Preliminaries

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 25 April 2013

Preliminaries

The graded Hecke algebra

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

$$H_{\alpha }=(+1\hspace{0.17em}\text{eigenspace of}\hspace{0.17em}{s}_{\alpha })=\{x\in {𝔥}_{ℝ}^{*}\hspace{0.17em}|\hspace{0.17em}{\alpha }^{\vee }\left(x\right)=0\},$$

where we fix the linear function ${\alpha }^{\vee }\in {𝔥}_{ℝ}={\text{Hom}}_{ℝ}({𝔥}_{ℝ}^{*},ℝ)$ 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

$$\begin{array}{cc}{s}_{\alpha }x=x-⟨x,{\alpha }^{\vee }⟩\alpha ,\text{for all}\hspace{0.17em}x\in {𝔥}_{ℝ}^{*},\hspace{0.17em}\text{where}\hspace{0.17em}⟨x,{\alpha }^{\vee }⟩={\alpha }^{\vee }\left(x\right)\text{.}& \text{(2.1)}\end{array}$$

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 ${𝔥}_{ℂ}={\text{Hom}}_{ℂ}({𝔥}_{ℂ}^{*},ℂ)\text{.}$

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

$$c_{\alpha }=c_{w\alpha },\text{for}\hspace{0.17em}w\in W\text{.}$$

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

$$ℂW=ℂ\text{-span}\left\{t_w\hspace{0.17em}\right|\hspace{0.17em}w\in W\}\text{with multiplication}{t}_w{t}_{w\prime }=t_{ww\prime }\text{.}$$

The graded Hecke algebra is

$$ℍ=ℂW\otimes S\left({𝔥}_{ℂ}^{*}\right)$$

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

$$\begin{array}{cc}x{t}_{s_i}=t_{s_i}{s}_i\left(x\right)+c_{{\alpha }_i}⟨x,{\alpha }_i^{\vee }⟩,\text{for}\hspace{0.17em}x\in {𝔥}_{ℂ}^{*},& \text{(2.2)}\end{array}$$

where ${\alpha }_1^{\vee },\dots ,{\alpha }_n^{\vee }\in {𝔥}_{ℝ}$ are the simple co-roots. More generally, it follows that for any $p\in S\left({𝔥}_{ℂ}^{*}\right),$

$$p{t}_{s_i}=t_{s_i}{s}_i\left(p\right)+c_{{\alpha }_i}{\Delta }_i\left(p\right)\text{and}{t}_{s_i}p=s_i\left(p\right)t_{s_i}+c_{{\alpha }_i}{\Delta }_i\left(p\right),$$

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

$${\Delta }_i\left(p\right)=\frac{p-s_i\left(p\right)}{{\alpha }_i}\text{for}\hspace{0.17em}p\in S\left({𝔥}_{ℂ}^{*}\right)\text{.}$$

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 $$p{t}_{s_i}=t_{s_i}{s}_i\left(p\right)+c_{{\alpha }_i}\frac{p-s_i\left(p\right)}{{\alpha }_i}=t_{s_i}p+0=t_{s_i}p,$$ 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 $$\mu p=\sum _{w\in W}\mu p_w{t}_w\text{equals}p\mu =\sum _{w\in W}{p}_w{t}_w\mu =\sum _{w\in W}{p}_w(\left(w\mu \right)t_w+\sum _{u<w}{c}_{u,w}^{\mu }{t}_u),$$ where $c_{u,w}^{\mu }\in ℂ\text{.}$ Comparing coefficients of $t_v$ yields $$\mu p_v=p_v·\left(v\mu \right)\text{.}$$ 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 $$p{t}_{s_i}=s_i\left(p\right)t_{s_i}+c_{{\alpha }_i}\frac{p-s_i\left(p\right)}{{\alpha }_i}$$ 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

$$⟨P,Q⟩=\left(P\left(\partial \right)Q\right){|}_{x_i=0},\text{for}\hspace{0.17em}P,Q\in S\left({𝔥}_{ℂ}^{*}\right),$$

where $P\left(\partial \right)=P(\frac{\partial }{\partial x_1},\dots ,\frac{\partial }{\partial x_n})$ and ${|}_{x_i=0}$ denotes specializing the variables to 0 (or, equivalently, taking the constant term). The monomials are an orthogonal basis of $S\left({𝔥}_{ℂ}^{*}\right),$

$$\begin{array}{ccc}⟨x_1^{{\lambda }_1}\dots x_n^{{\lambda }_n},x_1^{{\mu }_1}\dots x_n^{{\mu }_n}⟩& =& \left({\left(\frac{\partial }{\partial x_1}\right)}^{{\lambda }_1}\dots {\left(\frac{\partial }{\partial x_n}\right)}^{{\lambda }_n}{x}_1^{{\mu }_1}\dots x_n^{{\mu }_n}\right){|}_{x_i=0}\\ & =& {\delta }_{{\lambda }_1{\mu }_1}\dots {\delta }_{{\lambda }_n{\mu }_n}{\lambda }_1!{\lambda }_2!\dots {\lambda }_n!,\end{array}$$

and so the bilinear form $⟨,⟩$ is nondegenerate. The vector space $\mathscr{H}$ 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,

$$\mathscr{H}={\left(⟨f\in S{\left({𝔥}_{ℂ}^{*}\right)}^W\hspace{0.17em}|\hspace{0.17em}f\left(0\right)=0⟩\right)}^{\perp },\text{and}S\left({𝔥}_{ℂ}^{*}\right)=S{\left({𝔥}_{ℂ}^{*}\right)}^W\otimes \mathscr{H},$$

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

$$f=\sum _w{p}_w{h}_w,p_w\in S{\left({𝔥}_{ℂ}^{*}\right)}^W\text{.}$$

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 $W,$

$$\begin{array}{cc}{P}_W\left(t\right)=\sum _{k\ge 0}\text{dim}\left({\mathscr{H}}^k\right)t^k=\prod _{i=1}^n\frac{1-t^{d_i}}{1-t}=\sum _{w\in W}{t}^{\ell \left(w\right)},& \text{(2.3)}\end{array}$$

where $d_1,\dots ,d_n$ are the degrees of a set $f_1,\dots ,f_n$ of homogeneous generators of $S{\left({𝔥}_{ℂ}^{*}\right)}^W=ℂ[f_1,\dots ,f_n]$ and ${\mathscr{H}}^k$ is the $k\text{th}$ homogeneous component of $\mathscr{H}\text{.}$ In particular, $\text{dim}\left(\mathscr{H}\right)=\text{Card}\left(\left\{h_w\right\}\right)=P_W\left(1\right)=\left|W\right|$ and $S\left({𝔥}_{ℂ}^{*}\right)$ is a free module over $S{\left({𝔥}_{ℂ}^{*}\right)}^W$ of rank $\left|W\right|\text{.}$

