denotes specializing the variables to 0 (or, equivalently, taking the constant term). The monomials are an orthogonal basis of $S\left({𝔥}_{ℂ}^{*}\right),$

$⟨ x1λ1 … xnλn, x1μ1 … xnμn ⟩ = ( (∂∂x1)λ1 … (∂∂xn)λn x1μ1 … xnμn ) |xi=0 = δλ1μ1 … δλnμn λ1! λ2! …λn!,$

and so the bilinear form $⟨,⟩$ is nondegenerate. The vector space $ℋ$ of harmonic polynomials is the set of polynomials orthogonal to the ideal of $S\left({𝔥}_{ℂ}^{*}\right)$ generated by $W\text{-invariants}$ in $S\left({𝔥}_{ℂ}^{*}\right)$ with constant term 0,

$ℋ= ( ⟨ f∈S(𝔥ℂ*)W | f(0)=0 ⟩ ) ⊥ ,andS(𝔥ℂ*) =S(𝔥ℂ*)W ⊗ℋ,$

as vector spaces. More precisely, if $\left\{{h}_{w}\right\}$ is a $ℂ\text{-basis}$ of $ℋ,$ then any $f\in S\left({𝔥}_{ℂ}^{*}\right)$ can be written uniquely in the form

$f=∑wpwhw, pw∈S (𝔥ℂ*)W.$

If the basis $\left\{{h}_{w}\right\}$ consists of homogeneous polynomials, then the number and the degrees of these polynomials are determined by the Poincaré polynomial of $W,$

$PW(t)= ∑k≥0dim (ℋk)tk= ∏i=1n 1-tdi1-t =∑w∈W tℓ(w), (2.3)$

where ${d}_{1},\dots ,{d}_{n}$ are the degrees of a set ${f}_{1},\dots ,{f}_{n}$ of homogeneous generators of $S{\left({𝔥}_{ℂ}^{*}\right)}^{W}=ℂ\left[{f}_{1},\dots ,{f}_{n}\right]$ and ${ℋ}^{k}$ is the $k\text{th}$ homogeneous component of $ℋ\text{.}$ In particular, $\text{dim}\left(ℋ\right)=\text{Card}\left(\left\{{h}_{w}\right\}\right)={P}_{W}\left(1\right)=|W|$ and $S\left({𝔥}_{ℂ}^{*}\right)$ is a free module over $S{\left({𝔥}_{ℂ}^{*}\right)}^{W}$ of rank $|W|\text{.}$

The following useful lemma is well known (see, for example, [Kat1982-2, Lemma 2.11]). Other related results can be found in [Ste1975] and [Hul1974].

Lemma 2.2 Let ${\left\{{b}_{w}\right\}}_{w\in W}$ be a basis for the vector space $ℋ$ of harmonic polynomials and let $X$ be the $|W|×|W|$ matrix given by

$X= (z-1bw) z,w∈W .$

Then

$det X=ξ· (∏α>0α) |W|/2 ,$

where $\xi$ is a nonzero constant in $ℂ\text{.}$

 Proof. Note that if ${b}_{w}^{\prime }$ is another basis of $ℋ$ and we write $bw′=∑v∈W cvwbv, cvw∈ℂ,$ then $X′= (z-1bw′) z,w∈W = (z-1bv) (cvw) and sodet X′= ξ det X,$ for some nonzero constant $\xi =\text{det}\left(\left({c}_{vw}\right)\right)\text{.}$ Thus, by changing basis if necessary, we may assume that the ${b}_{w}$ are homogeneous. Subtract row ${z}^{-1}{b}_{w}$ from row ${s}_{\alpha }{z}^{-1}{b}_{w}\text{.}$ Then this row is divisible by $\alpha \text{.}$ By doing this subtraction for each of the $|W|/2$ pairs $\left\{{z}^{-1},{s}_{\alpha }{z}^{-1}\right\}$ we conclude that $\text{det}\left(X\right)$ is divisible by ${\alpha }^{|W|/2}\text{.}$ Thus, since the roots are co-prime as elements of the polynomial ring $S\left({𝔥}_{ℂ}^{*}\right),$ $det(X)is divisible by (∏α>0α) |W|/2 .$ The degree of ${\prod }_{\alpha >0}{\alpha }^{|W|/2}$ is $\left(|W|/2\right)\text{Card}\left({R}^{+}\right)$ and, using (2.3), the degree of $\text{det}\left(X\right)$ is $deg(∏w∈Wbw) = ∑kk dim(ℋk)= (ddtPW(t)) |t=1=∑w∈W ℓ(w) = ∑w∈WCard(R(w)) =∑α∈R+ (|W|/2)= (|W|/2)Card(R+).$ Since these two polynomials are homogeneous of the same degree, it follows that the quotient $\text{det}\left(X\right)/{\left(\prod _{\alpha \in 0}\alpha \right)}^{|W|/2}$ is a constant. If $\text{det}\left(X\right)=0,$ then the columns of $X$ are linearly dependent. In particular, there exist constants ${c}_{w}\in ℂ,$ not all zero, such that $\sum _{w}{c}_{w}{b}_{w}=0\text{.}$ But this is a contradiction to the assumption that $\left\{{b}_{w}\right\}$ is a basis of $ℋ\text{.}$ So $\text{det}\left(X\right)\ne 0\text{.}$ $\square$

For each $1\le i\le n$ let ${\Delta }_{i}^{*}:S\left({𝔥}_{ℂ}^{*}\right)\to S\left({𝔥}_{ℂ}^{*}\right)$ be the operator which is adjoint to the BGG-operator ${\Delta }_{i}$ with respect to $⟨,⟩\text{.}$ A homogeneous basis $\left\{{b}_{w} | w\in W\right\}$ of the space of harmonic polynomials $ℋ$ can be constructed by setting

$bw=Δw*(1) ,whereΔw*= Δi1*… Δiℓ* for a reduced word w=si1 …siℓ.$

### Weights and calibrated representations

The group $W$ acts on

$𝔥ℂ=Hom(𝔥ℂ*,ℂ) by(wγ)(x) =γ(w-1x),$

for $w\in W,$ $\gamma \in {𝔥}_{ℂ}$ and $x\in {𝔥}_{ℂ}^{*}\text{.}$

The inversion set of an element $w\in W$ is

$R(w)= {α>0 | wα<0}. (2.4)$

The choice of the simple roots ${\alpha }_{1},\dots ,{\alpha }_{n}\in {𝔥}_{ℝ}^{*}$ determines a fundamental chamber

$C= { λ∈𝔥ℝ | ⟨αi,λ⟩ >0,1≤i≤n } (2.5)$

for the action of $W$ on ${𝔥}_{ℝ}\text{.}$ For a root $\alpha \in R,$ the positive side of the hyperplane ${H}_{\alpha }$ is the side towards $C,$ i.e. $\left\{\lambda \in {𝔥}_{ℝ} | ⟨\lambda ,\alpha ⟩>0\right\},$ and the negative side of ${H}_{\alpha }$ is the side away from $C\text{.}$ There is a bijection

$W ⟷ {fundamental chambers for W acting on 𝔥ℝ} w ⟼ w-1C (2.6)$

and the chamber ${w}^{-1}C$ is the unique chamber which is on the positive side of ${H}_{\alpha }$ for $\alpha \notin R\left(w\right)$ and on the negative side of ${H}_{\alpha }$ for $\alpha \in R\left(w\right)\text{.}$

