Type $A_2$

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

Last update: 9 March 2013

Type $A_2$

The type $A_2$ root system is

$$R=\{\pm {\alpha }_1,\pm {\alpha }_2,\pm ({\alpha }_1+{\alpha }_2)\},$$

where $⟨{\alpha }_1,{\alpha }_2^{\vee }⟩=-1=⟨{\alpha }_2,a_1^{\vee }⟩\text{.}$ Then $R$ is a root system as defined in (1.2.1), and the Weyl group $W_0\cong S_3\text{.}$ The simple roots are ${\alpha }_1$ and ${\alpha }_2,$ and ${\alpha }_1+{\alpha }_2$ is the only other positive root.

$$\begin{array}{c} α1 α2 α1+α2 ω2 ω1 \\ \text{The type}\hspace{0.17em}{A}_2\hspace{0.17em}\text{root system}\end{array}$$

The fundamental weights satisfy

$$\begin{array}{c}{\omega }_1=\frac{1}{3}(2{\alpha }_1+{\alpha }_2),{\alpha }_1=2{\omega }_1-{\omega }_2,\\ {\omega }_2=\frac{1}{3}(2{\alpha }_2+{\alpha }_1),{\alpha }_2=2{\omega }_2-{\omega }_1\text{.}\end{array}$$

Let

$$P=\mathbb{Z}\text{-span}\{{\omega }_1,{\omega }_2\}$$

be the weight lattice of $R\text{.}$

$$\begin{array}{c} Hα1 Hα2 ω1 ω2 0 \\ \text{The weight lattice}\hspace{0.17em}P\end{array}$$

The affine Hecke algebra $\stackrel{\sim }{H}$ (see 1.2.2) is generated as a $ℂ\text{-algebra}$ by $T_1,$ $T_2,$ and $X=\{X^{\lambda }\hspace{0.17em}\mid \hspace{0.17em}\lambda \in P\},$ with relations

$$\begin{array}{cccc}{X}^{\lambda }{X}^{\mu }& =& X^{\lambda +\mu },\hspace{0.17em}\text{for}\hspace{0.17em}\lambda ,\mu \in P& \text{(2.4)}\\ T_1{T}_2{T}_1& =& T_2{T}_1{T}_2& \text{(2.5)}\\ T_i^2& =& (q-q^{-1})T_i+1,\hspace{0.17em}\text{for}\hspace{0.17em}i=1,2& \text{(2.6)}\\ X^{{\omega }_1}{T}_1& =& T_1{X}^{{\omega }_2-{\omega }_1}+(q-q^{-1})X^{{\omega }_1}& \text{(2.7)}\\ X^{{\omega }_1}{T}_2& =& T_2{X}^{{\omega }_1-{\omega }_2}+(q-q^{-1})X^{{\omega }_2}& \text{(2.8)}\\ X^{{\omega }_1}{T}_2& =& T_2{X}^{{\omega }_1}& \text{(2.9)}\\ X^{{\omega }_2}{T}_1& =& T_1{X}^{{\omega }_2}& \text{(2.10)}\end{array}$$

Let

$$ℂ\left[X\right]=\{X^{\lambda }\hspace{0.17em}\mid \hspace{0.17em}\lambda \in P\},$$

a subalgebra of $\stackrel{\sim }{H},$ and let

$$T={\text{Hom}}_{ℂ\text{-alg}}(ℂ\left[X\right],ℂ)\text{.}$$

Then $W_0$ acts on $X$ by

$$\begin{array}{ccc}{s}_1·X^{{\omega }_1}& =& X^{{\omega }_2-{\omega }_1},\\ s_1·X^{{\omega }_2}& =& X^{{\omega }_2},\\ s_2·X^{{\omega }_1}& =& X^{{\omega }_1},& \text{and}\\ s_2·X^{{\omega }_2}& =& X^{{\omega }_1-{\omega }_2},\end{array}$$

