Classification of Irreducible Representations for Rank 2

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

Last update: 27 April 2013

Classification of Irreducible Representations for Rank 2

In this section we analyze the structure of all simple $ℍ\text{-modules}$ for rank 2 graded Hecke algebras $ℍ\text{.}$ The results, the classification of simple modules and various other data (central character $\gamma ,$ $P\left(\gamma \right),$ $Z\left(\gamma \right),$ dimension, calibrated or not calibrated, Langlands parameters), are listed in Table 1. An irreducible representation that is calibrated (see (2.10)) has all its weights of the form $w\gamma$ with $w\in {ℱ}^{(\gamma ,J)}$ for a unique $J,$ and this is the set that is displayed in the fourth column of Table 1. The notation ‘nc’ indicates that the representation is not calibrated. The Langlands parameters of a simple $ℍ\text{-module}$ of central character $\gamma$ consists of a pair $(U,I)$ where $I$ is a subset of $\{1,2\}$ and $U$ is a tempered representation of ${ℍ}_I$ (see Theorem 2.4). If $I$ is empty there is a unique tempered representation of ${ℍ}_I$ of central character $\gamma$ so we place the pair $(\gamma ,\varnothing )$ in the corresponding entry of column 5 of Table 1. If $I$ consists of one element, then ${ℍ}_I\cong ℍA_1$ and each ${ℍ}_I\text{-tempered}$ representation is naturally indexed by its maximal weight $\mu$ so we place $(\mu ,I)$ in column 5 of Table 1. If $I=\{1,2\},$ then the corresponding simple $ℍ\text{-module}$ is tempered.

The classification of the simple $ℍ\text{-modules}$ is accomplished in three steps:

1. The central character of a simple module is a $W\text{-orbit}$ in ${𝔥}_{ℂ},$ and we label the orbit by a representative element $\gamma \text{.}$ The structure of the simple modules with central character $\gamma$ is, in a large part, controlled by the sets $Z\left(\gamma \right)$ and $P\left(\gamma \right)$ and the first step is to classify the central characters $\gamma$ according to their sets $Z\left(\gamma \right)$ and $P\left(\gamma \right)\text{.}$ The resulting partition of the central characters is given in Table 1 and the derivation of this list presented in Section 3.2. The derivation is accomplished by considering, case by case, the possibilities (0, 1, or $\ge 2\text{)}$ for $\text{Card}\left(Z\left(\gamma \right)\right)\text{).}$
2. For each central character $\gamma$ we use the knowledge of $Z\left(\gamma \right)$ and $P\left(\gamma \right)$ and Lemma 2.7 and Corollary 2.6 to determine the simple modules of central character $\gamma$ and their weight space structure. This case by case analysis is in Section 3.3.
3. Finally, we determine the Langlands parameters for each simple $ℍ\text{-module.}$ Since the Langlands parameters depend on the weight space structure (in particular, the maximal weights, see Section 2.4) these are determined in conjunction with the derivation of the weight space structure of each simple module in Section 3.3.

The root system

The reflection group $I_2\left(n\right)$ is the dihedral group of order $2n\text{.}$ Let ${\epsilon }_1,$ ${\epsilon }_2$ be an orthonormal basis of ${𝔥}_{ℝ}^{*}={ℝ}^2$ and define

$${\beta }_k=\text{cos}\left(k\theta \right){\epsilon }_1+\text{sin}\left(k\theta \right){\epsilon }_2,\text{where}\hspace{0.17em}\theta =\pi /n\text{.}$$ $$\begin{array}{c} Hβ0 Hβ1 Hβ2 Hβ3 Hβ4 Hβ5 Hβ6 Hβ0 Hβ1 Hβ2 Hβ3 Hβ4 Hβ5 Hβ6 β0 β1 β2 β3 β4 β5 β6 β7 β8 β9 β10 β11 β12 β13 \\ Hβ0 Hβ1 Hβ2 Hβ3 Hβ4 Hβ5 Hβ6 Hβ7 Hβ0 Hβ1 Hβ2 Hβ3 Hβ4 Hβ5 Hβ6 Hβ7 β0 β1 β2 β3 β4 β5 β6 β7 β8 β9 β10 β11 β12 β13 β14 β15 \\ \text{Figure 1. Hyperplanes and roots for}\hspace{0.17em}{I}_2\left(7\right)\hspace{0.17em}\text{and}\hspace{0.17em}{I}_2\left(8\right)\end{array}$$

Fix the roots, positive roots and simple roots for the reflection group $I_2\left(n\right)$ by

$$\begin{array}{ccc}R& =& \left\{{\beta }_k\hspace{0.17em}\right|\hspace{0.17em}0\le k\le 2n-1\},\\ R^{+}& =& \left\{{\beta }_k\hspace{0.17em}\right|\hspace{0.17em}0\le k\le n-1\},\end{array}$$

and

$$\begin{array}{ccc}{\alpha }_1& =& {\beta }_0,\\ {\alpha }_2& =& {\beta }_{n-1}\text{.}\end{array}$$