If ${s}_{\alpha }$ is a reflection in $W$ which fixes $\gamma \in {𝔥}_{ℂ},$ then $⟨\gamma ,{\alpha }^{\vee }⟩=0\text{.}$ By [Ste1964, Theorem 1.5], [Bou1968, Ch. V §5 Ex. 8] the stabilizer ${W}_{\gamma }$ of $\gamma$ under the $W\text{-action}$ is generated by the reflections which stabilize $\gamma$ and so

$Wγ= ⟨sα | α∈Z(γ)⟩ whereZ(γ)= {α | γ(α)=0}.$

The orbit $W\gamma$ can be viewed in several different ways via the bijections

$Wγ ⟷ W/Wγ ⟷ { w∈W | R(w) ∩Z(γ)=∅ } ⟷ { chambers on the positive side of Hα for α∈Z(γ) } , (2.7)$

where the last bijection is the restriction of the map in equation (2.6). If $\gamma$ is real and dominant (i.e. $\gamma \left(\alpha \right)\in {ℝ}_{\ge 0}$ for all $\alpha \in R\text{),}$ then ${W}_{\gamma }$ is a parabolic subgroup of $W$ and $\left\{w\in W | R\left(w\right)\cap Z\left(\gamma \right)=\varnothing \right\}$ is the set of minimal length coset representatives of the cosets in $W/{W}_{\gamma }\text{.}$

Let M be a simple $ℍ\text{-module.}$ Dixmier’s version of Schur’s lemma (see [Wal1988]) implies that $Z\left(ℍ\right)$ acts on $M$ by scalars. Let $\gamma \in {𝔥}_{ℂ}$ be such that

$pm=γ(p)m,for all m∈M,p∈S (𝔥ℂ*)W.$

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

Since $ℍ=ℂW\otimes S\left({𝔥}_{ℂ}^{*}\right)=ℂW\otimes S{\left({𝔥}_{ℂ}^{*}\right)}^{W}\otimes ℋ,$ the graded Hecke algebra $ℍ$ is a free module over $Z\left(ℍ\right)=S{\left({𝔥}_{ℂ}^{*}\right)}^{W}$ of rank $\text{dim}\left(ℂW\right)\text{dim}\phantom{\rule{0.2em}{0ex}}\left(ℋ\right)={|W|}^{2}\text{.}$ Since $Z\left(ℍ\right)$ acts on a simple $ℍ\text{-module}$ by scalars, every simple $ℍ\text{-module}$ is finite dimensional of dimension $\le {|W|}^{2}\text{.}$ Proposition 2.8(a) below will show that, in fact, the dimension of a simple $ℍ\text{-module}$ is $\le |W|\text{.}$

Let $M$ be a finite dimensional $ℍ\text{-module}$ and let $\gamma \in {𝔥}_{ℂ}\text{.}$ The $\gamma \text{-weight}$ space and the generalized $\gamma \text{-weight}$ space of $M$ are

$Mγ = { m∈M | xm=γ (x)m for all x∈ 𝔥ℂ* } , (2.8) Mγgen = { m∈M | for all x∈ 𝔥ℂ*, (x-γ(x))k m=0 for some k∈ ℤ>0 } . (2.9)$

Then

$M=⨁γ∈𝔥ℂ Mγgen,$

and we say that $\gamma$ is a weight of $M$ if ${M}_{\gamma }^{\text{gen}}\ne 0\text{.}$ Note that ${M}_{\gamma }^{\text{gen}}\ne 0$ if and only if ${M}_{\gamma }\ne 0\text{.}$ A finite dimensional $ℍ\text{-module}$

$M is calibrated if Mγgen= Mγfor all γ∈ 𝔥ℂ. (2.10)$

### Tempered representations and the Langlands classification.

A weight $\lambda \in {\text{Hom}}_{ℂ}\left({𝔥}_{ℂ}^{*},ℂ\right)$ is determined by its values $⟨\lambda ,{\alpha }_{i}⟩$ on the simple roots. Define $\text{Re}\left(\lambda \right)$ and $\text{Im}\left(\lambda \right)$ in ${𝔥}_{ℝ}={\text{Hom}}_{ℝ}\left({𝔥}_{ℝ}^{*},ℝ\right)$ by $⟨\text{Re}\left(\lambda \right),{\alpha }_{i}⟩=\text{Re}\left(⟨\lambda ,{\alpha }_{i}⟩\right)$ and $⟨\text{Im}\left(\lambda \right),{\alpha }_{i}⟩=\text{Im}\left(⟨\lambda ,{\alpha }_{i}⟩\right),$ and write

$λ=Re(λ)+i Im(λ).$

For any simple reflection ${s}_{j},$ we have

$sjλ=Re(λ)-Re (⟨λ,αj⟩) αj∨+iIm(λ)-i Im(⟨λ,αj⟩) αj∨=sjRe(λ) +isjIm(λ),$

and so

$R(wλ)=wRe(λ) ,for all w∈W.$

Let ${\omega }_{i}^{\vee }$ be the dual basis to ${\alpha }_{i}^{\vee }$ in ${𝔥}_{ℝ}$ and let $\stackrel{‾}{C}$ be the closure of the fundamental chamber $C\subseteq {𝔥}_{ℝ}$ defined in (2.5). For $\lambda \in {𝔥}_{ℂ}$ let ${\lambda }_{0}$ be the point of $\stackrel{‾}{C}$ which is closest to $\text{Re}\left(\lambda \right)\text{.}$ This point is uniquely defined because of the convexity of the region $C\text{.}$ Since ${\lambda }_{0}\in \stackrel{‾}{C}$ and the ${\omega }_{i}^{\vee }$ are on the boundary of $\stackrel{‾}{C},$ there is a uniquely determined set $I$ such that

$λ0=∑j∉I cjωj∨with cj>0,$

and we say that the weight $\lambda$ is I-tempered. For each $I$ the set $\left\{{\omega }_{j}^{\vee },{\alpha }_{i}^{\vee } | j\notin I,i\in I\right\}$ is a basis of ${𝔥}_{ℝ}$ and ${\lambda }_{0}$ and $I$ can, alternatively, be determined by the unique expansion

$Re(λ)=∑j∉I cjωj∨+ ∑i∈Idi αi∨with cj>0 and di ≤0. (2.11)$

Proposition 2.3 ((Lemma of Langlands) [Lan1989, Corollary 4.6], [Kna1986, Lemma 8.59]). Let $\lambda \ge \mu$ denote the dominance ordering on ${𝔥}_{ℝ}\text{.}$ If $\lambda ,\mu \in {𝔥}_{ℝ}$ such that $\lambda \ge \mu ,$ then ${\lambda }_{0}\ge {\mu }_{0}\text{.}$

For any subset $I\subseteq \left\{1,\dots ,n\right\},$ let ${ℍ}_{I}$ be the subalgebra of $ℍ$ generated by ${t}_{{s}_{i}},i\in I,$ and all $x\in {𝔥}_{ℂ}^{*}\text{.}$ An ${ℍ}_{I}\text{-module}$ $M$ is tempered if all weights of $M$ are $I\text{-tempered.}$