and $W_0$ acts on $T$ by

$$(w·t)\left(X^{\lambda }\right)=t\left(X^{w^{-1}\lambda }\right)\text{.}$$

Let

$$Q=\mathbb{Z}\text{-span}\left(R\right)$$

be the root lattice of $R\text{.}$ Let

$$ℂ\left[Q\right]=\{X^{\lambda }\hspace{0.17em}\mid \hspace{0.17em}\lambda \in Q\}$$

and let

$$T_Q={\text{Hom}}_{ℂ\text{-alg}}(ℂ\left[Q\right],ℂ)\text{.}$$

Define

$$\begin{array}{ccccc}{t}_{z,w}:& ℂ\left[Q\right]& ⟶& ℂ& \text{by}\\ & X^{{\alpha }_1}& ⟼& z\\ & X^{{\alpha }_2}& ⟼& w\text{.}\end{array}$$

We can also visualize $T_Q$ using the following picture. Here, the hyperplane $H_{\alpha }=\{t\in T_Q\hspace{0.17em}\mid \hspace{0.17em}t\left(X^{\alpha }\right)=1\}$ is drawn as a solid line. The hyperplanes $H_{\alpha +\delta }=\{t\in T_Q\hspace{0.17em}\mid \hspace{0.17em}t\left(X^{\alpha }\right)=q^2\}$ are drawn in this picture as dashed lines. The weight $t_{q^x,q^y}$ is the point $x$ units away from $H_{{\alpha }_1}$ and $y$ units away from $H_{{\alpha }_2}\text{.}$ The action of $W_0$ is also visible in this picture, as $s_i$ is given by reflection in the hyperplane $H_{{\alpha }_i}\text{.}$

$$\begin{array}{c} Hα1 Hα2 Hα1+α2 \\ T_Q\end{array}$$

For each $t_{z,w}\in T_Q,$ there are 3 elements $t\in T$ with $t{\mid }_Q=t_{z,w},$ determined by

$$t{\left(X^{{\omega }_1}\right)}^3=z^{2}w\text{and}t\left(X^{{\omega }_2}\right)=t\left(X^{-{\omega }_1}\right)·zw\text{.}$$

The dimension of the modules with central character $t$ and the submodule structure of $M\left(t\right)$ depends only on $t{\mid }_Q\text{.}$ Thus we begin by examining the $W_0\text{-orbits}$ in $T_Q\text{.}$ The structure of the modules with weight $t$ depends virtually exclusively on $P\left(t\right)=\{\alpha \in R^{+}\hspace{0.17em}\mid \hspace{0.17em}t\left(X^{\alpha }\right)=q^{\pm 2}\}$ and $Z\left(t\right)=\{\alpha \in R^{+}\hspace{0.17em}\mid \hspace{0.17em}t\left(X^{\alpha }\right)=1\}\text{.}$ For a generic weight $t,$ $P\left(t\right)$ and $Z\left(t\right)$ are empty, so we examine only the non-generic orbits.

Proposition 2.3. If $t\in T_Q,$ and $P\left(t\right)\cup Z\left(t\right)\ne \varnothing ,$ then $t$ is in the $W_0\text{-orbit}$ of one of the following weights:

$$t_{1,1},t_{1,q^2},t_{q^2,1},t_{q^2,q^2},\{t_{1,z}\hspace{0.17em}\mid \hspace{0.17em}z\in {ℂ}^{\times }z\ne 1,q^{\pm 2}\},\hspace{0.17em}\text{or}\hspace{0.17em}\{t_{q^2,z}\hspace{0.17em}\mid \hspace{0.17em}z\in {ℂ}^{\times }z\ne 1,q^{\pm 2},q^{-4}\}$$