For $0\le k\le n-1,$ $-{\beta }_k={\beta }_{n+k},$ $s_1{\beta }_k={\beta }_{n-k}$ and $s_2{\beta }_k={\beta }_{n-2-k}\text{.}$ If $n$ is even there are two orbits of roots, $\{\pm {\beta }_{2k}\hspace{0.17em}|\hspace{0.17em}0\le k<n/2\}$ and $\{\pm {\beta }_{2k+1}\hspace{0.17em}|\hspace{0.17em}0\le k<n/2\}\text{.}$ Let $c_k=c_{{\beta }_k}$ be a choice of parameters for the graded Hecke algebra $ℍ\text{.}$ When $n$ is odd all of the $c_k$ are equal and, when $n$ is even, there are two, possibly unequal, parameters $c_0=c_{2k}$ and $c_1=c_{2k+1}\text{.}$ Figure 1 displays the roots ${\beta }_k$ and hyperplanes $H_{{\beta }_k}=\{x\in {ℝ}^2\hspace{0.17em}|\hspace{0.17em}⟨{\beta }_k,x⟩=0\}$ for $I_2\left(7\right)$ and $I_2\left(8\right)\text{.}$ When $n$ is even each root ${\beta }_k$ lies on the hyperplane $H_{{\beta }_{k+n/2}}$ and this is why, in the picture of hyperplanes and roots for $I_2\left(8\right)$ there are multiple labels on each line.

Figure 2 displays, using thin and thick lines, the hyperplanes

$$H_{{\beta }_k}=\{x\in {ℝ}^2\hspace{0.17em}|\hspace{0.17em}⟨{\beta }_k,x⟩=0\}\text{and}{H}_{{\beta }_k\pm \delta }=\{x\in {ℝ}^2\hspace{0.17em}|\hspace{0.17em}⟨{\beta }_k,x⟩=\pm c_k\}$$

for $I_2\left(7\right)$ and $I_2\left(8\right)$ (and a particular choice of the parameters $c_k\text{).}$

The central characters

Using the orthonormal basis ${\epsilon }_1,{\epsilon }_2$ we can identify ${𝔥}_{ℝ}$ with ${ℝ}^2$ and ${𝔥}_{ℂ}$ with ${ℂ}^2\text{.}$ If $\gamma \in {𝔥}_{ℂ},$ then

$$Z\left(\gamma \right)=\{{\beta }_k\in R^{+}\hspace{0.17em}|\hspace{0.17em}⟨\gamma ,{\beta }_k⟩=0\}\text{and}P\left(\gamma \right)=\{{\beta }_k\in R^{+}\hspace{0.17em}|\hspace{0.17em}⟨\gamma ,{\beta }_k⟩=\pm c_k\}\text{.}$$

In terms of the pictures in Figure 2, if $\gamma$ is a point in ${ℝ}^2,$ then the elements of $Z\left(\gamma \right)$ label the $H_{{\beta }_k}$ (thin lines) that $\gamma$ is on and the elements of $P\left(\gamma \right)$ label the set of $H_{{\beta }_k\pm \delta }$ (thick lines) that $\gamma$ is on.

Let us analyze the possibilities for $Z\left(\gamma \right)$ and $P\left(\gamma \right)$ as $\gamma$ runs over representatives of $W\text{-orbits}$ in ${𝔥}_{ℂ}\text{.}$

If $\gamma \left(\alpha \right)=c_{\alpha },$ then $(1/c_{\alpha })\gamma \left(\alpha \right)=1$ and so we may, without loss of generality, assume that $c_k=1$ for all $k$ when $n$ is odd, and $c_{2k}=1$ and $c_{2k+1}=c$ when $n$ is even.