Theorem 2.4. Let $L$ be a simple $ℍ\text{-module.}$

1. There is a subset $I\subseteq \left\{1,2,\dots ,n\right\}$ and a simple tempered ${ℍ}_{I}\text{-module}$ $U$ such that $L$ is the unique simple quotient of $ℍ{\otimes }_{{ℍ}_{I}}U\text{.}$ If $I$ and $I\prime$ are subsets of $\left\{1,2,\dots ,n\right\}$ and $U$ and $U\prime$ are simple tempered ${ℍ}_{I}$ and ${ℍ}_{I\prime }\text{-modules,}$ respectively, such that $L$ is a quotient of both $H{\otimes }_{{ℍ}_{I}}U$ and $ℍ{\otimes }_{{ℍ}_{I\prime }},$ $U\prime ,$ then $I=I\prime$ and $U\cong U\prime$ as ${ℍ}_{I}\text{-modules.}$

 Proof. Let $L$ be a simple $ℍ\text{-module.}$ Let $\lambda$ be a weight of $L$ such that $λ0 is a maximal element of {μ0 | μ is a weight of L} (2.12)$ with respect to the dominance ordering on ${𝔥}_{ℝ}\text{.}$ Let $I\subseteq \left\{1,2,\dots ,n\right\}$ be determined by $λ0=∑j∉I cjωj∨$ and let $V$ be the ${ℍ}_{I}\text{-submodule}$ of $L$ generated by a nonzero vector ${m}_{\lambda }$ in ${L}_{\lambda }\text{.}$ Let ${W}_{I}$ be the subgroup of $W$ generated by ${s}_{i},$ $i\in I\text{.}$ The weights of $V$ are of the form $w\lambda$ with $w\in {W}_{I}\text{.}$ If $w\in {W}_{I},$ then there are some constants ${\alpha }_{i}\in ℝ$ such that $Re(wλ)= ∑j∉Icj ωj∨+ ∑αi≤0,i∈I aiαi∨+ ∑αi>0,i∈I aiαi∨≥ ∑j∉Icj ωj∨+ ∑αi≤0,i∈I aiαi∨,$ since $\text{Re}\left(\lambda \right)$ is as in (2.11). So, by Proposition 2.3, $(wλ)0≥ ( ∑j∉I cjωj∨+ ∑αi≤0 aiαi∨ ) 0 =∑j∉I cjωj∨=λ0.$ Thus, by the maximality of ${\lambda }_{0},{\mu }_{0}={\lambda }_{0}$ for all weights $\mu$ of $V\text{.}$ So $V$ is tempered. Let $U$ be a simple ${ℍ}_{I}\text{-submodule}$ of $V\text{.}$ All weights of $H{\otimes }_{{ℍ}_{I}}$ are of the form $w\mu$ with $w\in W$ and $\mu$ a weight of $U\text{.}$ Let ${W}^{I}$ denote the set of minimal length coset representatives of cosets in $W/{W}_{I}\text{.}$ If $w\mu$ is a weight and $w={w}_{1}{w}_{2}$ with ${w}_{1}\in {W}^{I}$ and ${w}_{2}\in {W}_{I},$ then by the argument just given ${w}_{2}\mu$ is $I\text{-tempered}$ and so $Re(w2μ)=∑j∉I cjωj∨+∑i∈I aiαi∨with cj>0,ai≤0.$ Recall that ${W}^{I}=\left\{{w}_{1}\in W | R\left({w}_{1}\right)\cap {\left\{{\alpha }_{i}\right\}}_{i\in I}=\varnothing \right\}\text{.}$ Thus, for each $i\in I,$ ${w}_{1}{\alpha }_{i}^{\vee }$ is a positive co-root and $Re(w1w2μ)= w1(w2μ)0+ ∑i∈Iaiw1 αi∨≤w1 (w2μ)0.$ If ${w}_{1}\ne 1,$ then ${w}_{1}{\omega }_{j}^{\vee }\le {\omega }_{j}^{\vee }$ for all $j\notin I$ and ${w}_{1}{\omega }_{j}^{\vee }<{\omega }_{j}^{\vee }$ for some $j\notin I\text{.}$ So $Re(w1w2μ)≤ w1(w2μ)0 <(w2μ)0$ and thus, by Proposition 2.3, $(w1w2μ)0 <(w2μ)0 whenw1≠1. (2.13)$ Let $\nu$ be a weight of $U$ such that, among weights of $U,$ ${\nu }_{0}$ is maximal. If $N$ is an $ℍ\text{-submodule}$ of $H{\otimes }_{{ℍ}_{I}}U$ such that ${N}_{\nu }\ne 0,$ then, by (2.13), ${N}_{\nu }\subseteq {U}_{\nu }$ and so $N\cap U\ne 0\text{.}$ Since $U$ is simple as an ${ℍ}_{I}\text{-module,}$ any vector of $U$ generates all of $H{\otimes }_{{ℍ}_{I}}U$ and so $N=H{\otimes }_{{ℍ}_{I}}U\text{.}$ This shows that if $Mmax= ( sum of all ℍ-submodules N of H⊗ℍIU such that Nν=0 ) ,$ then ${M}_{\text{max}}$ is equal to the sum of all proper submodules of $H{\otimes }_{{ℍ}_{I}}U$ and is the (unique) maximal proper submodule of $H{\otimes }_{{ℍ}_{I}}U\text{.}$ So $H{\otimes }_{{ℍ}_{I}}U$ has a unique simple quotient. Since $U$ is an ${ℍ}_{I}\text{-submodule}$ of $L$ and induction is the adjoint functor to restriction, there is an $ℍ\text{-module}$ homomorphism $ϕU: H⊗ℍIU ⟶ L u ⟼ u for u∈U. (2.14)$ Thus, since $L$ is simple, $L\cong \left(H{\otimes }_{{ℍ}_{I}}U\right)/{M}_{\text{max}}\text{.}$ This proves (a) and shows that for any tempered ${ℍ}_{I}\text{-module}$ $U$ the module $H{\otimes }_{{ℍ}_{I}}U$ has a unique simple quotient. To prove (b) let us analyze the freedom of the choices that are made in the above construction of $H{\otimes }_{{ℍ}_{I}}U\text{.}$ Equation (2.13) and Proposition 2.3 show that ${\nu }_{0}\le {\lambda }_{0}$ for all weights $\nu$ of $H{\otimes }_{{ℍ}_{I}}U\text{.}$ In particular, all weights $\nu$ of $L$ satisfy ${\nu }_{0}\le {\lambda }_{0}$ and so ${\lambda }_{0}$ is the same for all weights $\lambda$ of $L$ which satisfy (2.12). This shows that there is a unique choice of $I$ in the construction of $H{\otimes }_{{ℍ}_{I}}U\text{.}$ If $U\prime$ is another simple ${ℍ}_{I}\text{-submodule}$ of $V,$ then either $U\cap U\prime =0$ or $U=U\prime \text{.}$ Suppose that $U\cap U\prime =0\text{.}$ Then $U\oplus U\prime$ is a tempered ${ℍ}_{I}\text{-submodule}$ of $L\text{.}$ Let $\nu$ be a weight of $U\text{.}$ Suppose $\mu$ is a weight of $L$ with ${\mu }_{0}={\nu }_{0}\text{.}$ By equations (2.13) and (2.14), the only elements of the $\mu \text{-weight}$ space of the image of the homomorphism ${\varphi }_{U}:H{\otimes }_{{ℍ}_{I}}U\to L$ are elements of $U\text{.}$ Thus $\text{im}\left({\varphi }_{U}\right)\cap U\prime =0\text{.}$ But this is impossible because $L$ is simple and ${\varphi }_{U}$ is surjective. Thus $U=U\prime \text{.}$ $\square$

Theorem 2.4 gives us a way to classify simple $H\text{-modules.}$ The Langlands parameters $\left(U,I\right)$ of the simple module $L$ are the pair determined by Theorem 2.4.