		Proof.
	First, assume generic $q\text{.}$ Case 1: If $Z\left(t\right)$ contains two positive roots, then it must contain the third. This implies $t=t_{1,1}\text{.}$ Case 2: If $Z\left(t\right)$ contains only one root, by applying an element of $W_0,$ assume that it is ${\alpha }_1\text{.}$ Then $t\left(X^{{\alpha }_2}\right)=t\left(X^{{\alpha }_1+{\alpha }_2}\right),$ so either $P\left(t\right)=\varnothing$ or $P\left(t\right)=\{{\alpha }_2,{\alpha }_1+{\alpha }_2\}\text{.}$ The first central character is $t_{1,z}$ for some $z\ne 1$ or $q^{\pm 2}\text{.}$ (If $z=1$ or $z=\pm 2,$ either $P\left(t\right)$ or $Z\left(t\right)$ would be larger.) For the second case, there are two potential choices for the orbit, arising from choosing $t\left(X^{{\alpha }_2}\right)=q^2$ or $q^{-2}\text{.}$ However, $t_{1,q^{-2}}$ is in the same orbit as $t_{q^2,1}\text{.}$ Case 3: Now assume that $Z\left(t\right)=\varnothing \text{.}$ If $P\left(t\right)$ is not empty, assume that ${\alpha }_1\in P\left(t\right)$ and $t\left(X^{{\alpha }_1}\right)=q^2\text{.}$ Then $t\left(X^{{\alpha }_2}\right)\ne q^{-2}$ by assumption on $Z\left(t\right)\text{.}$ Then it is possible that ${\alpha }_2\in P\left(t\right),$ in which case $t=t_{q^2,q^2}\text{.}$ If ${\alpha }_1+{\alpha }_2\in P\left(t\right)$ then $t\left(X^{{\alpha }_2}\right)=q^{-4}$ and $t=t_{q^2,q^{-4}}=s_2{s}_1{t}_{q^2{q}^2}\text{.}$ Otherwise, $t=t_{q^2,z}$ for some $z\ne 1,q^{\pm 2},q^{-4}\text{.}$ $\square$

Remark: There is some redundancy present in the list of central characters above for specific values of $q\text{.}$ If $q^2=-1,$ then $t_{1,q^2},$ $t_{q^2,1},$ and $t_{q^2,q^2}$ are all in the same $W_0\text{-orbit.}$ Also in this case, $t_{q^2,z}=t_{-1,z}=s_1{t}_{-1,-z}\text{.}$ If $q^2=1,$ then $t_{1,1}=t_{q^2,1}=t_{1,q^2}=t_{q^2,q^2},$ and $t_{1,z}=t_{q^2,z}\text{.}$ Also note that for every generic weight $t_{z,w},$ there are six weights in its $W_0\text{-orbits,}$ all of which are of course generic.

It is helpful to draw a picture of the weights $\{t_{q^x,q^y}\hspace{0.17em}\mid \hspace{0.17em}x,y\in ℝ\}$ for various values of $q\text{.}$ Solid lines in these pictures show sets of the form

$$H_{\alpha }=\{t\in T_Q\hspace{0.17em}\mid \hspace{0.17em}t\left(X^{\alpha }\right)=1\},$$

for $\alpha \in R^{+},$ while dashed lines denote sets of the form

$$H_{\alpha \pm \delta }=\{t\in T_Q\hspace{0.17em}\mid \hspace{0.17em}t\left(X^{\alpha }\right)=q^{\pm 2}\},$$

for $\alpha \in R^{+}\text{.}$

$$\begin{array}{c} Hα1 Hα2 Hα1+α2 Hα1+α2+δ Hα1+δ Hα2+δ t1,1 t1,q2 t1,z tz,w tq2,1 tq2,q2 tq2,z \\ \text{Figure 1: Representatives of some central characters of modules over}\hspace{0.17em}\stackrel{\sim }{H}\text{, with general}\hspace{0.17em}q\text{.}\\ \hspace{0.17em}\\ Hα1 Hα1+δ Hα1-δ Hα1 Hα2-δ Hα2+δ Hα2 Hα1+α2 Hα1+α2-δ Hα1+α2+δ t1,1 t1,q2 t1,z tz,w tq2,1 tq2,q2 tq2,z \\ \text{Figure 2:}\hspace{0.17em}{q}^2\hspace{0.17em}\text{a primitive third root of unity.}\\ \hspace{0.17em}\\ Hα1 Hα1±δ Hα2±δ Hα2 Hα1+α2 Hα1+α2±δ t1,1 t1,q2 t1,z tz,w tq2,z \\ \text{Figure 3:}\hspace{0.17em}{q}^2-1\text{.}\\ \hspace{0.17em}\\ Hα1 Hα2 Hα1+α2 t1,1 tz,w tq2,z \\ \text{Figure 4:}\hspace{0.17em}{q}^2=1\text{.}\end{array}$$