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 $\mathscr{H}$ of harmonic polynomials and let $X$ be the $\left|W\right|\times \left|W\right|$ matrix given by

$$X={\left(z^{-1}{b}_w\right)}_{z,w\in W}\text{.}$$

Then

$$\text{det}\hspace{0.17em}X=\xi ·{\left(\prod _{\alpha >0}\alpha \right)}^{\left|W\right|/2},$$

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

		Proof.
	Note that if $b_w^{\prime }$ is another basis of $\mathscr{H}$ and we write $$b_w^{\prime }=\sum _{v\in W}{c}_{vw}{b}_v,c_{vw}\in ℂ,$$ then $$X\prime ={\left(z^{-1}{b}_w^{\prime }\right)}_{z,w\in W}=\left(z^{-1}{b}_v\right)\left(c_{vw}\right)\text{and so}\text{det}\hspace{0.17em}X\prime =\xi \hspace{0.17em}\text{det}\hspace{0.17em}X,$$ 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 $\left|W\right|/2$ pairs $\{z^{-1},s_{\alpha }{z}^{-1}\}$ we conclude that $\text{det}\left(X\right)$ is divisible by ${\alpha }^{\left|W\right|/2}\text{.}$ Thus, since the roots are co-prime as elements of the polynomial ring $S\left({𝔥}_{ℂ}^{*}\right),$ $$\text{det}\left(X\right)\text{is divisible by}{\left(\prod _{\alpha >0}\alpha \right)}^{\left|W\right|/2}\text{.}$$ The degree of ${\prod }_{\alpha >0}{\alpha }^{\left|W\right|/2}$ is $(\left|W\right|/2)\text{Card}\left(R^{+}\right)$ and, using (2.3), the degree of $\text{det}\left(X\right)$ is $$\begin{array}{ccc}\text{deg}\left(\prod _{w\in W}{b}_w\right)& =& \sum _{k}k\hspace{0.17em}\text{dim}\left({\mathscr{H}}^k\right)=\left(\frac{d}{dt}{P}_W\left(t\right)\right){|}_{t=1}=\sum _{w\in W}\ell \left(w\right)\\ & =& \sum _{w\in W}\text{Card}\left(R\left(w\right)\right)=\sum _{\alpha \in R^{+}}(\left|W\right|/2)=(\left|W\right|/2)\text{Card}\left(R^{+}\right)\text{.}\end{array}$$ Since these two polynomials are homogeneous of the same degree, it follows that the quotient $\text{det}\left(X\right)/{\left(\prod _{\alpha \in 0}\alpha \right)}^{\left|W\right|/2}$ 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 $\mathscr{H}\text{.}$ 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 $\left\{b_w\hspace{0.17em}\right|\hspace{0.17em}w\in W\}$ of the space of harmonic polynomials $\mathscr{H}$ can be constructed by setting

$$b_w={\Delta }_w^{*}\left(1\right),\text{where}{\Delta }_w^{*}={\Delta }_{i_1}^{*}\dots {\Delta }_{i_{\ell }}^{*}\hspace{0.17em}\text{for a reduced word}\hspace{0.17em}w=s_{i_1}\dots s_{i_{\ell }}\text{.}$$

Weights and calibrated representations

The group $W$ acts on

$${𝔥}_{ℂ}=\text{Hom}({𝔥}_{ℂ}^{*},ℂ)\text{by}\left(w\gamma \right)\left(x\right)=\gamma \left(w^{-1}x\right),$$

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

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

$$\begin{array}{cc}R\left(w\right)=\{\alpha >0\hspace{0.17em}|\hspace{0.17em}w\alpha <0\}\text{.}& \text{(2.4)}\end{array}$$

The choice of the simple roots ${\alpha }_1,\dots ,{\alpha }_n\in {𝔥}_{ℝ}^{*}$ determines a fundamental chamber

$$\begin{array}{cc}C=\{\lambda \in {𝔥}_{ℝ}\hspace{0.17em}|\hspace{0.17em}⟨{\alpha }_i,\lambda ⟩>0,1\le i\le n\}& \text{(2.5)}\end{array}$$

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. $\{\lambda \in {𝔥}_{ℝ}\hspace{0.17em}|\hspace{0.17em}⟨\lambda ,\alpha ⟩>0\},$ and the negative side of $H_{\alpha }$ is the side away from $C\text{.}$ There is a bijection

$$\begin{array}{cc}\begin{array}{ccc}W& ⟷& \left\{\text{fundamental chambers for}\hspace{0.17em}W\hspace{0.17em}\text{acting on}\hspace{0.17em}{𝔥}_{ℝ}\right\}\\ w& ⟼& w^{-1}C\end{array}& \text{(2.6)}\end{array}$$

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

$$W_{\gamma }=⟨s_{\alpha }\hspace{0.17em}|\hspace{0.17em}\alpha \in Z\left(\gamma \right)⟩\text{where}Z\left(\gamma \right)=\left\{\alpha \hspace{0.17em}\right|\hspace{0.17em}\gamma \left(\alpha \right)=0\}\text{.}$$

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

$$\begin{array}{cc}\begin{array}{ccccc}W\gamma & ⟷& W/W_{\gamma }& ⟷& \{w\in W\hspace{0.17em}|\hspace{0.17em}R\left(w\right)\cap Z\left(\gamma \right)=\varnothing \}\\ & & & ⟷& \left\{\genfrac{}{}{0ex}{}{\text{chambers on the positive}}{\text{side of}\hspace{0.17em}{H}_{\alpha }\hspace{0.17em}\text{for}\hspace{0.17em}\alpha \in Z\left(\gamma \right)}\right\},\end{array}& \text{(2.7)}\end{array}$$