### $\tau$ operators

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

Proposition 2.5. Let $M$ be a finite dimensional $ℍ\text{-module.}$ Fix $i,$ let $\gamma \in {𝔥}_{ℂ}$ be such that $\gamma \left({\alpha }_{i}\right)\ne 0$ and define

$τi: Mγgen ⟶ Msiγgen m ⟼ (tsi-cαiαi)m.$
1. The map ${\tau }_{i}:{M}_{\gamma }^{\text{gen}}\to {M}_{{s}_{i}\gamma }^{\text{gen}}$ is well defined.
2. As operators on ${M}_{\gamma }^{\text{gen}},$ $x{\tau }_{i}={\tau }_{i}{s}_{i}\left(x\right)$ for all $x\in S\left({𝔥}_{ℂ}^{*}\right)\text{.}$
3. As operators on ${M}_{\gamma }^{\text{gen}},$ ${\tau }_{i}{\tau }_{i}=\frac{\left({c}_{{\alpha }_{i}}+{\alpha }_{i}\right)\left({c}_{{\alpha }_{i}}-{\alpha }_{i}\right)}{\left({\alpha }_{i}\right)\left(-{\alpha }_{i}\right)}\text{.}$
4. Both maps ${\tau }_{i}:{M}_{\gamma }^{\text{gen}}\to {M}_{{s}_{i}\gamma }^{\text{gen}}$ and ${\tau }_{i}:{M}_{{s}_{i}\gamma }^{\text{gen}}\to {M}_{\gamma }^{\text{gen}}$ are invertible if and only if $\gamma \left({\alpha }_{i}\right)\ne ±{c}_{{\alpha }_{i}}\text{.}$
5. If $1\le i,j\le n,i\ne j,$ let ${m}_{ij}$ be the order of ${s}_{i}{s}_{j}$ in $W\text{.}$ Then $τiτjτi… ⏟mijfactors = τjτiτj… ⏟mijfactors ,$ whenever both sides are well defined operators on ${M}_{\gamma }^{\text{gen}}\text{.}$

 Proof. Since ${\alpha }_{i}$ acts on ${M}_{\gamma }^{\text{gen}}$ by $\gamma \left({\alpha }_{i}\right)$ times a unipotent transformation, the operator ${\alpha }_{i}$ on ${M}_{\gamma }^{\text{gen}}$ has nonzero determinant and is invertible. Since ${c}_{{\alpha }_{i}}/{\alpha }_{i}$ is not an element of $S\left({𝔥}_{ℂ}^{*}\right)$ or $ℍ$ it will be viewed only as an operator on ${M}_{\gamma }^{\text{gen}}$ in the following calculations. If $x\in {𝔥}_{ℂ}^{*}$ and $m\in {M}_{\gamma }^{\text{gen}},$ then $xτim = x(tsi-cαiαi) m = ( tsisi(x)+ cαi⟨x,αi∨⟩ -cαixαi ) m = ( tsisi(x)- cαi x-⟨x,αi∨⟩αi αi ) m = ( tsisi(x)- cαisi(x)αi ) m = (tsi-cαiαi) si(x)m = τisi(x)m.$ This proves (a) and (b). $τiτim = ( tsi2- cαiαi tsi-tsi cαiαi +cαi2αi2 ) m = ( 1-cαiαi tsi- cαi-αi tsi-cαi ( cαiαi- cαi-αi ) αi +cαi2αi2 ) m = ( 1+ cαi2 (αi) (-αi) ) m= ( (cαi+αi) (cαi-αi) (αi)(-αi) ) m,$ proving (c). (d) Since ${\alpha }_{i}$ acts on ${M}_{\gamma }^{\text{gen}}$ by $\gamma \left({\alpha }_{i}\right)$ times a unipotent transformation, $\text{det}\left(\left({c}_{{\alpha }_{i}}+{\alpha }_{i}\right)\left({c}_{{\alpha }_{i}}-{\alpha }_{i}\right)\right)=0$ if and only if $\gamma \left({\alpha }_{i}\right)=±{c}_{{\alpha }_{i}}\text{.}$ Thus ${\tau }_{i}{\tau }_{i},$ and each factor in this composition, is invertible if and only if $\gamma \left({\alpha }_{i}\right)\ne ±{c}_{{\alpha }_{i}}\text{.}$ (e) Let $w={s}_{{i}_{1}}\dots {s}_{{i}_{\ell }}$ be a reduced word for $w\in W$ and set ${\tau }_{w}={\tau }_{{i}_{1}}\dots {\tau }_{{i}_{\ell }}\text{.}$ Using the definition ${\tau }_{i}={t}_{{s}_{i}}-\frac{{c}_{{\alpha }_{i}}}{{\alpha }_{i}}$ and the defining relation (2.2) for $ℍ$ yields an expansion $τw=tw+∑z where the ${R}_{z}$ are rational functions of $\alpha \in R\text{.}$ We shall show that this expansion of ${\tau }_{w}$ does not depend on the choice of reduced word of $w\text{.}$ Let $1$ be the trivial $ℂW\text{-module}$ and let $e={\sum }_{w\in W}{t}_{w}\text{.}$ View the $ℍ\text{-module}$ $ℍe\cong {\text{Ind}}_{ℂW}^{ℍ}\left(1\right)=ℍ{\otimes }_{ℂW}1=S\left({𝔥}_{ℂ}^{*}\right)\otimes ℂW{\otimes }_{ℂW}1=S\left({𝔥}_{ℂ}^{*}\right)\otimes 1$ simply as $S\left({𝔥}_{ℂ}^{*}\right)\text{.}$ Let us first show that this $ℍ\text{-module}$ $S\left({𝔥}_{ℂ}^{*}\right)={\text{Ind}}_{ℂW}^{ℍ}\left(1\right)$ is faithful. Assume that $h={\sum }_{z\in W}{P}_{z}{t}_{z}$ in $ℍ=S\left({𝔥}_{ℂ}^{*}\right)\otimes ℂW$ satisfies $h\left(p\right)=0$ for all $p\in S\left({𝔥}_{ℂ}^{*}\right)\text{.}$ We must show that $h=0\text{.}$ Since $0=h\left(1\right)={\sum }_{z}{P}_{z},$ and this is true degree by degree, we may assume that the polynomials ${P}_{z}$ are homogeneous of the same degree. Use the notations of Lemma 2.2 so that $\left\{{b}_{w} | w\in W\right\}$ is a basis of the space of harmonic polynomials $ℋ$ consisting of homogeneous polynomials. Then, for each $w\in W,$ $0=h(bw)= ∑z∈WPz tzbw(1) = ( ∑z∈WPz (z-1bw) tz+lower degree terms ) (1) = ∑z∈WPz (z-1bw)+ lower degree terms.$ where, by definition, each ${t}_{z}$ is degree 0. Focusing on top degree terms in this equality, $0={\sum }_{z\in W}{P}_{z}\left({z}^{-1}{b}_{w}\right),$ for each $w\in W\text{.}$ By Lemma 2.2, the matrix ${\left({z}^{-1}{b}_{w}\right)}_{z,w\in W}$ is invertible, so there is a nonzero $\xi \in ℂ$ with $\xi ·{\left({\prod }_{\alpha >0}\alpha \right)}^{|W|/2}{P}_{z}=0,$ for every $z\in W\text{.}$ Since $S\left({𝔥}_{ℂ}^{*}\right)$ is an integral domain, ${P}_{z}=0$ for each $z\in W,$ and hence $h=0\text{.}$ So the $ℍ\text{-module}$ ${\text{Ind}}_{ℂW}^{ℍ}\left(1\right)\cong S\left({𝔥}_{ℂ}^{*}\right)$ is faithful. Let ${\stackrel{\sim }{\tau }}_{i}={t}_{{s}_{i}}{\alpha }_{i}-{c}_{{\alpha }_{i}}\in ℍ\text{.}$ As operators on ${\text{Ind}}_{ℂW}^{ℍ}\left(1\right)\cong S\left({𝔥}_{ℂ}^{*}\right),$ $τiαi=τ∼i, τ∼i(1)= (-αitsi+cαi) (1)=(-αi+cαi) ,andτ∼ip= (sip)τ∼i,$ for any polynomial $p\in S\left({𝔥}_{ℂ}^{*}\right)\text{.}$ Using the fact [Bou1968, Chapt. VI §1.11 Prop. 33] that, for a reduced word $w={s}_{{i}_{1}}\dots {s}_{{i}_{\ell }},$ $R\left(w\right)=\left\{{\alpha }_{{i}_{\ell }},{s}_{{i}_{\ell }}{\alpha }_{{i}_{\ell -1}},\dots ,{s}_{{i}_{\ell }}\dots {s}_{{i}_{2}}{\alpha }_{{i}_{1}}\right\},$ $(τi1…τiℓ) (∏α∈R(w)α) (p) = ( t∼i1… t∼iℓ ) p(1)=(wp) ( t∼i1… t∼iℓ ) (1) = (wp) ( ∏α∈R(w) (-α+cα) ) .$ Thus, since $S\left({𝔥}_{ℂ}^{*}\right)$ is an integral domain and ${\text{Ind}}_{ℂW}^{ℍ}\left(1\right)$ is faithful, ${\tau }_{{i}_{1}}\dots {\tau }_{{i}_{\ell }}$ does not depend on the choice of the reduced word $w={s}_{{i}_{1}}\dots {s}_{{i}_{\ell }}\text{.}$ $\square$