Analysis of the characters

Proposition 2.4. Fix $\epsilon =e^{2\pi i/3}\text{.}$ The 1-dimensional $\stackrel{\sim }{H}\text{-modules}$ are

$$\begin{array}{ccc}\begin{array}{cccc}{L}_{q^2,q^2}:& \stackrel{\sim }{H}& ⟶& ℂ\\ & T_1& ⟼& q\\ & T_2& ⟼& q\\ & X^{{\omega }_1}& ⟼& q^2\\ & X^{{\omega }_2}& ⟼& q^2\end{array}& \begin{array}{cccc}{L}_{\epsilon q^2,{\epsilon }^2{q}^2}:& \stackrel{\sim }{H}& ⟶& ℂ\\ & T_1& ⟼& q\\ & T_2& ⟼& q\\ & X^{{\omega }_1}& ⟼& \epsilon q^2\\ & X^{{\omega }_2}& ⟼& {\epsilon }^2{q}^2\end{array}& \begin{array}{cccc}{L}_{{\epsilon }^2{q}^2,\epsilon q^2}:& \stackrel{\sim }{H}& ⟶& ℂ\\ & T_1& ⟼& q\\ & T_2& ⟼& q\\ & X^{{\omega }_1}& ⟼& {\epsilon }^2{q}^2\\ & X^{{\omega }_2}& ⟼& \epsilon q^2\end{array}\\ \begin{array}{cccc}{L}_{q^{-2},q^{-2}}:& \stackrel{\sim }{H}& ⟶& ℂ\\ & T_1& ⟼& -q^{-1}\\ & T_2& ⟼& -q^{-1}\\ & X^{{\omega }_1}& ⟼& q^{-2}\\ & X^{{\omega }_2}& ⟼& q^{-2}\end{array}& \begin{array}{cccc}{L}_{\epsilon q^{-2},{\epsilon }^2{q}^{-2}}:& \stackrel{\sim }{H}& ⟶& ℂ\\ & T_1& ⟼& -q^{-1}\\ & T_2& ⟼& -q^{-1}\\ & X^{{\omega }_1}& ⟼& \epsilon q^{-2}\\ & X^{{\omega }_2}& ⟼& {\epsilon }^2{q}^{-2}\end{array}& \begin{array}{cccc}{L}_{{\epsilon }^2{q}^{-2},\epsilon q^{-2}}:& \stackrel{\sim }{H}& ⟶& ℂ\\ & T_1& ⟼& -q^{-1}\\ & T_2& ⟼& -q^{-1}\\ & X^{{\omega }_1}& ⟼& {\epsilon }^2{q}^{-2}\\ & X^{{\omega }_2}& ⟼& \epsilon q^{-2}\end{array}\end{array}$$