a.	If $Z\left(\gamma \right)$ contains 2 roots or more, then $\gamma =0,$ since any two distinct positive roots are linearly independent. This is the central character ${\gamma }_0$ in Table 1.
b.	If $Z\left(\gamma \right)$ contains one root, then, by choosing our representative $\gamma$ of the $W\text{-orbit}$ Wγ appropriately, we may arrange that $Z\left(\gamma \right)=\left\{{\beta }_0\right\}$ when $n$ is odd, and $Z\left(\gamma \right)=\left\{{\beta }_0\right\}$ or $Z\left(\gamma \right)=\left\{{\beta }_{n-1}\right\},$ when $n$ is even. When $n$ is even there is an automorphism $\tau$ of the root system (and of the graded Hecke algebra) which switches ${\alpha }_1={\beta }_0$ and ${\alpha }_2={\beta }_{n-1}\text{.}$ The automorphism $\tau$ extends linearly to ${𝔥}_{ℂ}$ and if $Z\left(\gamma \right)=\left\{{\beta }_{n-1}\right\},$ then $Z\left(\tau \left(\gamma \right)\right)=\left\{{\beta }_0\right\}$ and $\tau \left(P\left(\gamma \right)\right)=P\left(\tau \gamma \right)\text{.}$ Thus, even when $n$ is even, it will be sufficient to analyze the case $Z\left(\gamma \right)=\left\{{\beta }_0\right\}\text{.}$
b'.	If $Z\left(\gamma \right)=\left\{{\beta }_0\right\}$ and ${\beta }_k\in P\left(\gamma \right),$ then the equations $0=\gamma \left({\beta }_0\right)=\gamma \left({\epsilon }_1\right)$ and $$\begin{array}{cc}{c}_k=\gamma \left({\beta }_k\right)=\gamma (\text{cos}\left(k\theta \right){\epsilon }_1+\text{sin}\left(k\theta \right){\epsilon }_2)=\text{sin}\left(k\theta \right)\gamma \left({\epsilon }_2\right)& \text{(3.1)}\end{array}$$ uniquely determine $\gamma \text{.}$ Since $\text{sin}\left(k\theta \right)=\text{sin}\left((n-k)\theta \right),$ ${\beta }_{n-l}$ must also be in $P\left(\gamma \right)\text{.}$ This happens for the central characters ${\gamma }_{b,k},{\gamma }_{b,n/2}$ and ${\gamma }_q$ in Table 1.
b''.	If $Z\left(\gamma \right)=\left\{{\beta }_0\right\},$ ${\beta }_k,{\beta }_{\ell }\in P\left(\gamma \right)$ and $\ell \ne n-k,$ then equation (3.1) for $k$ and $\ell$ forces $c_k\ne c_{\ell }$ which forces $n$ even and $k$ and $\ell$ to be of different parity. Furthermore, the parameters must satisfy $c_k/c_{\ell }=\text{sin}\left(k\theta \right)/\text{sin}\left(\ell \theta \right)$ and, when this happens, it happens for a unique choice of the 4-tuple $(k,\ell ,n-k,n-\ell )\text{.}$ Thus, the only possible option is $P\left(\gamma \right)=\{{\beta }_k,{\beta }_{n-k},{\beta }_{\ell },{\beta }_{n-\ell }\}$ (if $\ell =n/2,$ then $P\left(\gamma \right)=\{{\beta }_{n/2},{\beta }_k,{\beta }_{n-k}\}\text{).}$ This is the central character ${\gamma }_q$ in Table 1.
c.	If $Z\left(\gamma \right)=\varnothing$ and ${\beta }_k,{\beta }_{\ell }\in P\left(\gamma \right)$ such that $c_k=c_{\ell }=c,$ then $\gamma$ is uniquely determined by the equations $$c=\text{cos}\left(k\theta \right)\gamma \left({\epsilon }_1\right)+\text{sin}\left(k\theta \right)\gamma \left({\epsilon }_2\right)=\text{cos}\left(\ell \theta \right)\gamma \left({\epsilon }_1\right)+\text{sin}\left(\ell \theta \right)\gamma \left({\epsilon }_2\right)\text{.}$$ These equations force ${\beta }_{(n+k+\ell )/2}\in Z\left(\gamma \right)$ if $(n+k+\ell )$ is even (the easiest way to see this is to look at the pictures in Figure 2). Since we assumed $Z\left(\gamma \right)=\varnothing ,$ it follows that $n+k+\ell$ is odd. If $P\left(\gamma \right)$ contains 3 elements, then at least two of them would satisfy $n+k+\ell$ even, and so it follows that $P\left(\gamma \right)$ contains a maximum of two elements. By appropriately choosing our representative $\gamma$ of the orbit $W_{\gamma }$ we can assume that $P\left(\gamma \right)=\{{\beta }_{k-1},{\beta }_{n-k}\}$ for some $1\le k\le n/2\text{.}$ This case corresponds to the central character ${\gamma }_{c,k}$ in Table 1.

This analysis shows that Table 1 covers all $(P\left(\gamma \right),Z\left(\gamma \right))$ possibilities.

$$\begin{array}{c} \\ \\ \text{Figure 2. Hyperplanes for}\hspace{0.17em}{I}_2\left(7\right)\hspace{0.17em}\text{and}\hspace{0.17em}{I}_2\left(n\right)\text{.}\end{array}$$

The irreducible representations

The following analysis determines the structure of each of the irreducible $ℍ\text{-modules:}$ the dimensions of each generalized weight space and the Langlands parameters. The derivation of the irreducible representations proceeds by considering, separately, each central character $\gamma \text{.}$ In each case we have included a picture showing the local regions $(\gamma ,J)\text{.}$ In these pictures the solid lines correspond to hyperplanes $H_{\alpha }$ for $\alpha \in Z\left(\gamma \right)$ and the dotted lines correspond to hyperplanes $H_{\alpha }$ for $\alpha \in P\left(\gamma \right)\text{.}$ Each local region is labeled by the corresponding set $J$ of roots which determines its location in the picture (see the discussion before Corollary 2.6).

The Langlands parameters of an irreducible $ℍ\text{-module}$ $M$ are determined by the real parts of weights of $M\text{.}$ This means that, according to the labeling of the simple modules as in Table 1, the Langlands parameters can depend on the choice of the parameters $c_k\text{.}$ In our calculations of Langlands parameters, and in the Langlands data displayed in Table 1, we assume that all $c_k\in {ℝ}_{>0}$ (this assumption is used only in the analysis of Langlands parameters).

In the case when $n$ is even not all roots are in the orbit of ${\alpha }_1={\beta }_0$ and one should really consider central characters $\gamma$ which have $Z\left(\gamma \right)=\left\{{\beta }_{n-1}\right\}=\left\{{\alpha }_2\right\}$ (see the remark in Section 3.2(b)). These central characters $\tau \left({\gamma }_a\right),$ $\tau \left({\gamma }_{b,k}\right),$ $\tau \left({\gamma }_{c,k}\right)$ are the images of the central characters ${\gamma }_a,$ ${\gamma }_{b,k}$ and ${\gamma }_{c,k}$ under the automorphism $\tau$ of the root system which switches ${\alpha }_1$ and ${\alpha }_2\text{.}$ This automorphism extends to an automorphism of $ℍ$ and thus it follows that the modules with central characters $\tau \left({\gamma }_a\right),$ $\tau \left({\gamma }_{b,k}\right),$ $\tau \left({\gamma }_{c,k}\right)$ have exactly the same structures as the modules with central characters ${\gamma }_a,$ ${\gamma }_b$ and ${\gamma }_{c,k},$ respectively.