Let $\gamma \in {𝔥}_{ℂ}$ and define

$Z(γ)= {α>0 | γ(α)=0} andP(γ)= {α>0 | γ(α)=±cα}. (2.15)$

If $J\subseteq P\left(\gamma \right),$ define

$ℱ(γ,J)= { w∈W | R(w)∩Z(γ) =∅ and R(w) ∩P(γ)=J } . (2.16)$

A local region is a pair $\left(\gamma ,J\right)$ such that $\gamma \in {𝔥}_{ℂ},$ $J\subseteq P\left(\gamma \right),$ and $ℱ\left(\gamma ,J\right)\ne \varnothing \text{.}$ Under the bijection (2.6) the set ${ℱ}^{\left(\gamma ,J\right)}$ maps to the set of points $x\in {𝔥}_{ℝ}$ which are

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

In this way the local region $\left(\gamma ,J\right)$ 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 $w\in {ℱ}^{\left(\gamma ,J\right)}$ index the chambers ${w}^{-1}C$ in the local region and the sets ${ℱ}^{\left(\gamma ,J\right)}$ form a partition of the set $\left\{w\in W | R\left(w\right)\cap Z\left(\gamma \right)=\varnothing \right\}$ (which, by (2.7), indexes the cosets in $W/{W}_{\gamma }\text{).}$

Corollary 2.6. Let $M$ be a finite dimensional $ℍ\text{-module.}$ Let $\gamma \in {𝔥}_{ℂ}$ and let $J\subseteq P\left(\gamma \right)\text{.}$ Then

$dim(Mwγgen)= dim(Mw′γgen) for w,w′∈ ℱ(γ,J),$

where ${ℱ}^{\left(\gamma ,J\right)}$ is given by (2.16).

 Proof. Suppose $w,{s}_{i}w\in {ℱ}^{\left(\gamma ,J\right)}\text{.}$ We may assume ${s}_{i}w>w\text{.}$ Then $\alpha ={w}^{-1}{\alpha }_{i}>0,$ $\alpha \notin R\left(w\right)$ and $\alpha \in R\left({s}_{i}w\right)\text{.}$ Now, $R\left(w\right)\cap P\left(\gamma \right)=R\left({s}_{i}w\right)\cap P\left(\gamma \right)$ implies $\gamma \left(\alpha \right)\ne ±{c}_{{\alpha }_{i}}\text{.}$ Since ${c}_{\alpha }={c}_{w\alpha }={c}_{{\alpha }_{i}},$ $w\gamma \left({\alpha }_{i}\right)=\gamma \left({w}^{-1}{\alpha }_{i}\right)=\gamma \left(\alpha \right)\ne 0$ and $w\gamma \left({\alpha }_{i}\right)\ne ±{c}_{{\alpha }_{i}}$ and thus, by Proposition 2.5(d), the map ${\tau }_{i}:{M}_{w\gamma }^{\text{gen}}\to {M}_{{s}_{i}w\gamma }^{\text{gen}}$ is well defined and invertible. It remains to note that if $w,w\prime \in {ℱ}^{\left(\gamma ,J\right)},$ then $w\prime ={s}_{{i}_{1}}\dots {s}_{{i}_{\ell }}w$ where ${s}_{{i}_{k}}\dots {s}_{{i}_{\ell }}w\in {ℱ}^{\left(\gamma ,J\right)}$ for all $1\le k\le \ell \text{.}$ This follows from the fact that $\left(\gamma ,J\right)$ corresponds to a connected convex region in ${𝔥}_{ℝ}\text{.}$ $\square$

The following lemma will be used in the classification in Section 3 to analyze weight spaces for representations with nonregular central character.

Lemma 2.7. Let $\gamma {𝔥}_{ℂ}$ such that $\gamma \left({\alpha }_{i}\right)=0\text{.}$ Let $M$ be an $ℍ\text{-module}$ such that ${M}_{\gamma }^{\text{gen}}\ne 0$ and let $w\in {ℱ}^{\left(\gamma ,\varnothing \right)}\text{.}$ Then

1. $\text{dim} {M}_{w\gamma }^{\text{gen}}\ge 2,$
2. if ${M}_{{s}_{j}w\gamma }^{\text{gen}}=0,$ then $\left(w\gamma \right)\left({\alpha }_{j}\right)=±{c}_{{\alpha }_{j}}$ and $⟨{w}^{-1}{\alpha }_{j},{\alpha }_{i}^{\vee }⟩=0\text{.}$