		Proof.
	A straightforward check shows that the maps above respect the defining relations for $\stackrel{\sim }{H}$ (2.4) - (2.10), so that the maps are homomorphisms. Let $ℂv$ be any 1-dimensional $\stackrel{\sim }{H}\text{-module.}$ By (2.6) and (2.5), $$T_{i}v=qv\text{or}{T}_{i}v=-q^{-1}v\text{and}{T}_{1}v=T_{2}v\text{.}$$ Case 1: $T_{1}v=T_{2}v=qv\text{.}$ By (2.7) and (2.8), $$X^{2{\omega }_1}v=q^2{X}^{{\omega }_2}v,\text{and}{X}^{2{\omega }_2}v=q^2{X}^{{\omega }_1}v\text{.}$$ Thus $$X^{{\alpha }_1}v=X^{2{\omega }_1-{\omega }_2}v=q^{2}v,\text{and}{X}^{{\alpha }_2}v=X^{2{\omega }_2-{\omega }_1}=q^{2}v,$$ so that $$X^{3{\omega }_1}v=X^{2{\alpha }_1+{\alpha }_2}v=q^{6}v,\text{and}{X}^{3{\omega }_2}v=X^{{\alpha }_1+2{\alpha }_2}v=q^{6}v\text{.}$$ Hence, $X^{{\omega }_1}v={\epsilon }^i{q}^{2}v,$ where $i=0,1,$ or 2, and $X^{{\omega }_2}v={\epsilon }^{2i}{q}^{2}v\text{.}$ Case 2: $T_{1}v=T_{2}v=-q^{-1}v\text{.}$ Then $$X^{2{\omega }_1}v=q^{-2}{X}^{{\omega }_2}v\text{and}{X}^{2{\omega }_2}v=q^{-2}{X}^{{\omega }_1}v\text{.}$$ This implies that $$X^{{\alpha }_1}v=X^{2{\omega }_1-{\omega }_2}v=q^{-2}v,\text{and}{X}^{{\alpha }_2}v=X^{2{\omega }_2-{\omega }_1}v=q^{-2}v,$$ so that $$X^{3{\omega }_1}v=X^{2{\alpha }_1+{\alpha }_2}v=q^{-6}v,\text{and}{X}^{3{\omega }_2}v=X^{{\alpha }_1+2{\alpha }_2}v=q^{-6}v\text{.}$$ Hence, $X^{{\omega }_1}v={\epsilon }^i{q}^{-2}v,$ where $i=0,1,$ or 2, and $X^{{\omega }_2}v={\epsilon }^{2i}{q}^{-2}v\text{.}$ $\square$

Principal Series Modules and Local Regions

Let $t\in T\text{.}$ The principal series module is

$$M\left(t\right)={\text{Ind}}_{ℂ\left[X\right]}^{\stackrel{\sim }{H}}{ℂ}_t=\stackrel{\sim }{H}{\otimes }_{ℂ\left[X\right]}{ℂ}_t,$$

where ${ℂ}_t$ is the one-dimensional $ℂ\left[X\right]\text{-module}$ given by

$${ℂ}_t=\text{span}\left\{v_t\right\}\text{and}{X}^{\lambda }{v}_t=t\left(X^{\lambda }\right)v_t\text{.}$$

By (1.6), every irreducible $\stackrel{\sim }{H}$ module is a quotient of some principal series module $M\left(t\right)\text{.}$ Thus, finding all the composition factors of $M\left(t\right)$ for all central characters $t$ will find all the irreducible $\stackrel{\sim }{H}\text{-modules.}$

Assume for now that $q^2\ne 1\text{.}$

Case 1: $P\left(t\right)$ empty. By 1.8, if $P\left(t\right)=\{\alpha \in R^{+}\hspace{0.17em}\mid \hspace{0.17em}t\left(X^{{\alpha }_1}\right)=1\}$ is empty, then $M\left(t\right)$ is irreducible and is the only irreducible module with central character $t\text{.}$ This case includes the central characters $t_{1,1},$ $t_{1,z},$ and $t_{z,w}$ for generic $z,w\text{.}$

Since $P\left(t\right)=\varnothing ,$ there is one local region

$${ℱ}^{t,\varnothing }=W_0/W_t,$$

the set of minimal length coset representatives of $W_t$ cosets in $W_0,$ where $W_t$ is the stabilizer of $t$ in $W_0\text{.}$ If $w$ and $s_{i}w$ are both in ${ℱ}^{(t,\varnothing )}$ then ${\tau }_i:M{\left(t\right)}_{wt}^{\text{gen}}\to M{\left(t\right)}_{s_{i}wt}^{\text{gen}}$ is a bijection. The following pictures show ${ℱ}^{(t,\varnothing )}$ with one dot in the chamber $w^{-1}C$ for each basis element of $M{\left(t\right)}_{wt}^{\text{gen}}\text{.}$