Central character ${\gamma }_a\text{.}$ $Z\left({\gamma }_a\right)=\varnothing ,$ $P\left({\gamma }_a\right)=\varnothing \text{.}$

By Theorem 2.10 the principal series module $M\left({\gamma }_a\right)$ is irreducible and, by Proposition 2.8(a), this is the unique irreducible module with central character ${\gamma }_a\text{.}$ Since ${\gamma }_a$ is regular, $M\left({\gamma }_a\right)$ is calibrated.

Central character ${\gamma }_{b,k}\text{.}$ $Z\left({\gamma }_{b,k}\right)=\left\{{\beta }_0\right\},$ $P\left({\gamma }_{b,k}\right)=\{{\beta }_k,{\beta }_{n-k}\},$ $1\le k\le (n-1)/2\text{.}$

The weight ${\gamma }_{b,k}$ is uniquely determined by the fact that ${\gamma }_{b,k}\left({\beta }_0\right)=\gamma \left({\epsilon }_1\right)=0$ and $c_k=\gamma \left({\beta }_k\right)=\text{sin}\left(k\theta \right)\gamma \left({\epsilon }_2\right),$ where $\theta =\pi /n\text{.}$

$$Hβ0 Hβk Hβn-k k chambers k chambers k chambers k chambers n-2k chambers n-2k chambers J=∅ J={βn-k} J={βk,βn-k}$$

Suppose $M$ is an irreducible module with central character $\gamma$ and $M_{{\gamma }_{b,k}}^{\text{gen}}\ne 0\text{.}$ Then by Lemma 2.7(a), for all $w\in {ℱ}^{({\gamma }_{b,k},\varnothing )},$

$$2\le \text{dim}\hspace{0.17em}{M}_{w{\gamma }_{b,k}}^{\text{gen}}\le \text{dim}M{\left({\gamma }_{b,k}\right)}_{w{\gamma }_{b,k}}^{\text{gen}}=2,\text{and so dim}\hspace{0.17em}{M}_{w{\gamma }_{b,k}}^{\text{gen}}=2\text{.}$$

Now apply $\tau$ operators of the form $\dots {\tau }_1{\tau }_2$ to $M_{{\gamma }_{b,k}}^{\text{gen}}\text{.}$ If $w\in {ℱ}^{({\gamma }_{b,k},\varnothing )}$ but $s_{j}w\in {ℱ}^{({\gamma }_{b,k},\left\{{\beta }_{n-k}\right\})},$ then $⟨w^{-1}{\alpha }_j,{\alpha }_1^{\vee }⟩\ne 0$ and $\text{ker}\hspace{0.17em}{\tau }_j\ne 0\text{.}$ Therefore, by Lemma 2.7(b),

$$1\le \text{dim}\hspace{0.17em}{M}_{s_{j}w{\gamma }_{b,k}}^{\text{gen}}\le 1\text{.}$$

Thus, by Corollary 2.6,

$$\text{dim}\hspace{0.17em}{M}_{w{\gamma }_{b,k}}^{\text{gen}}=1\text{for all}\hspace{0.17em}w\in {ℱ}^{({\gamma }_{b,k},\left\{{\beta }_{n-k}\right\})}\text{.}$$

By applying more $\tau$ operators, if $w\in {ℱ}^{({\gamma }_{b,k},\left\{{\beta }_{n-k}\right\})}$ but $s_{j}w\in {ℱ}^{({\gamma }_{b,k},\{{\beta }_k,{\beta }_{n-k}\})}\text{.}$ then $\text{dim}\hspace{0.17em}{M}_{s_{j}w{\gamma }_{b,k}}^{\text{gen}}=0\text{.}$ Thus, by Corollary 2.6,

$$\text{dim}\hspace{0.17em}{M}_{w{\gamma }_{b,k}}^{\text{gen}}=0\text{for all}\hspace{0.17em}w\in {ℱ}^{({\gamma }_{b,k},\{{\beta }_k,{\beta }_{n-k}\})}\text{.}$$

The module $M$ is $n\text{-dimensional.}$

Similar reasoning applied to an irreducible module $N$ with central character ${\gamma }_{b,k}$ and $N_{w{\gamma }_{b,k}}^{\text{gen}}\ne 0$ for some $w\in {ℱ}^{({\gamma }_{b,k},\{{\beta }_k,{\beta }_{n-k}\})}$ yields the dimensions of the generalized weight spaces of $N,$ which sum to $n\text{.}$ Thus the decomposition of the principal series module $M\left({\gamma }_{b,k}\right)$ consists of two irreducible modules $M$ and $N$ with central character ${\gamma }_{b,k}$ and

$$\begin{array}{cc}\text{dim}\left(M_{w{\gamma }_{b,k}}^{\text{gen}}\right)=2& \text{for}\hspace{0.17em}w\in {ℱ}^{({\gamma }_{b,k},\varnothing )},\\ \text{dim}\left(M_{w{\gamma }_{b,k}}^{\text{gen}}\right)=1& \text{for}\hspace{0.17em}w\in {ℱ}^{({\gamma }_{b,k},\left\{{\beta }_{n-k}\right\})},\\ \text{dim}\left(M_{w{\gamma }_{b,k}}^{\text{gen}}\right)=2& \text{for}\hspace{0.17em}w\in {ℱ}^{({\gamma }_{b,k},\left\{{\beta }_{n-k}\right\})},\\ \text{dim}\left(M_{w{\gamma }_{b,k}}^{\text{gen}}\right)=1& \text{for}\hspace{0.17em}w\in {ℱ}^{\left({\gamma }_{b,k}\{{\beta }_k,{\beta }_{n-k}\}\right)},\end{array}$$

and all other weight spaces of $M$ and $N$ are 0. Neither of the two irreducible modules $M$ and $N$ with central character ${\gamma }_{b,k}$ are calibrated.