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 $\{w\in W\hspace{0.17em}|\hspace{0.17em}R\left(w\right)\cap Z\left(\gamma \right)=\varnothing \}$ 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

$$pm=\gamma \left(p\right)m,\text{for all}\hspace{0.17em}m\in M,p\in S{\left({𝔥}_{ℂ}^{*}\right)}^W\text{.}$$

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

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

$$\begin{array}{cccc}{M}_{\gamma }& =& \{m\in M\hspace{0.17em}|\hspace{0.17em}xm=\gamma \left(x\right)m\hspace{0.17em}\text{for all}\hspace{0.17em}x\in {𝔥}_{ℂ}^{*}\},& \text{(2.8)}\\ M_{\gamma }^{\text{gen}}& =& \{m\in M\hspace{0.17em}|\hspace{0.17em}\text{for all}\hspace{0.17em}x\in {𝔥}_{ℂ}^{*},{(x-\gamma \left(x\right))}^{k}m=0\hspace{0.17em}\text{for some}\hspace{0.17em}k\in {\mathbb{Z}}_{>0}\}\text{.}& \text{(2.9)}\end{array}$$

Then

$$M=\underset{\gamma \in {𝔥}_{ℂ}}{⨁}{M}_{\gamma }^{\text{gen}},$$

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 $ℍ\text{-module}$

$$\begin{array}{cc}M\hspace{0.17em}\text{is}\hspace{0.17em}\text{calibrated}\hspace{0.17em}\text{if}\hspace{0.17em}{M}_{\gamma }^{\text{gen}}=M_{\gamma }\text{for all}\hspace{0.17em}\gamma \in {𝔥}_{ℂ}\text{.}& \text{(2.10)}\end{array}$$

Tempered representations and the Langlands classification.

A weight $\lambda \in {\text{Hom}}_{ℂ}({𝔥}_{ℂ}^{*},ℂ)$ 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 ${𝔥}_{ℝ}={\text{Hom}}_{ℝ}({𝔥}_{ℝ}^{*},ℝ)$ 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

$$\lambda =\text{Re}\left(\lambda \right)+i\text{Im}\left(\lambda \right)\text{.}$$

For any simple reflection $s_j,$ we have

$$s_j\lambda =\text{Re}\left(\lambda \right)-\text{Re}\left(⟨\lambda ,{\alpha }_j⟩\right){\alpha }_j^{\vee }+i\text{Im}\left(\lambda \right)-i\text{Im}\left(⟨\lambda ,{\alpha }_j⟩\right){\alpha }_j^{\vee }=s_j\text{Re}\left(\lambda \right)+i{s}_j\text{Im}\left(\lambda \right),$$

and so

$$R\left(w\lambda \right)=w\text{Re}\left(\lambda \right),\text{for all}\hspace{0.17em}w\in W\text{.}$$

Let ${\omega }_i^{\vee }$ be the dual basis to ${\alpha }_i^{\vee }$ in ${𝔥}_{ℝ}$ and let $\bar{C}$ 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 $\bar{C}$ 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 ${\lambda }_0\in \bar{C}$ and the ${\omega }_i^{\vee }$ are on the boundary of $\bar{C},$ there is a uniquely determined set $I$ such that

$${\lambda }_0=\sum _{j\notin I}{c}_j{\omega }_j^{\vee }\text{with}\hspace{0.17em}{c}_j>0,$$

and we say that the weight $\lambda$ is I-tempered. For each $I$ the set $\{{\omega }_j^{\vee },{\alpha }_i^{\vee }\hspace{0.17em}|\hspace{0.17em}j\notin I,i\in I\}$ is a basis of ${𝔥}_{ℝ}$ and ${\lambda }_0$ and $I$ can, alternatively, be determined by the unique expansion

$$\begin{array}{cc}\text{Re}\left(\lambda \right)=\sum _{j\notin I}{c}_j{\omega }_j^{\vee }+\sum _{i\in I}{d}_i{\alpha }_i^{\vee }\text{with}\hspace{0.17em}{c}_j>0\hspace{0.17em}\text{and}\hspace{0.17em}{d}_i\le 0\text{.}& \text{(2.11)}\end{array}$$

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 $I\subseteq \{1,\dots ,n\},$ 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 $I\subseteq \{1,2,\dots ,n\}$ 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 $\{1,2,\dots ,n\}$ 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 $$\begin{array}{cc}{\lambda }_0\hspace{0.17em}\text{is a maximal element of}\left\{{\mu }_0\hspace{0.17em}\right|\hspace{0.17em}\mu \hspace{0.17em}\text{is a weight of}\hspace{0.17em}L\}& \text{(2.12)}\end{array}$$ with respect to the dominance ordering on ${𝔥}_{ℝ}\text{.}$ Let $I\subseteq \{1,2,\dots ,n\}$ be determined by $${\lambda }_0=\sum _{j\notin I}{c}_j{\omega }_j^{\vee }$$ 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 $$\text{Re}\left(w\lambda \right)=\sum _{j\notin I}{c}_j{\omega }_j^{\vee }+\sum _{{\alpha }_i\le 0,i\in I}{a}_i{\alpha }_i^{\vee }+\sum _{{\alpha }_i>0,i\in I}{a}_i{\alpha }_i^{\vee }\ge \sum _{j\notin I}{c}_j{\omega }_j^{\vee }+\sum _{{\alpha }_i\le 0,i\in I}{a}_i{\alpha }_i^{\vee },$$ since $\text{Re}\left(\lambda \right)$ is as in (2.11). So, by Proposition 2.3, $${\left(w\lambda \right)}_0\ge {(\sum _{j\notin I}{c}_j{\omega }_j^{\vee }+\sum _{{\alpha }_i\le 0}{a}_i{\alpha }_i^{\vee })}_0=\sum _{j\notin I}{c}_j{\omega }_j^{\vee }={\lambda }_0\text{.}$$ 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 $$\text{Re}\left(w_2\mu \right)=\sum _{j\notin I}{c}_j{\omega }_j^{\vee }+\sum _{i\in I}{a}_i{\alpha }_i^{\vee }\text{with}{c}_j>0,a_i\le 0\text{.}$$ Recall that $W^I=\{w_1\in W\hspace{0.17em}|\hspace{0.17em}R\left(w_1\right)\cap {\left\{{\alpha }_i\right\}}_{i\in I}=\varnothing \}\text{.}$ Thus, for each $i\in I,$ $w_1{\alpha }_i^{\vee }$ is a positive co-root and $$\text{Re}\left(w_1{w}_2\mu \right)=w_1{\left(w_2\mu \right)}_0+\sum _{i\in I}{a}_i{w}_1{\alpha }_i^{\vee }\le w_1{\left(w_2\mu \right)}_0\text{.}$$ 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 $$\text{Re}\left(w_1{w}_2\mu \right)\le w_1{\left(w_2\mu \right)}_0<{\left(w_2\mu \right)}_0$$ and thus, by Proposition 2.3, $$\begin{array}{cc}{\left(w_1{w}_2\mu \right)}_0<{\left(w_2\mu \right)}_0\text{when}{w}_1\ne 1\text{.}& \text{(2.13)}\end{array}$$ 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 $$M_{\text{max}}=\left(\genfrac{}{}{0ex}{}{\text{sum of all}\hspace{0.17em}ℍ\text{-submodules}\hspace{0.17em}N\hspace{0.17em}\text{of}\hspace{0.17em}H{\otimes }_{{ℍ}_I}U}{\text{such that}\hspace{0.17em}{N}_{\nu }=0}\right),$$ 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 $$\begin{array}{cc}\begin{array}{cccc}{\varphi }_U:& H{\otimes }_{{ℍ}_I}U& ⟶& L\\ & u& ⟼& u& \text{for}\hspace{0.17em}u\in U\text{.}\end{array}& \text{(2.14)}\end{array}$$ 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 $(U,I)$ 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