These pictures also show the weight space structure of $M\left(t\right)\text{.}$ For $t=t_{1,1},$ all of $M\left(t\right)$ is in the $t$ weight space, so all the dots lie in the same chamber. For $t=t_{1,z},$ $s_{1}t=t,$ so that the weights of $M\left(t\right)$ are $t,$ $s_{2}t,$ and $s_1{s}_{2}t,$ and each weight space is 2-dimensional. The weight $t=t_{z,w}$ is regular, so there are six different weights. In each case, all the dots lie in the same local region, a region bounded by solid and/or dashed lines.

$$\begin{array}{ccc} (M(t)) tgen =M(t) & (M(t)) tgen (M(t)) s2tgen (M(t)) s1s2tgen & (M(t))t (M(t))s2t (M(t))s1s2t (M(t))s1s2s1t (M(t))s2s1t (M(t))s1t \\ t_{1,1},q^2\ne 1& t_{1,z},q^2\ne 1& t_{z,w}\end{array}$$

Case 2: $Z\left(t\right)=\varnothing ,$ $P\left(t\right)\ne \varnothing \text{.}$ This case includes the central characters $t_{q^2,q^2},$ and $t_{q^2,z}\text{.}$ If $Z\left(t\right)$ is empty, then $M\left(t\right)$ is calibrated and the irreducible modules with central character $t$ are in one-to-one correspondence with the local regions and the components of the calibration graph. Each local region in the following pictures is a set of chambers between two dashed lines.

For the weight $t_{q^2,q^2}$ for generic $q,$ there are four local regions. If $q^6=1,$ each weight is in a local region by itself and thus there are six different representations in that case. For the weight $t_{q^2,z},$ there are two local regions. These local regions are bounded by dashed lines, which also serve as barriers of sorts between the composition factors. This is a combinatorial reflection of the fact that if the $\tau$ operator between two weight spaces of $M\left(t\right)$ is not invertible, those weight spaces will be in different composition factors. Weight spaces with invertible $\tau$ operators between them are separated by dotted lines since they are in the same composition factor and local region.

$$\begin{array}{ccc} (M(t))t (M(t))s2t (M(t))s1s2t (M(t))s1s2s1t (M(t))s2s1t (M(t))s1t & (M(t))t (M(t))s2t (M(t))s1s2t (M(t))s1s2s1t (M(t))s2s1t (M(t))s1t & (M(t))t (M(t))s2t (M(t))s1s2t (M(t))s1s2s1t (M(t))s2s1t (M(t))s1t \\ t_{q^2,q^2},q^4\ne 1,q^6\ne 1& t_{q^2,q^2},q^2\hspace{0.17em}\text{a primitive third root of unity}& t_{q^2,z},q^2\ne 1\end{array}$$

Case 3: $P\left(t\right)\ne \varnothing ,$ $Z\left(t\right)\ne \varnothing \text{.}$ The only central characters with both $Z\left(t\right)$ and $P\left(t\right)$ nonempty are $t_{1,z}=t_{q^2,z}$ when $q^2=1,$ and $t_{q^2,1}$ and $t_{1,q^2}$ in all cases. If $q^4=1,$ then $t_{q^2,1}$ and $t_{1,q^2}$ are in the same orbit, and are in the same orbit as $t_{q^2,q^2}\text{.}$ If $q^2=1,$ then $t_{q^2,1}=t_{1,q^2}=t_{1,1}\text{.}$ In each of these pictures, there are three local regions, but the structure of $M\left(t\right)$ is slightly more complicated in this case.