The maximal weight of $M$ is ${\gamma }_{b,k}$ which is dominant and on the hyperplane $H_{{\alpha }_1}\text{.}$ The Langlands set for this weight is $I=\left\{1\right\}\text{.}$ The maximal weight of $N$ is on the hyperplane $H_{{\beta }_k}$ if $k$ is even, and on the hyperplane $H_{{\beta }_{n-(k+1)}}$ if $k$ is odd. This observation determines the set $I$ in the Langlands decomposition of the (real part) of the maximal weight of $N$ (equation (2.11)).

Central character ${\gamma }_{b,n/2}\text{.}$ $n$ even, $Z\left({\gamma }_{b,n/2}\right)=\left\{{\beta }_0\right\},$ $P\left({\gamma }_{b,n/2}\right)=\left\{{\beta }_{n/2}\right\}\text{.}$

$$Hβ0 Hβn/2 n/2 chambers n/2 chambers n/2 chambers n/2 chambers J=∅ J={βn/2}$$

We use Lemma 2.7 and an argument similar to that for central character ${\gamma }_{b,k}$ to decompose the principal series module $M\left({\gamma }_{b,n/2}\right)$ and conclude that there are two irreducible modules $M$ and $N$ with central character ${\gamma }_{b,n/2}$ with

$$\begin{array}{cc}\text{dim}\left(M_{w{\gamma }_{b,n/2}}^{\text{gen}}\right)=2& \text{for}\hspace{0.17em}w\in {ℱ}^{({\gamma }_{b,n/2},\varnothing )},\\ \text{dim}\left(N_{w{\gamma }_{b,n/2}}^{\text{gen}}\right)=2& \text{for}\hspace{0.17em}w\in {ℱ}^{({\gamma }_{b,k},\left\{{\beta }_{n/2}\right\})}\text{.}\end{array}$$

All other weight spaces of $M$ and $N$ are 0. Neither of the two irreducible modules $M$ and $N$ with central character ${\gamma }_{b,n/2}$ are calibrated.

The maximal weight of $M$ is ${\gamma }_{b,n/2}$ which is dominant and on the hyperplane $H_{{\alpha }_1}\text{.}$ The Langlands set for this weight is $I=\left\{1\right\}\text{.}$ The module $N$ is tempered with maximal weight $\underset{\underset{n/2\hspace{0.17em}\text{factors}}{⏟}}{\dots s_1{s}_2}\hspace{0.17em}{\gamma }_{b,n/2}\text{.}$

Central character ${\gamma }_q\text{.}$ $Z\left({\gamma }_q\right)=\left\{{\beta }_0\right\},$ $P\left({\gamma }_q\right)=\{{\beta }_k,{\beta }_{n-k},{\beta }_{\ell },{\beta }_{n-\ell }\}\text{.}$

It may be that $\ell =n/2=n-\ell$ so that the hyperplanes $H_{{\beta }_{\ell }}$ and $H_{{\beta }_{n-\ell }}$ are the same and $P\left(\gamma \right)$ contains only 3 roots. We do not have to consider this situation separately.

In some sense, the special central character ${\gamma }_q$ occurs when the parameters are exactly right so that the central characters ${\gamma }_{b,k}$ and ${\gamma }_{b,\ell }$ “coalesce”. This occurs only if $n$ is even, $k$ and $\ell$ are of different parity, and the parameters satisfy $c_k/c_{\ell }=\text{sin}\left(k\theta \right)/\text{sin}\left(\ell \theta \right)\text{.}$ For a fixed choice of parameters, there is at most one choice of the quadruple $(k,\ell ,n-k,n-\ell )\text{.}$

$$Hβ0 Hβk Hβn-k Hβℓ Hβn-ℓ k chambers ℓ-k chambers n-2ℓ chambers ℓ-k chambers k chambers J=∅ J={βn-k} J={βn-k,βn-ℓ} J={βℓ,βn-ℓ,βn-k} J=P(γq)$$

We use Lemma 2.7 and Corollary 2.6 in an argument similar to that for central character ${\gamma }_{b,k}$ to see that there are five nonisomorphic irreducible $ℍ\text{-modules}$ $L,$ $M,$ $N,$ $P$ and $Q$ with central character ${\gamma }_q,$ unless $\ell =n/2,$ in which case there are only four $\text{(}N$ has dimension 0).