$$\begin{array}{cccc}{\tau }_i:& M_{\gamma }^{\text{gen}}& ⟶& M_{s_i\gamma }^{\text{gen}}\\ & m& ⟼& (t_{s_i}-\frac{c_{{\alpha }_i}}{{\alpha }_i})m\text{.}\end{array}$$

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 $M_{\gamma }^{\text{gen}},$ ${\tau }_i{\tau }_i=\frac{(c_{{\alpha }_i}+{\alpha }_i)(c_{{\alpha }_i}-{\alpha }_i)}{\left({\alpha }_i\right)(-{\alpha }_i)}\text{.}$
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 $\gamma \left({\alpha }_i\right)\ne \pm c_{{\alpha }_i}\text{.}$
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 $$\underset{\underset{m_{ij}\text{factors}}{⏟}}{{\tau }_i{\tau }_j{\tau }_i\dots }=\underset{\underset{m_{ij}\text{factors}}{⏟}}{{\tau }_j{\tau }_i{\tau }_j\dots },$$ 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 $$\begin{array}{ccccc}x{\tau }_{i}m& =& x(t_{s_i}-\frac{c_{{\alpha }_i}}{{\alpha }_i})m& =& (t_{s_i}{s}_i\left(x\right)+c_{{\alpha }_i}⟨x,{\alpha }_i^{\vee }⟩-c_{{\alpha }_i}\frac{x}{{\alpha }_i})m\\ & =& (t_{s_i}{s}_i\left(x\right)-c_{{\alpha }_i}\frac{x-⟨x,{\alpha }_i^{\vee }⟩{\alpha }_i}{{\alpha }_i})m& =& (t_{s_i}{s}_i\left(x\right)-c_{{\alpha }_i}\frac{s_i\left(x\right)}{{\alpha }_i})m\\ & =& (t_{s_i}-\frac{c_{{\alpha }_i}}{{\alpha }_i})s_i\left(x\right)m& =& {\tau }_i{s}_i\left(x\right)m\text{.}\end{array}$$ This proves (a) and (b). $$\begin{array}{ccc}{\tau }_i{\tau }_{i}m& =& (t_{s_i}^2-\frac{c_{{\alpha }_i}}{{\alpha }_i}{t}_{s_i}-t_{s_i}\frac{c_{{\alpha }_i}}{{\alpha }_i}+\frac{c_{{\alpha }_i}^2}{{\alpha }_i^2})m\\ & =& (1-\frac{c_{{\alpha }_i}}{{\alpha }_i}{t}_{s_i}-\frac{c_{{\alpha }_i}}{-{\alpha }_i}{t}_{s_i}-c_{{\alpha }_i}\frac{(\frac{c_{{\alpha }_i}}{{\alpha }_i}-\frac{c_{{\alpha }_i}}{-{\alpha }_i})}{{\alpha }_i}+\frac{c_{{\alpha }_i}^2}{{\alpha }_i^2})m\\ & =& (1+\frac{c_{{\alpha }_i}^2}{\left({\alpha }_i\right)(-{\alpha }_i)})m=\left(\frac{(c_{{\alpha }_i}+{\alpha }_i)(c_{{\alpha }_i}-{\alpha }_i)}{\left({\alpha }_i\right)(-{\alpha }_i)}\right)m,\end{array}$$ proving (c). (d) Since ${\alpha }_i$ acts on $M_{\gamma }^{\text{gen}}$ by $\gamma \left({\alpha }_i\right)$ times a unipotent transformation, $\text{det}\left((c_{{\alpha }_i}+{\alpha }_i)(c_{{\alpha }_i}-{\alpha }_i)\right)=0$ if and only if $\gamma \left({\alpha }_i\right)=\pm c_{{\alpha }_i}\text{.}$ Thus ${\tau }_i{\tau }_i,$ and each factor in this composition, is invertible if and only if $\gamma \left({\alpha }_i\right)\ne \pm c_{{\alpha }_i}\text{.}$ (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 $${\tau }_w=t_w+\sum _{z<w}{R}_z{t}_z,$$ 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 $\left\{b_w\hspace{0.17em}\right|\hspace{0.17em}w\in W\}$ is a basis of the space of harmonic polynomials $\mathscr{H}$ consisting of homogeneous polynomials. Then, for each $w\in W,$ $$0=h\left(b_w\right)=\sum _{z\in W}{P}_z{t}_z{b}_w\left(1\right)=(\sum _{z\in W}{P}_z\left(z^{-1}{b}_w\right)t_z+\text{lower degree terms})\left(1\right)=\sum _{z\in W}{P}_z\left(z^{-1}{b}_w\right)+\text{lower degree terms.}$$ 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 $\xi ·{\left({\prod }_{\alpha >0}\alpha \right)}^{\left|W\right|/2}{P}_z=0,$ 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 ${\text{Ind}}_{ℂW}^{ℍ}\left(1\right)\cong S\left({𝔥}_{ℂ}^{*}\right),$ $${\tau }_i{\alpha }_i={\stackrel{\sim }{\tau }}_i,{\stackrel{\sim }{\tau }}_i\left(1\right)=(-{\alpha }_i{t}_{s_i}+c_{{\alpha }_i})\left(1\right)=(-{\alpha }_i+c_{{\alpha }_i}),\text{and}{\stackrel{\sim }{\tau }}_{i}p=\left(s_{i}p\right){\stackrel{\sim }{\tau }}_i,$$ 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 $w=s_{i_1}\dots s_{i_{\ell }},$ $R\left(w\right)=\{{\alpha }_{i_{\ell }},s_{i_{\ell }}{\alpha }_{i_{\ell -1}},\dots ,s_{i_{\ell }}\dots s_{i_2}{\alpha }_{i_1}\},$ $$\begin{array}{ccc}\left({\tau }_{i_1}\dots {\tau }_{i_{\ell }}\right)\left(\prod _{\alpha \in R\left(w\right)}\alpha \right)\left(p\right)& =& \left({\stackrel{\sim }{t}}_{i_1}\dots {\stackrel{\sim }{t}}_{i_{\ell }}\right)p\left(1\right)=\left(wp\right)\left({\stackrel{\sim }{t}}_{i_1}\dots {\stackrel{\sim }{t}}_{i_{\ell }}\right)\left(1\right)\\ & =& \left(wp\right)\left(\prod _{\alpha \in R\left(w\right)}(-\alpha +c_{\alpha })\right)\text{.}\end{array}$$ 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