 Proof. Let $ℍ{A}_{1}$ be the subalgebra of $ℍ$ generated by ${t}_{{s}_{i}}$ and all $x\in S\left({𝔥}_{ℂ}^{*}\right)\text{.}$ Let $ℂ{v}_{\gamma }$ be the one-dimensional representation of $S\left({𝔥}_{ℂ}^{*}\right)$ defined by $x{v}_{\gamma }=\gamma \left(x\right){v}_{\gamma }$ and let $M\left(\gamma \right)={\text{Ind}}_{S\left({𝔥}_{ℂ}^{*}\right)}^{ℍ{A}_{1}}\left(ℂ{v}_{\gamma }\right)=ℍ{A}_{1}{\otimes }_{S\left({𝔥}_{ℂ}^{*}\right)}ℂ{v}_{\gamma }\text{.}$ This module is irreducible and has basis $\left\{{v}_{\gamma },{t}_{{s}_{i}}{v}_{\gamma }\right\}$ and, with respect to this basis, the action of $x\in {𝔥}_{ℂ}^{*}$ on $M\left(\gamma \right)$ is given by the matrix $ργ(x)= ( γ(x) cαi⟨x,αi∨⟩ 0 γ(x) ) . (2.17)$ Let ${n}_{\gamma }$ be a nonzero vector in ${M}_{\gamma }\text{.}$ As an $S\left({𝔥}_{ℂ}^{*}\right)\text{-module}$ $ℂ{n}_{\gamma }\cong ℂ{v}_{\gamma }$ and, since induction is the adjoint functor to restriction, there is a unique $ℍ{A}_{1}\text{-module}$ homomorphism given by $M(γ)⟶M vγ⟼nγ$ Since $M\left(\gamma \right)$ is irreducible, this homomorphism is injective, and the vectors ${n}_{\gamma },{t}_{{s}_{i}}{n}_{\gamma }$ span a two-dimensional subspace of ${M}_{\gamma }^{\text{gen}}$ on which the action of $x\in {𝔥}_{ℂ}^{*}$ is given by the matrix in (2.17). Let $w={s}_{{i}_{1}}\dots {s}_{{i}_{p}}$ be a reduced word for $w\text{.}$ Proposition 2.5(d) and the assumption that $w\in {ℱ}^{\left(\gamma ,\varnothing \right)}$ guarantee that the map $τw=τi1… τiℓ: Mγgen→ Mwγgen$ is well-defined and bijective. Thus ${\tau }_{w}{n}_{\gamma }$ and ${\tau }_{w}{t}_{{s}_{i}}{n}_{\gamma }$ span a two-dimensional subspace of ${M}_{w\gamma }^{\text{gen}}$ and, by Proposition 2.5(b), the $ℍ{A}_{1}$ action of $x\in X$ on this subspace is given by $ρwγ(x)= ( γ(w-1x) cαi⟨w-1x,αi∨⟩ 0 γ(w-1x) ) .$ This proves (a). Assume ${M}_{{s}_{j}w\gamma }^{\text{gen}}=0\text{.}$ Then part (a) implies ${s}_{j}w\gamma \ne w\gamma ,$ so $\left(w\gamma \right)\left({\alpha }_{j}\right)=\gamma \left({w}^{-1}{\alpha }_{j}\right)\ne 0\text{.}$ So the matrix ${\rho }_{w\gamma }\left({\alpha }_{j}\right)$ is invertible and $ρwγ (1αj)= 1γ(w-1αj)2 ( γ(w-1αj) -cαi⟨w-1αj,αi∨⟩ 0 γ(w-1αj) ) .$ Since ${M}_{{s}_{j}w\gamma }^{\text{gen}}=0,$ the map ${\tau }_{j}:{M}_{w\gamma }^{\text{gen}}\to {M}_{{s}_{j}w\gamma }^{\text{gen}}$ is the zero map and $ρwγ(tsj) =ρwγ (cαjαj)= cαjγ(w-1αj)2 ( γ(w-1αj) -cαj⟨w-1αj,αi∨⟩ 0 γ(w-1αj) ) .$ Since ${t}_{{s}_{j}}^{2}-1=\left({t}_{{s}_{j}}-1\right)\left({t}_{{s}_{j}}+1\right)=0,{\rho }_{w\gamma }\left({t}_{{s}_{j}}\right)$ must have Jordan blocks of size 1 and eigenvalues $±1\text{.}$ Since ${c}_{{\alpha }_{i}}\ne 0,$ it follows that $\gamma \left({w}^{-1}{\alpha }_{j}\right)=±{c}_{{\alpha }_{j}}$ and $⟨{w}^{-1}{\alpha }_{j},{\alpha }_{i}^{\vee }⟩=0\text{.}$ $\square$

### Principal series modules

For $\gamma \in {𝔥}_{ℂ}$ let $ℂ{v}_{\gamma }$ be the one-dimensional $S\left({𝔥}_{ℂ}^{*}\right)\text{-module}$ given by

$xvγ=γ(x) vγfor x∈ 𝔥ℂ*.$

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

$M(γ)=ℍ ⊗S(𝔥ℂ*) ℂvγ= IndS(𝔥ℂ*)ℍ (ℂvγ). (2.18)$

The module $M\left(\gamma \right)$ has basis $\left\{{t}_{w}\otimes {v}_{\gamma } | w\in W\right\}$ with $ℂW$ acting by left multiplication. By the defining relations for $ℍ,$ for $x\in {𝔥}_{ℂ}^{*},$ $w\in W,$

$xtwvγ= (wγ)(x) tw⊗vγ+ ∑z

Thus, if $\gamma \in {𝔥}_{ℂ}$ is regular all the $w\gamma$ are distinct and

$M(γ)=⨁w∈W m(γ)wγwith dim(M(γ)wγ) =1.$

Thus, if $\gamma \in {𝔥}_{ℂ}$ is regular, there is a unique basis $\left\{{v}_{w\gamma } | w\in W\right\}$ of $M\left(\gamma \right)$ determined by

$xvwγ=(wγ) (x)vwγ for all w∈W and x∈ 𝔥ℂ*, (2.19) vwγ=tw⊗ vγ+∑u

Alternatively,

$vwγ=τwvγ (2.21)$

where ${\tau }_{w}={\tau }_{{i}_{1}}{\tau }_{{i}_{2}}\dots {\tau }_{{i}_{p}}$ for a reduced word $w={s}_{{i}_{1}}\dots {s}_{{i}_{p}}$ of $w\text{.}$ The uniqueness of the element ${v}_{w\gamma }$ given by the conditions (2.19) and (2.20) shows that ${v}_{w\gamma }={\tau }_{w}{v}_{\gamma }$ does not depend on the reduced decomposition which is chosen for $w\text{.}$

Part (a) of the following proposition implies that the dimension of every irreducible $ℍ\text{-module}$ is less than or equal to $|W|\text{.}$ In combination, part (a) and part (b) show that every irreducible $ℍ\text{-module}$ with regular central character is calibrated. Part (c) is a graded Hecke analogue of a result of Rogawski [Rog1985, Proposition 2.3].

Proposition 2.8.

1. If $M$ is an irreducible finite dimensional $ℍ\text{-module}$ with ${M}_{\gamma }^{\text{gen}}\ne 0,$ then $M$ is a quotient of $M\left(\gamma \right)\text{.}$
2. If $\gamma \in {𝔥}_{ℂ}$ is regular, then $M\left(\gamma \right)$ is calibrated.
3. For fixed $\gamma \in {𝔥}_{ℂ}$ and any $w\in W,$ $M\left(\gamma \right)$ and $M\left(w\gamma \right)$ have the same composition factors.