$$\begin{array}{cc}\text{dim}\left(L_{w{\gamma }_q}^{\text{gen}}\right)=2& \text{for}\hspace{0.17em}w\in {ℱ}^{({\gamma }_q,\varnothing )},\\ \text{dim}\left(L_{w{\gamma }_q}^{\text{gen}}\right)=1& \text{for}\hspace{0.17em}w\in {ℱ}^{({\gamma }_q,\left\{{\beta }_{n-k}\right\})},\\ \text{dim}\left(M_{w{\gamma }_q}^{\text{gen}}\right)=1& \text{for}\hspace{0.17em}w\in {ℱ}^{({\gamma }_q,\left\{{\beta }_{n-k}\right\})},\\ \text{dim}\left(N_{w{\gamma }_q}^{\text{gen}}\right)=1& \text{for}\hspace{0.17em}w\in {ℱ}^{({\gamma }_q,\{{\beta }_{n-k},{\beta }_{n-\ell }\})},\\ \text{dim}\left(P_{w{\gamma }_q}^{\text{gen}}\right)=1& \text{for}\hspace{0.17em}w\in {ℱ}^{({\gamma }_q,\{{\beta }_{\ell },{\beta }_{n-k},{\beta }_{n-\ell }\})},\\ \text{dim}\left(Q_{w{\gamma }_q}^{\text{gen}}\right)=1& \text{for}\hspace{0.17em}w\in {ℱ}^{({\gamma }_q,\{{\beta }_{\ell },{\beta }_{n-k},{\beta }_{n-\ell }\})},\\ \text{dim}\left(Q_{w{\gamma }_q}^{\text{gen}}\right)=2& \text{for}\hspace{0.17em}w\in {ℱ}^{({\gamma }_q,\{{\beta }_k,{\beta }_{\ell },{\beta }_{n-k},{\beta }_{n-\ell }\})},\end{array}$$

and all other weight spaces of these modules are 0.

Both modules $P$ and $Q$ are tempered and have the same maximal weight $\underset{\underset{n-\ell \hspace{0.17em}\text{factors}}{⏟}}{\dots s_1{s}_2}{\gamma }_q\text{.}$

Central character ${\gamma }_{c,k}\text{.}$ $Z\left({\gamma }_{c,k}\right)=\varnothing ,$ $P\left({\gamma }_{c,k}\right)=\{{\beta }_{k-1},{\beta }_{n-k}\},$ $1\le k\le (n-1)/2\text{.}$

The weight ${\gamma }_{c,k}$ is uniquely determined by $\gamma \left({\beta }_{k-1}\right)=c_k-1$ and $\gamma \left({\beta }_{n-k}\right)=c_{n-k}\text{.}$

$$Hβ0 Hβk-1 Hβn-k k chambers k-1 chambers n-2k+1 chambers n-2k+1 chambers k chambers k-1 chambers J=∅ J={βn-k} J={βk-1} J=P(γc,k)$$

The dashed line in this picture is for reference only, it does not correspond to a root in $Z\left(\gamma \right)$ or $P\left(\gamma \right)\text{.}$

Since ${\gamma }_{c,k}$ is regular, the irreducible $ℍ\text{-modules}$ with central character ${\gamma }_{c,k}$ are calibrated and can be indexed by the sets $J\text{.}$ The irreducible calibrated module ${ℍ}^{({\gamma }_{c,k},J)}$ indexed by the set $J$ has

$$\text{dim}{\left({ℍ}^{({\gamma }_{c,k},J)}\right)}_{w{\gamma }_{c,k}}=1\text{for}\hspace{0.17em}w\in {ℱ}^{({\gamma }_{c,k},J)}$$

and all other weight spaces 0. A construction of ${ℍ}^{({\gamma }_{c,k},J)}$ is given in Theorem 4.5.

To compute the Langlands parameters of these modules we first assume that $n$ is odd and $m=\frac{n-1}{2}\text{.}$ If $J=\left\{{\beta }_{k-1}\right\},$ the maximal weight of the module ${ℍ}^{({\gamma }_{c,k},J)}$ is in the same chamber as ${\beta }_{m-k}$ if $k$ is even, and in the same chamber as ${\beta }_{m+k}$ if $k$ is odd. If $J=\left\{{\beta }_{n-k}\right\},$ the maximal weight of ${ℍ}^{({\gamma }_{c,k},J)}$ is in the same chamber as ${\beta }_{m-k}$ if $k$ is odd, and in the same chamber as ${\beta }_{m+k}$ if $k$ is even. In each case this information determines the set $I$ in the Langlands parameters. If $J=\{{\beta }_{k-1},{\beta }_{n-k}\},$ the module ${ℍ}^{({\gamma }_{c,k},J)}$is tempered with maximal weights

$$\underset{\underset{n-k+1\hspace{0.17em}\text{factors}}{⏟}}{\dots s_2{s}_1}{\gamma }_{c,k}\text{and}\underset{\underset{k\hspace{0.17em}\text{factors}}{⏟}}{\dots s_1{s}_2}{\gamma }_{c,k}\text{.}$$

If $n$ is even and all parameters $c_k$ are equal, then the Langlands parameters are as in the previous paragraph. In the case that $n$ is even and $c_{2k}\ne c_{2k+1},$ then it may happen that ${\gamma }_{c,k}$ is not in the dominant chamber. The structure of the modules with central character ${\gamma }_{c,k}$ does not change but the Langlands parameters of the representations may change significantly. One of the four irreducibles with central character ${\gamma }_{c,k}$ will always be tempered, but which one (and thus the dimension of the tempered module with this central character) depends on the values of the parameters $c_{2k}$ and $c_{2k+1}\text{.}$