$$\begin{array}{cc}Z\left(\gamma \right)=\{\alpha >0\hspace{0.17em}|\hspace{0.17em}\gamma \left(\alpha \right)=0\}\text{and}P\left(\gamma \right)=\{\alpha >0\hspace{0.17em}|\hspace{0.17em}\gamma \left(\alpha \right)=\pm c_{\alpha }\}\text{.}& \text{(2.15)}\end{array}$$

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

$$\begin{array}{cc}{ℱ}^{(\gamma ,J)}=\{w\in W\hspace{0.17em}|\hspace{0.17em}R\left(w\right)\cap Z\left(\gamma \right)=\varnothing \hspace{0.17em}\text{and}\hspace{0.17em}R\left(w\right)\cap P\left(\gamma \right)=J\}\text{.}& \text{(2.16)}\end{array}$$

A local region is a pair $(\gamma ,J)$ such that $\gamma \in {𝔥}_{ℂ},$ $J\subseteq P\left(\gamma \right),$ and $ℱ(\gamma ,J)\ne \varnothing \text{.}$ Under the bijection (2.6) the set ${ℱ}^{(\gamma ,J)}$ 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 $\alpha \in P\left(\gamma \right)\backslash J,$ and
3. on the negative side of the hyperplanes $H_{\alpha }$ for $\alpha \in J\text{.}$

In this way the local region $(\gamma ,J)$ 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 $w\in {ℱ}^{(\gamma ,J)}$ index the chambers $w^{-1}C$ in the local region and the sets ${ℱ}^{(\gamma ,J)}$ form a partition of the set $\{w\in W\hspace{0.17em}|\hspace{0.17em}R\left(w\right)\cap Z\left(\gamma \right)=\varnothing \}$ (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

$$\text{dim}\left(M_{w\gamma }^{\text{gen}}\right)=\text{dim}\left(M_{w\prime \gamma }^{\text{gen}}\right)\text{for}\hspace{0.17em}w,w\prime \in {ℱ}^{(\gamma ,J)},$$

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

		Proof.
	Suppose $w,s_{i}w\in {ℱ}^{(\gamma ,J)}\text{.}$ 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 $\gamma \left(\alpha \right)\ne \pm c_{{\alpha }_i}\text{.}$ 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 $w\gamma \left({\alpha }_i\right)\ne \pm c_{{\alpha }_i}$ 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 $w,w\prime \in {ℱ}^{(\gamma ,J)},$ then $w\prime =s_{i_1}\dots s_{i_{\ell }}w$ where $s_{i_k}\dots s_{i_{\ell }}w\in {ℱ}^{(\gamma ,J)}$ for all $1\le k\le \ell \text{.}$ This follows from the fact that $(\gamma ,J)$ 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 $w\in {ℱ}^{(\gamma ,\varnothing )}\text{.}$ Then

1. $\text{dim}\hspace{0.17em}{M}_{w\gamma }^{\text{gen}}\ge 2,$
2. if $M_{s_{j}w\gamma }^{\text{gen}}=0,$ then $\left(w\gamma \right)\left({\alpha }_j\right)=\pm c_{{\alpha }_j}$ and $⟨w^{-1}{\alpha }_j,{\alpha }_i^{\vee }⟩=0\text{.}$