 Proof. (a) Since $S\left({𝔥}_{ℂ}^{*}\right)$ is commutative, an irreducible $S\left({𝔥}_{ℂ}^{*}\right)$ submodule must be one-dimensional. Thus there exists a nonzero vector ${m}_{\gamma }$ in ${M}_{\gamma }$ and, as an $S\left({𝔥}_{ℂ}^{*}\right)\text{-module,}$ $ℂ{m}_{\gamma }\cong ℂ{v}_{\gamma }\text{.}$ Since induction is the adjoint functor to restriction there is a unique $ℍ\text{-module}$ homomorphism given by $M(γ) ⟶M vγ⟼ mγ$ and, since $M$ is irreducible, this homomorphism is surjective. Thus $M$ is a quotient of $M\left(\gamma \right)\text{.}$ (b) Since $\gamma$ is regular, ${W}_{\gamma }=\left\{1\right\},$ and by equation (2.19), $M(γ)=⨁w∈W M(γ)wγand dim(M(γ)wγ) =1$ for all $w\in W\text{.}$ Since $M{\left(\gamma \right)}_{w\gamma }$ is nonzero whenever $M{\left(\gamma \right)}_{w\gamma }^{\text{gen}}$ is nonzero and $\text{dim}\left(M{\left(\gamma \right)}_{w\gamma }^{\text{gen}}\right)=1,$ $M{\left(\gamma \right)}_{w\gamma }=M{\left(\gamma \right)}_{w\gamma }^{\text{gen}}$ for all $w\in W\text{.}$ (c) Let ${s}_{i}$ be a simple reflection such that ${s}_{i}\gamma \ne \gamma \text{.}$ Then $\gamma \left({\alpha }_{i}\right)\ne 0$ and the operator $\tau$ is well defined on $M{\left({s}_{i}\gamma \right)}_{{s}_{i}\gamma }^{\text{gen}}\text{.}$ The vector ${v}_{{s}_{i}\gamma }$ is a weight vector in $M{\left({s}_{i}\gamma \right)}_{{s}_{i}\gamma }$ and, by Proposition 2.5(b), ${\tau }_{i}{v}_{{s}_{i}\gamma }$ is a weight vector of weight $\gamma$ (it is nonzero since ${t}_{{s}_{i}}{v}_{{s}_{i}\gamma }$ and $\left({s}_{i}\gamma \right)\left({c}_{{\alpha }_{i}}/{\alpha }_{i}\right){v}_{{s}_{i}\gamma }$ are linearly independent in $M\left({s}_{i}\gamma \right)\text{).}$ Thus, there is an $ℍ\text{-module}$ homomorphism $A(si,γ): M(γ) ⟶ M(siγ) hvγ ⟼ hτivsiγ, h∈ℍ.$ The modules $M\left(\gamma \right)$ and $M\left({s}_{i}\gamma \right)$ have bases ${ tw (tsi+1) vγ, tw (tsi-1) vγ } siw>w , { tw (tsi+1) vsiγ, tw (tsi-1) vsiγ } siw>w , (2.22)$ respectively. Since $\left({t}_{{s}_{i}}+1\right){t}_{{s}_{i}}={t}_{{s}_{i}}+1$ and $\left({t}_{{s}_{i}}-1\right){t}_{{s}_{i}}=-\left({t}_{{s}_{i}}-1\right),$ $A(si,γ) (tw(tsi+1)vγ) = tw(tsi+1) ( tsi- cαiαi ) vsiγ = tw(tsi+1) (1-cαiαi) vsiγ = ( siγ ( αi-cαi αi ) ) tw(tsi+1) vsiγ A(si,γ) (tw(tsi-1)vγ) = tw(tsi-1) ( tsi- cαiαi ) vsiγ = tw(tsi-1) ( -1- cαiαi ) vsiγ = ( siγ ( αi+cαi -αi ) ) tw(tsi-1) vsiγ$ and so the matrix of $A\left({s}_{i},\gamma \right)$ with respect to the bases in (2.22) is diagonal with $|W|/2$ diagonal entries equal to $\left({s}_{i}\gamma \right)\left(\left({\alpha }_{i}-{c}_{{\alpha }_{i}}\right)/{\alpha }_{i}\right)$ and $|W|/2$ diagonal entries equal to $\left({s}_{i}\gamma \right)\left(\left({\alpha }_{i}+{c}_{{\alpha }_{i}}\right)/\left(-{\alpha }_{i}\right)\right)\text{.}$ If $\gamma \left({\alpha }_{i}\right)\ne ±{c}_{{\alpha }_{i}},$ then $A\left({s}_{i},\gamma \right)$ is an isomorphism and so $M\left(\gamma \right)$ and $M\left({s}_{i}\gamma \right)$ have the same composition factors. If $\gamma \left({\alpha }_{i}\right)=±{c}_{{\alpha }_{i}},$ then $\text{dim}\left(\text{ker} A\left({s}_{i},\gamma \right)\right)=|W|/2\text{.}$ In this case $A\left({s}_{i},{s}_{i}\gamma \right)A\left({s}_{i},\gamma \right)=0$ and so the sequence $M(γ) ⟶A(si,γ) M(siγ) ⟶A(si,siγ) M(γ)$ is exact. Then $M(siγ)⊇ ker(A(si,siγ)) ⊇0$ is a filtration of $M\left({s}_{i}\gamma \right)$ where the first factor is isomorphic to a submodule of $M\left(\gamma \right),$ $M(siγ)/ ker(A(si,siγ)) ≅im(A(si,siγ)) ⊆M(γ),$ and the second factor is isomorphic to a quotient of $M\left(\gamma \right),$ $ker(A(si,siγ))≅ M(γ)/ker(A(si,γ)).$ Since $\text{dim}\left(\text{ker}\left(A\left({s}_{i},{s}_{i}\gamma \right)\right)\right)+\text{dim}\left(\text{im}\left(A\left({s}_{i},{s}_{i}\gamma \right)\right)\right)=|W|/2+|W|/2=\text{dim}\left(M\left({s}_{i}\gamma \right)\right)=\text{dim}\left(M\left(\gamma \right)\right),$ it follows that $M\left(\gamma \right)$ and $M\left({s}_{i}\gamma \right)$ must have the same composition factors. $\square$

Our next goal is to prove Theorem 2.10 which determines exactly when the principal series module $M\left(\gamma \right)$ is irreducible. Let $\gamma \in {𝔥}_{ℂ}$ and let $M\left(\gamma \right)=ℍ{\otimes }_{S\left({𝔥}_{ℂ}^{*}\right)}ℂ{v}_{\gamma }$ be the corresponding principal series module for $ℍ\text{.}$ The spherical vector in $M\left(\gamma \right)$ is

$1γ= ∑w∈Wtw vγ. (2.23)$

Up to multiplication by constants this is the unique vector in $M\left(\gamma \right)$ such that ${t}_{w}{1}_{\gamma }={1}_{\gamma }$ for all $w\in W\text{.}$ The following proposition provides a graded Hecke analogue of the results in [Kat1982-2, Proposition 1.20] and [Kat1982-2, Lemma 2.3]. Mention of this analogue was made in [Opd1995].

Proposition 2.9.