Central character ${\gamma }_d\text{.}$ $Z\left({\gamma }_d\right)=\varnothing ,$ $P\left({\gamma }_d\right)=\left\{{\beta }_0\right\}$

$$Hβ0 n chambers n chambers J=∅ J={β0}$$

Since ${\gamma }_d$ is regular the irreducible modules with central character ${\gamma }_d$ are calibrated and can be indexed by the sets $J\text{.}$ The module ${ℍ}^{({\gamma }_d,J)}$ has

$$\text{dim}{\left({ℍ}^{({\gamma }_{c,k},J)}\right)}_{w{\gamma }_{c,k}}=1\text{for}\hspace{0.17em}w\in {ℱ}^{({\gamma }_{c,k},J)}$$

and all other weight spaces 0. A construction of ${ℍ}^{({\gamma }_{d,k},J)}$ is given in Theorem 4.5.

The Langlands parameters given in Table 1 for irreducible representations with central character ${\gamma }_d$ assume that ${\gamma }_d\notin W{\gamma }_{d\prime }$ where $n$ is odd and ${\gamma }_{d\prime }=\xi ·{\beta }_{(n-1)/2},$ $\xi \in {ℝ}_{>0}\text{.}$ In the particular case where $n$ is odd and ${\gamma }_d\in W{\gamma }_{d\prime }$ the irreducible module indexed by the set $J=\left\{{\beta }_0\right\}$ is tempered.

Tempered representations and the Springer correspondence

The Springer correspondence for Weyl groups (see [BMo1989, p. 34]) associates to each tempered representation $M$ of $ℍ$ with real central character, the unique “maximal” irreducible $W\text{-module}$ which is contained in $M\text{.}$ For Weyl groups (crystallographic reflection groups) this is a one-to-one correspondence between tempered representations of $ℍ$ and irreducible representations of $W\text{.}$ Using our classification of $ℍ\text{-modules}$ in Table 1, we can establish a similar correspondence for the noncrystallographic groups $I_2\left(n\right)\text{.}$

If $n$ is odd, then the group $I_2\left(n\right)$ has 2 one-dimensional irreducible representations and $(n-1)/2$ two-dimensional irreducible representations. The trivial (resp. sign) representation of $I_2\left(n\right)$ corresponds to the tempered irreducible $ℍ\text{-module}$ with central character ${\gamma }_0$ (resp. ${\gamma }_{c,1}\text{).}$ The two-dimensional representations of $I_2\left(n\right)$ correspond to the tempered $ℍ\text{-modules}$ with central characters ${\gamma }_d\in W{\gamma }_d^{\prime }$ and ${\gamma }_{c,k},1\le k\le (n-1)/2\text{.}$ Note that ${\gamma }_0,{\gamma }_d$ and ${\gamma }_{c,k},1\le k\le (n-1)/2,$ can all be taken to be multiples of the root ${\beta }_{(n-1)/2}$ and in the dominant chamber. In this normalization the 1-dimensional representations correspond to the two extreme elements of this chain of weights.

If $n$ is even and the parameters $c_k$ are all equal, the trivial (resp. sign) representation of $I_2\left(n\right)$ corresponds to the tempered irreducible $ℍ\text{-module}$ with central character ${\gamma }_0$ (resp. ${\gamma }_{c,1}\text{)}$ and the other two 1-dimensional representations of $I_2\left(n\right)$ correspond to the tempered $ℍ\text{-modules}$ with central characters ${\gamma }_{b,n/2}$ and $\tau \left({\gamma }_{b,n/2}\right),$ where $\tau$ is the involution that switches ${\alpha }_1={\beta }_0$ and ${\alpha }_2={\beta }_{n-1}\text{.}$ The 2-dimensional $I_2\left(n\right)\text{-modules}$ correspond to the tempered $ℍ\text{-modules}$ with central characters ${\gamma }_{c,k},2\le k\le n/2\text{.}$ As in the case where $n$ is odd, the central characters ${\gamma }_0$ and ${\gamma }_{c,k},1\le k\le (n-1)/2,$ can be taken to be in the dominant chamber and on the line through the origin and the point ${\beta }_{n/2}+{\beta }_{n/2-1}\text{.}$ In this normalization the trivial and the sign representations correspond to the two extreme elements of this chain of weights. In the case when the parameters are unequal, two of the points on this chain may coalesce in the weight ${\gamma }_q$ and “become” the two tempered representations of $ℍ$ with central character ${\gamma }_q\text{.}$ The case where $P\left({\gamma }_q\right)$ contains only 3 roots comes from one of the central characters ${\gamma }_{b,n/2}$ or $\tau \left({\gamma }_{b,n/2}\right)$ coalescing with one of the ${\gamma }_{c,k}\text{.}$