		Proof.
	Let $ℍA_1$ be the subalgebra of $ℍ$ generated by $t_{s_i}$ and all $x\in S\left({𝔥}_{ℂ}^{*}\right)\text{.}$ Let $ℂv_{\gamma }$ be the one-dimensional representation of $S\left({𝔥}_{ℂ}^{*}\right)$ defined by $x{v}_{\gamma }=\gamma \left(x\right)v_{\gamma }$ and let $M\left(\gamma \right)={\text{Ind}}_{S\left({𝔥}_{ℂ}^{*}\right)}^{ℍA_1}\left(ℂv_{\gamma }\right)=ℍA_1{\otimes }_{S\left({𝔥}_{ℂ}^{*}\right)}ℂv_{\gamma }\text{.}$ This module is irreducible and has basis $\{v_{\gamma },t_{s_i}{v}_{\gamma }\}$ and, with respect to this basis, the action of $x\in {𝔥}_{ℂ}^{*}$ on $M\left(\gamma \right)$ is given by the matrix $$\begin{array}{cc}{\rho }_{\gamma }\left(x\right)=\left(\begin{array}{cc}\gamma \left(x\right)& c_{{\alpha }_i}⟨x,{\alpha }_i^{\vee }⟩\\ 0& \gamma \left(x\right)\end{array}\right)\text{.}& \text{(2.17)}\end{array}$$ Let $n_{\gamma }$ be a nonzero vector in $M_{\gamma }\text{.}$ As an $S\left({𝔥}_{ℂ}^{*}\right)\text{-module}$ $ℂn_{\gamma }\cong ℂv_{\gamma }$ and, since induction is the adjoint functor to restriction, there is a unique $ℍA_1\text{-module}$ homomorphism given by $$\begin{array}{ccc}M\left(\gamma \right)& ⟶& M\\ v_{\gamma }& ⟼& n_{\gamma }\end{array}$$ 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 $w\in {ℱ}^{(\gamma ,\varnothing )}$ guarantee that the map $${\tau }_w={\tau }_{i_1}\dots {\tau }_{i_{\ell }}:M_{\gamma }^{\text{gen}}\to M_{w\gamma }^{\text{gen}}$$ 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 $ℍA_1$ action of $x\in X$ on this subspace is given by $${\rho }_{w\gamma }\left(x\right)=\left(\begin{array}{cc}\gamma \left(w^{-1}x\right)& c_{{\alpha }_i}⟨w^{-1}x,{\alpha }_i^{\vee }⟩\\ 0& \gamma \left(w^{-1}x\right)\end{array}\right)\text{.}$$ 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 $${\rho }_{w\gamma }\left(\frac{1}{{\alpha }_j}\right)=\frac{1}{\gamma {\left(w^{-1}{\alpha }_j\right)}^2}\left(\begin{array}{cc}\gamma \left(w^{-1}{\alpha }_j\right)& -c_{{\alpha }_i}⟨w^{-1}{\alpha }_j,{\alpha }_i^{\vee }⟩\\ 0& \gamma \left(w^{-1}{\alpha }_j\right)\end{array}\right)\text{.}$$ 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 $${\rho }_{w\gamma }\left(t_{s_j}\right)={\rho }_{w\gamma }\left(\frac{c_{{\alpha }_j}}{{\alpha }_j}\right)=\frac{c_{{\alpha }_j}}{\gamma {\left(w^{-1}{\alpha }_j\right)}^2}\left(\begin{array}{cc}\gamma \left(w^{-1}{\alpha }_j\right)& -c_{{\alpha }_j}⟨w^{-1}{\alpha }_j,{\alpha }_i^{\vee }⟩\\ 0& \gamma \left(w^{-1}{\alpha }_j\right)\end{array}\right)\text{.}$$ Since $t_{s_j}^2-1=(t_{s_j}-1)(t_{s_j}+1)=0,{\rho }_{w\gamma }\left(t_{s_j}\right)$ must have Jordan blocks of size 1 and eigenvalues $\pm 1\text{.}$ Since $c_{{\alpha }_i}\ne 0,$ it follows that $\gamma \left(w^{-1}{\alpha }_j\right)=\pm c_{{\alpha }_j}$ and $⟨w^{-1}{\alpha }_j,{\alpha }_i^{\vee }⟩=0\text{.}$ $\square$

Principal series modules

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

$$x{v}_{\gamma }=\gamma \left(x\right)v_{\gamma }\text{for}\hspace{0.17em}x\in {𝔥}_{ℂ}^{*}\text{.}$$

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

$$\begin{array}{cc}M\left(\gamma \right)=ℍ{\otimes }_{S\left({𝔥}_{ℂ}^{*}\right)}ℂv_{\gamma }={\text{Ind}}_{S\left({𝔥}_{ℂ}^{*}\right)}^{ℍ}\left(ℂv_{\gamma }\right)\text{.}& \text{(2.18)}\end{array}$$

The module $M\left(\gamma \right)$ has basis $\{t_w\otimes v_{\gamma }\hspace{0.17em}|\hspace{0.17em}w\in W\}$ with $ℂW$ acting by left multiplication. By the defining relations for $ℍ,$ for $x\in {𝔥}_{ℂ}^{*},$ $w\in W,$

$$x{t}_w{v}_{\gamma }=\left(w\gamma \right)\left(x\right)t_w\otimes v_{\gamma }+\sum _{z<w}{c}_{zw}\left(x\right)t_z\otimes v_{\gamma }\text{with}\hspace{0.17em}{c}_{zw}\left(x\right)\in ℂ\text{.}$$

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

$$M\left(\gamma \right)=\underset{w\in W}{⨁}m{\left(\gamma \right)}_{w\gamma }\text{with}\hspace{0.17em}\text{dim}\left(M{\left(\gamma \right)}_{w\gamma }\right)=1\text{.}$$

Thus, if $\gamma \in {𝔥}_{ℂ}$ is regular, there is a unique basis $\left\{v_{w\gamma }\hspace{0.17em}\right|\hspace{0.17em}w\in W\}$ of $M\left(\gamma \right)$ determined by

$$\begin{array}{cc}x{v}_{w\gamma }=\left(w\gamma \right)\left(x\right)v_{w\gamma }\text{for all}\hspace{0.17em}w\in W\hspace{0.17em}\text{and}\hspace{0.17em}x\in {𝔥}_{ℂ}^{*},& \text{(2.19)}\\ v_{w\gamma }=t_w\otimes v_{\gamma }+\sum _{u<w}{a}_{wu}\left(\gamma \right)(t_u\otimes v_{\gamma })\text{where}\hspace{0.17em}{a}_{wu}\left(\gamma \right)\in ℂ\text{.}& \text{(2.20)}\end{array}$$

Alternatively,

$$\begin{array}{cc}{v}_{w\gamma }={\tau }_w{v}_{\gamma }& \text{(2.21)}\end{array}$$