1. If $\gamma$ is a generic element of ${𝔥}_{ℂ}$ and ${v}_{w\gamma },$ $w\in W,$ is the basis of $M\left(\gamma \right)$ defined in (2.21), then $1γ=∑z∈W γ(cz)vzγ wherecz= ∏α∈R(w0z) α+cαα.$
2. The spherical vector ${1}_{\gamma }$ generates $M\left(\gamma \right)$ if and only if ${\prod }_{\alpha >0}\left(\gamma \left(\alpha \right)+{c}_{\alpha }\right)\ne 0\text{.}$
3. For $\gamma \in {𝔥}_{ℂ},$ the principal series module $M\left(\gamma \right)$ is irreducible if and only if ${1}_{w\gamma }$ generates $M\left(w\gamma \right)$ for all $w\in W\text{.}$

 Proof. (a) Suppose that ${\xi }_{z}\in ℂ$ are constants such that $1γ= (∑w∈Wtw) vγ=∑z∈W ξzvzγ.$ We shall prove that the $\xi$ are given by the formula in the statement of the proposition. Since ${t}_{{s}_{i}}\left({\sum }_{w\in W}{t}_{w}\right)={\sum }_{w\in W}{t}_{w},$ $1γ = tsi1γ= ( τi+ cαiαi ) ∑z∈W ξzvzγ= ( τi+ cαiαi ) ∑siz>z ( ξzvzγ+ ξsiz vsizγ ) = ∑siz>z ( ξzvsizγ+ ξzcαiγ(z-1αi) vzγ+ξsiz τi2vzγ+ ξsiz cαiγ(-z-1αi) vsizγ ) .$ Comparing coefficients of ${v}_{{s}_{i}z\gamma }$ on each side of this expression gives $ξsiz=ξz+ ξsiz cαi γ(-z-1αi) ,$ and so $ξzξsiz=γ ( z-1αi+cαi z-1αi ) ,if siz>z.$ Using this formula inductively gives $ξw = ξsi1…sip =γ ( sip…si2αi1 sip…si2αi1+cαi ) …γ ( αip αip+cαip ) ξ1 = γ ( ∏α∈R(w) αα+cα ) ξ1.$ Since the transition matrix between the basis $\left\{{t}_{w}{v}_{\gamma }\right\}$ and the basis $\left\{{v}_{w\gamma }\right\}$ is upper unitriangular with respect to Bruhat order, ${\xi }_{{w}_{0}}=1\text{.}$ Thus, the last equation implies that $ξ1=γ ( ∏α>0 α+cαα )$ and $ξw=γ ( ∏α∈R(w) αα+cα ) ·ξ1=γ ( ∏α∈R(w0w) α+cαα ) .$ (b) By expanding ${v}_{z\gamma }={\tau }_{z}{v}_{\gamma }={\tau }_{{i}_{1}}\dots {\tau }_{{i}_{p}}{v}_{\gamma }$ for a reduced word ${s}_{{i}_{1}}\dots {s}_{{i}_{p}}=z$ it follows that there exist rational functions ${m}_{uz}$ such that $vzγ=∑u∈W γ(muz)tu vγ,$ for all generic $\gamma \in {𝔥}_{ℂ}\text{.}$ Furthermore, the matrix $M={\left({m}_{uz}\right)}_{u,z\in W}$ with these rational functions as entries is upper unitriangular. Let ${b}_{w},$ $w\in W,$ be a basis of harmonic polynomials and define polynomials ${q}_{uy}\in S\left({𝔥}_{ℂ}^{*}\right),$ $u,y\in W,$ by $by(∑w∈Wtw) =∑u∈Wtu quy,y∈W,$ where these equations are equalities in $ℍ\text{.}$ Then, $by1γ=by (∑w∈Wtw) =∑u∈Wγ (quy) (tu⊗vγ),$ and part (a) implies that if $\gamma$ is generic, then $by1γ = by∑z∈Wγ (cz)vzγ =∑z∈Wγ (cz(z-1by)) vzγ = ∑z,u∈Wγ (cz(z-1by)muz) (tu⊗vγ).$ Since these two expressions are equal for all generic $\gamma \in {𝔥}_{ℂ},$ it follows that $quy=∑z∈W muz·cz· (z-1by), u,y∈W, (2.24)$ as rational functions (in fact, both sides are polynomials). Since ${t}_{w},$ $w\in W,$ and $p\in Z\left(ℍ\right)=S{\left({𝔥}_{ℂ}^{*}\right)}^{W}$ act on ${1}_{\gamma }$ by constants, the $ℍ\text{-module}$ $M\left(\gamma \right)$ is generated by ${1}_{\gamma }$ if and only if there exist constants ${p}_{yw}\in ℂ$ such that $tw⊗vγ=∑y∈W pywby1γ for each w∈W.$ If these constants exist, then, for each $w\in W,$ $tw⊗vγ= ∑y∈Wpyw by1γ= ∑y,z,u∈Wγ ( muzcz (z-1by) pyw ) tu⊗vγ,$ where, by (2.24), there is no restriction that $\gamma$ be generic. If $M=(muz)u,z∈W, C=diag(cz)z∈W, X=(z-1by)z,y∈W, P=(pyw)y,w∈W,$ then $P={\left(\gamma \left(MCX\right)\right)}^{-1}$ and so $P$ exists if and only if $\text{det}\left(\gamma \left(MCX\right)\right)\ne 0\text{.}$ Now $\text{det}\left(M\right)=1,$ and, by Lemma 2.2 and part (a), $det(X)=ξ· ∏α>0 α|W|/2$ and $det(C)=∏z∈W ∏α∈R(w0z) α+cαα= ( ∏α>0 α+cαα ) |W|/2 ,$ where $\xi \in ℂ$ is nonzero. Thus $P$ exists if and only if ${\prod }_{\alpha >0}\left(\gamma \left(\alpha \right)+{c}_{\alpha }\right)\ne 0\text{.}$ (c) $⟹:$ If $M\left(\gamma \right)$ is irreducible, then, by Proposition 2.8(c), $M\left(w\gamma \right)$ is irreducible for all $w\in W\text{.}$ Hence $M\left(w\gamma \right)$ is generated by ${1}_{w\gamma }\text{.}$ $⟸:$ Suppose that ${1}_{w\gamma }$ generates $M\left(w\gamma \right)$ for all $w\in W\text{.}$ Let $E$ be a nonzero irreducible submodule of $M\left(\gamma \right)$ and let $w\in W$ be such that the weight space ${E}_{w\gamma }$ is nonzero. Then, by Proposition 2.8(a), there is a nonzero surjective $ℍ\text{-module}$ homomorphism $\phi :M\left(w\gamma \right)\to E\text{.}$ Since ${1}_{w\gamma }$ generates $M\left(w\gamma \right),$ $\phi \left({1}_{w\gamma }\right)$ is a nonzero vector in $E$ such that ${t}_{v}\phi \left({1}_{w\gamma }\right)=\phi \left({1}_{w\gamma }\right)$ for all $v\in W\text{.}$ Since there is a unique, up to constant multiples, spherical vector in $M\left(\gamma \right),$ $\varphi \left({1}_{w\gamma }\right)$ is a multiple of ${1}_{\gamma }$ and ${1}_{\gamma }$ is nonzero. This implies that $E=M\left(\gamma \right)$since ${1}_{\gamma }$ generates $M\left(\gamma \right)\text{.}$ $\square$

Together the three parts of Proposition 2.9 prove the following graded Hecke algebra analogue of [Kat1982-2, Theorem 2.1].

Theorem 2.10. Let $\gamma \in {𝔥}_{ℂ}$ and let $P\left(\gamma \right)=\left\{\alpha >0 | \gamma \left(\alpha \right)=±{c}_{\alpha }\right\}\text{.}$ The principal series $ℍ\text{-module}$

$M(γ) is irreducible if and only if P(γ)=∅.$

## 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.