This analysis establishes the “Springer correspondence” for all dihedral groups and all choices of the parameters $c_k$ of $ℍ$ with $c_k\in {ℝ}_{>0}\text{.}$

$$\begin{array}{ccccc}\multicolumn{5}{c}{\text{Table 1. Irreducible representations of}\hspace{0.17em}ℍI_2\left(n\right)\hspace{0.17em}\text{up to outer automorphism}}\\ \\ \text{Character}& Z\left(\gamma \right),P\left(\gamma \right)& \text{Dimension}& J& \text{Langlands Parameters}\\ \\ {\gamma }_0=0& R^{+},\varnothing & 2n& \text{nc}& \text{tempered}\\ \\ {\gamma }_a& \left\{{\beta }_0\right\},\varnothing & 2n& \text{nc}& ({\gamma }_a,\left\{1\right\})\\ \\ {\gamma }_{b,k}& \left\{{\beta }_0\right\},\{{\beta }_k,{\beta }_{n-k}\}& n& \text{nc}& ({\gamma }_{b,k},\left\{1\right\})\\ 1\le k<n/2& & n& \text{nc}& (\underset{\underset{k\hspace{0.17em}\text{factors}}{⏟}}{\dots s_1{s}_2}{\gamma }_{b,k},\left\{1\right\})\hspace{0.17em}k\hspace{0.17em}\text{even}\\ & & & & (\underset{\underset{k\hspace{0.17em}\text{factors}}{⏟}}{\dots s_1{s}_2}{\gamma }_{b,k},\left\{2\right\})\hspace{0.17em}k\hspace{0.17em}\text{odd}\\ \\ {\gamma }_{b,n/2}& \left\{{\beta }_0\right\},\left\{{\beta }_{n/2}\right\}& n& \text{nc}& ({\gamma }_{b,n/2},\left\{1\right\})\\ \left(n\hspace{0.17em}\text{even}\right)& & n& \text{nc}& \text{tempered}\\ \\ {\gamma }_q& \left\{{\beta }_0\right\},& \ell +k& \text{nc}& ({\gamma }_q,\left\{1\right\})\\ \left(n\hspace{0.17em}\text{even}\right)& \{{\beta }_k,{\beta }_{n-k},{\beta }_{\ell },{\beta }_{n-\ell }\}& \ell -k& \left\{{\beta }_{n-k}\right\}& (\underset{\underset{k\hspace{0.17em}\text{factors}}{⏟}}{\dots s_1{s}_2}{\gamma }_q,\left\{1\right\})\hspace{0.17em}k\hspace{0.17em}\text{even}\\ 0<k<\ell \le n/2& & & & (\underset{\underset{k\hspace{0.17em}\text{factors}}{⏟}}{\dots s_1{s}_2}{\gamma }_q,\left\{2\right\})\hspace{0.17em}k\hspace{0.17em}\text{odd}\\ & & n-2\ell & \{{\beta }_{n-k},{\beta }_{n-\ell }\}& (\underset{\underset{\ell \hspace{0.17em}\text{factors}}{⏟}}{\dots s_1{s}_2}{\gamma }_q,\left\{1\right\})\hspace{0.17em}k\hspace{0.17em}\text{even}\\ & & & & (\underset{\underset{\ell \hspace{0.17em}\text{factors}}{⏟}}{\dots s_1{s}_2}{\gamma }_q,\left\{2\right\})\hspace{0.17em}k\hspace{0.17em}\text{odd}\\ & & \ell -k& \{{\beta }_{n-k},{\beta }_{n-\ell },{\beta }_{\ell }\}& \text{tempered}\\ & & \ell +k& \text{nc}& \text{tempered}\\ \\ {\gamma }_{c,k}& \varnothing ,\{{\beta }_{k-1},{\beta }_{n-k}\}& 2k-1& \varnothing & ({\gamma }_{c,k},\varnothing )\\ 1\le k\le n/2& & n-2k+1& \left\{{\beta }_{k-1}\right\}& (\underset{\underset{k\hspace{0.17em}\text{factors}}{⏟}}{\dots s_2{s}_1}{\gamma }_{c,k},\left\{1\right\})\hspace{0.17em}k\hspace{0.17em}\text{odd}\\ & & & & (\underset{\underset{k\hspace{0.17em}\text{factors}}{⏟}}{\dots s_2{s}_1}{\gamma }_{c,k},\left\{2\right\})\hspace{0.17em}k\hspace{0.17em}\text{even}\\ & & n-2k+1& \left\{{\beta }_{n-k}\right\}& (\underset{\underset{k\hspace{0.17em}\text{factors}}{⏟}}{\dots s_1{s}_2}{\gamma }_{c,k},\left\{1\right\})\hspace{0.17em}k\hspace{0.17em}\text{even}\\ & & & & (\underset{\underset{k\hspace{0.17em}\text{factors}}{⏟}}{\dots s_1{s}_2}{\gamma }_{c,k},\left\{2\right\})\hspace{0.17em}k\hspace{0.17em}\text{odd}\\ & & 2k-1& \{{\beta }_{k-1},{\beta }_{n-k}\}& \text{tempered}\\ \\ {\gamma }_d& \varnothing ,\left\{{\beta }_0\right\}& n& \varnothing & ({\gamma }_d,\varnothing )\\ & & n& \left\{{\beta }_0\right\}& {(s_1{\gamma }_d,\left\{1\right\})}^{†}\\ \\ {\gamma }_{\text{gen}}& \varnothing ,\varnothing & 2n& \varnothing & ({\gamma }_{\text{gen}},\varnothing )\\ \\ \multicolumn{5}{c}{{}^{†}\hspace{0.17em}\text{This module is tempered if}\hspace{0.17em}n\hspace{0.17em}\text{is odd and}\hspace{0.17em}{\gamma }_d\in W{\gamma }_d^{\prime }\text{, with}\hspace{0.17em}{\gamma }_d^{\prime }=\xi ·{\beta }_{(n-1)/2},\xi \in {ℝ}_{>0}\text{.}}\end{array}$$

Notes and References

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

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

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