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 $\left|W\right|\text{.}$ 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 $S\left({𝔥}_{ℂ}^{*}\right)\text{-module,}$ $ℂm_{\gamma }\cong ℂv_{\gamma }\text{.}$ Since induction is the adjoint functor to restriction there is a unique $ℍ\text{-module}$ homomorphism given by $$\begin{array}{ccc}M\left(\gamma \right)& ⟶& M\\ v_{\gamma }& ⟼& m_{\gamma }\end{array}$$ 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), $$M\left(\gamma \right)=\underset{w\in W}{⨁}M{\left(\gamma \right)}_{w\gamma }\text{and}\text{dim}\left(M{\left(\gamma \right)}_{w\gamma }\right)=1$$ 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 $\left(s_i\gamma \right)(c_{{\alpha }_i}/{\alpha }_i)v_{s_i\gamma }$ are linearly independent in $M\left(s_i\gamma \right)\text{).}$ Thus, there is an $ℍ\text{-module}$ homomorphism $$\begin{array}{cccc}A(s_i,\gamma ):& M\left(\gamma \right)& ⟶& M\left(s_i\gamma \right)\\ & h{v}_{\gamma }& ⟼& h{\tau }_i{v}_{s_i\gamma },& h\in ℍ\text{.}\end{array}$$ The modules $M\left(\gamma \right)$ and $M\left(s_i\gamma \right)$ have bases $$\begin{array}{cc}\begin{array}{c}{\{t_w(t_{s_i}+1)v_{\gamma },t_w(t_{s_i}-1)v_{\gamma }\}}_{s_{i}w>w},\\ {\{t_w(t_{s_i}+1)v_{s_i\gamma },t_w(t_{s_i}-1)v_{s_i\gamma }\}}_{s_{i}w>w},\end{array}& \text{(2.22)}\end{array}$$ respectively. Since $(t_{s_i}+1)t_{s_i}=t_{s_i}+1$ and $(t_{s_i}-1)t_{s_i}=-(t_{s_i}-1),$ $$\begin{array}{ccc}A(s_i,\gamma )\left(t_w(t_{s_i}+1)v_{\gamma }\right)& =& t_w(t_{s_i}+1)(t_{s_i}-\frac{c_{{\alpha }_i}}{{\alpha }_i})v_{s_i\gamma }\\ & =& t_w(t_{s_i}+1)(1-\frac{c_{{\alpha }_i}}{{\alpha }_i})v_{s_i\gamma }\\ & =& \left(s_i\gamma \left(\frac{{\alpha }_i-c_{{\alpha }_i}}{{\alpha }_i}\right)\right)t_w(t_{s_i}+1)v_{s_i\gamma }\\ A(s_i,\gamma )\left(t_w(t_{s_i}-1)v_{\gamma }\right)& =& t_w(t_{s_i}-1)(t_{s_i}-\frac{c_{{\alpha }_i}}{{\alpha }_i})v_{s_i\gamma }\\ & =& t_w(t_{s_i}-1)(-1-\frac{c_{{\alpha }_i}}{{\alpha }_i})v_{s_i\gamma }\\ & =& \left(s_i\gamma \left(\frac{{\alpha }_i+c_{{\alpha }_i}}{-{\alpha }_i}\right)\right)t_w(t_{s_i}-1)v_{s_i\gamma }\end{array}$$ and so the matrix of $A(s_i,\gamma )$ with respect to the bases in (2.22) is diagonal with $\left|W\right|/2$ diagonal entries equal to $\left(s_i\gamma \right)(({\alpha }_i-c_{{\alpha }_i})/{\alpha }_i)$ and $\left|W\right|/2$ diagonal entries equal to $\left(s_i\gamma \right)(({\alpha }_i+c_{{\alpha }_i})/(-{\alpha }_i))\text{.}$ If $\gamma \left({\alpha }_i\right)\ne \pm c_{{\alpha }_i},$ then $A(s_i,\gamma )$ is an isomorphism and so $M\left(\gamma \right)$ and $M\left(s_i\gamma \right)$ have the same composition factors. If $\gamma \left({\alpha }_i\right)=\pm c_{{\alpha }_i},$ then $\text{dim}\left(\text{ker}\hspace{0.17em}A(s_i,\gamma )\right)=\left|W\right|/2\text{.}$ In this case $A(s_i,s_i\gamma )A(s_i,\gamma )=0$ and so the sequence $$\begin{array}{ccccc}M\left(\gamma \right)& \stackrel{A(s_i,\gamma )}{⟶}& M\left(s_i\gamma \right)& \stackrel{A(s_i,s_i\gamma )}{⟶}& M\left(\gamma \right)\end{array}$$ is exact. Then $$M\left(s_i\gamma \right)\supseteq \text{ker}\left(A(s_i,s_i\gamma )\right)\supseteq 0$$ is a filtration of $M\left(s_i\gamma \right)$ where the first factor is isomorphic to a submodule of $M\left(\gamma \right),$ $$M\left(s_i\gamma \right)/\text{ker}\left(A(s_i,s_i\gamma )\right)\cong \text{im}\left(A(s_i,s_i\gamma )\right)\subseteq M\left(\gamma \right),$$ and the second factor is isomorphic to a quotient of $M\left(\gamma \right),$ $$\text{ker}\left(A(s_i,s_i\gamma )\right)\cong M\left(\gamma \right)/\text{ker}\left(A(s_i,\gamma )\right)\text{.}$$ Since $\text{dim}\left(\text{ker}\left(A(s_i,s_i\gamma )\right)\right)+\text{dim}\left(\text{im}\left(A(s_i,s_i\gamma )\right)\right)=\left|W\right|/2+\left|W\right|/2=\text{dim}\left(M\left(s_i\gamma \right)\right)=\text{dim}\left(M\left(\gamma \right)\right),$ 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 $M\left(\gamma \right)=ℍ{\otimes }_{S\left({𝔥}_{ℂ}^{*}\right)}ℂv_{\gamma }$ be the corresponding principal series module for $ℍ\text{.}$ The spherical vector in $M\left(\gamma \right)$ is

$$\begin{array}{cc}{1}_{\gamma }=\sum _{w\in W}{t}_w{v}_{\gamma }\text{.}& \text{(2.23)}\end{array}$$

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 $$1_{\gamma }=\sum _{z\in W}\gamma \left(c_z\right)v_{z\gamma }\text{where}{c}_z=\prod _{\alpha \in R\left(w_{0}z\right)}\frac{\alpha +c_{\alpha }}{\alpha }\text{.}$$
2. The spherical vector $1_{\gamma }$ generates $M\left(\gamma \right)$ if and only if ${\prod }_{\alpha >0}(\gamma \left(\alpha \right)+c_{\alpha })\ne 0\text{.}$
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 $$1_{\gamma }=\left(\sum _{w\in W}{t}_w\right)v_{\gamma }=\sum _{z\in W}{\xi }_z{v}_{z\gamma }\text{.}$$ We shall prove that the $\xi$ are given by the formula in the statement of the proposition. Since $t_{s_i}\left({\sum }_{w\in W}{t}_w\right)={\sum }_{w\in W}{t}_w,$ $$\begin{array}{ccc}{1}_{\gamma }& =& t_{s_i}{1}_{\gamma }=({\tau }_i+\frac{c_{{\alpha }_i}}{{\alpha }_i})\sum _{z\in W}{\xi }_z{v}_{z\gamma }=({\tau }_i+\frac{c_{{\alpha }_i}}{{\alpha }_i})\sum _{s_{i}z>z}({\xi }_z{v}_{z\gamma }+{\xi }_{s_{i}z}{v}_{s_{i}z\gamma })\\ & =& \sum _{s_{i}z>z}({\xi }_z{v}_{s_{i}z\gamma }+{\xi }_z\frac{c_{{\alpha }_i}}{\gamma \left(z^{-1}{\alpha }_i\right)}{v}_{z\gamma }+{\xi }_{s_{i}z}{\tau }_i^2{v}_{z\gamma }+{\xi }_{s_{i}z}\frac{c_{{\alpha }_i}}{\gamma (-z^{-1}{\alpha }_i)}{v}_{s_{i}z\gamma })\text{.}\end{array}$$ Comparing coefficients of $v_{s_{i}z\gamma }$ on each side of this expression gives $${\xi }_{s_{i}z}={\xi }_z+{\xi }_{s_{i}z}\frac{c_{{\alpha }_i}}{\gamma (-z^{-1}{\alpha }_i)},$$ and so $$\frac{{\xi }_z}{{\xi }_{s_{i}z}}=\gamma \left(\frac{z^{-1}{\alpha }_i+c_{{\alpha }_i}}{z^{-1}{\alpha }_i}\right),\text{if}\hspace{0.17em}{s}_{i}z>z\text{.}$$ Using this formula inductively gives $$\begin{array}{ccc}{\xi }_w& =& {\xi }_{s_{i_1}\dots s_{i_p}}=\gamma \left(\frac{s_{i_p}\dots s_{i_2}{\alpha }_{i_1}}{s_{i_p}\dots s_{i_2}{\alpha }_{i_1}+c_{{\alpha }_i}}\right)\dots \gamma \left(\frac{{\alpha }_{i_p}}{{\alpha }_{i_p}+c_{{\alpha }_{i_p}}}\right){\xi }_1\\ & =& \gamma \left(\prod _{\alpha \in R\left(w\right)}\frac{\alpha }{\alpha +c_{\alpha }}\right){\xi }_1\text{.}\end{array}$$ 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 $${\xi }_1=\gamma \left(\prod _{\alpha >0}\frac{\alpha +c_{\alpha }}{\alpha }\right)$$ and $${\xi }_w=\gamma \left(\prod _{\alpha \in R\left(w\right)}\frac{\alpha }{\alpha +c_{\alpha }}\right)·{\xi }_1=\gamma \left(\prod _{\alpha \in R\left(w_{0}w\right)}\frac{\alpha +c_{\alpha }}{\alpha }\right)\text{.}$$ (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 $$v_{z\gamma }=\sum _{u\in W}\gamma \left(m_{uz}\right)t_u{v}_{\gamma },$$ 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 $$b_y\left(\sum _{w\in W}{t}_w\right)=\sum _{u\in W}{t}_u{q}_{uy},y\in W,$$ where these equations are equalities in $ℍ\text{.}$ Then, $$b_y{1}_{\gamma }=b_y\left({\sum }_{w\in W}{t}_w\right)=\sum _{u\in W}\gamma \left(q_{uy}\right)(t_u\otimes v_{\gamma }),$$ and part (a) implies that if $\gamma$ is generic, then $$\begin{array}{ccc}{b}_y{1}_{\gamma }& =& b_y\sum _{z\in W}\gamma \left(c_z\right)v_{z\gamma }=\sum _{z\in W}\gamma \left(c_z\left(z^{-1}{b}_y\right)\right)v_{z\gamma }\\ & =& \sum _{z,u\in W}\gamma \left(c_z\left(z^{-1}{b}_y\right)m_{uz}\right)(t_u\otimes v_{\gamma })\text{.}\end{array}$$ Since these two expressions are equal for all generic $\gamma \in {𝔥}_{ℂ},$ it follows that $$\begin{array}{cc}{q}_{uy}=\sum _{z\in W}{m}_{uz}·c_z·\left(z^{-1}{b}_y\right),u,y\in W,& \text{(2.24)}\end{array}$$ 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 $$t_w\otimes v_{\gamma }=\sum _{y\in W}{p}_{yw}{b}_y{1}_{\gamma }\text{for each}\hspace{0.17em}w\in W\text{.}$$ If these constants exist, then, for each $w\in W,$ $$t_w\otimes v_{\gamma }=\sum _{y\in W}{p}_{yw}{b}_y{1}_{\gamma }=\sum _{y,z,u\in W}\gamma \left(m_{uz}{c}_z\left(z^{-1}{b}_y\right)p_{yw}\right)t_u\otimes v_{\gamma },$$ where, by (2.24), there is no restriction that $\gamma$ be generic. If $$M={\left(m_{uz}\right)}_{u,z\in W},C=\text{diag}{\left(c_z\right)}_{z\in W},X={\left(z^{-1}{b}_y\right)}_{z,y\in W},P={\left(p_{yw}\right)}_{y,w\in W},$$ 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), $$\text{det}\left(X\right)=\xi ·\prod _{\alpha >0}{\alpha }^{\left|W\right|/2}$$ and $$\text{det}\left(C\right)=\prod _{z\in W}\prod _{\alpha \in R\left(w_{0}z\right)}\frac{\alpha +c_{\alpha }}{\alpha }={\left(\prod _{\alpha >0}\frac{\alpha +c_{\alpha }}{\alpha }\right)}^{\left|W\right|/2},$$ where $\xi \in ℂ$ is nonzero. Thus $P$ exists if and only if ${\prod }_{\alpha >0}(\gamma \left(\alpha \right)+c_{\alpha })\ne 0\text{.}$ (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 $P\left(\gamma \right)=\{\alpha >0\hspace{0.17em}|\hspace{0.17em}\gamma \left(\alpha \right)=\pm c_{\alpha }\}\text{.}$ The principal series $ℍ\text{-module}$

$$M\left(\gamma \right)\hspace{0.17em}\text{is irreducible if and only if}\hspace{0.17em}P\left(\gamma \right)=\varnothing \text{.}$$

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.

page history