The composition factors of $M\left(t\right)$ each have some weight $wt$ with $t\left(X^{{\alpha }_i}\right)=1$ for some simple root ${\alpha }_i\text{.}$ Then 1.11 shows that when $q^2\ne \pm 1,$ this composition factor must be at least 3-dimensional. In this case, these composition factors are shown with the two dots in that weight space $M{\left(t\right)}_{wt}$ connected to each other, as well as to a dot in the next chamber, since the basis vectors correpsonding to these dots must lie in the same irreducible. When $q^2=-1,$ the two dimensional weight space $M{\left(t\right)}_{wt}$ makes up an entire composition factor.

$$\begin{array}{ccc} (M(t))t (M(t))s2t (M(t))s1s2t & (M(t))t (M(t))s2s1t (M(t))s1t & (M(t))t (M(t))s2t (M(t))s1s2t \\ t_{1,q^2},q^4\ne 1& t_{q^2,1},q^4\ne 1& t_{1,q^2},q^2=-1\end{array}$$

To prove that these pictures do reflect the structure of $M\left(t\right),$ rather than analyzing $M\left(t\right)$ directly, it is easier to construct several irreducible modules with central character $t$ and show that they must account for all the composition factors of $M\left(t\right)\text{.}$ Let ${ℂ}_{q^2,1}$ be the 1-dimensional ${\stackrel{\sim }{H}}_{\left\{1\right\}}\text{-module}$ spanned by $v_t$ and let ${ℂ}_{1,q^{-2}}$ be the 1-dimensional ${\stackrel{\sim }{H}}_{\left\{2\right\}}\text{-module}$ spanned by $v_{s_2{s}_{1}t},$ given by

$$\begin{array}{c}{X}^{\lambda }{v}_t=t\left(X^{\lambda }\right)v_t,\text{and}{T}_1{v}_t=q{v}_t,\text{and}\\ X^{\lambda }{v}_{s_2{s}_1}t=\left(s_2{s}_{1}t\right)\left(X^{\lambda }\right)v_{s_2{s}_{1}t}\text{and}{T}_1{v}_{s_2{s}_{1}t}=-q^{-1}{v}_{s_2{s}_{1}t}\text{.}\end{array}$$

Then

$$M=\stackrel{\sim }{H}{\otimes }_{{\stackrel{\sim }{H}}_{\left\{1\right\}}}{ℂ}_{q^2,1}\text{and}N=\stackrel{\sim }{H}{\otimes }_{{\stackrel{\sim }{H}}_{\left\{2\right\}}}{ℂ}_{1,q^{-2}}$$

are 3-dimensional $\stackrel{\sim }{H}\text{-modules}$ with central character $t_{q^2,1}\text{.}$

Proposition 2.5. Let $M=\stackrel{\sim }{H}{\otimes }_{{\stackrel{\sim }{H}}_{\left\{1\right\}}}{ℂ}_{q^2,1}$ and $N=\stackrel{\sim }{H}{\otimes }_{{\stackrel{\sim }{H}}_{\left\{2\right\}}}{ℂ}_{1,q^{-2}\text{.}}$

1. If $q^4\ne 1$ then $M$ and $N$ are irreducible.
2. If $q^2=-1$ then $M_{s_{1}t}$ is an irreducible submodule of $M$ and $N_{s_{1}t}$ is an irreducible submodule of $N\text{.}$ The quotients $N/N_{s_{1}t}$ and $M/M_{s_{1}t}$ are irreducible.

		Proof.
	(a) Assume $q^4\ne 1\text{.}$ If either $M$ or $N$ were reducible, it would have a 1-dimensional submodule or quotient, which cannot happen since the 1-dimensional modules have central character $t_{q^2,q^2}\text{.}$ Thus both $M$ and $N$ are reducible. (b) If $q^2=-1,$ then note that the map $$\begin{array}{cccc}\theta :& \stackrel{\sim }{H}& ⟶& \stackrel{\sim }{H}\\ & T_1& ⟼& T_2\\ & T_2& ⟼& T_1\\ & X^{{\omega }_1}& ⟼& X^{{\omega }_2}\\ & X^{{\omega }_2}& ⟼& X^{{\omega }_1}\end{array}$$ is an automorphism of $\stackrel{\sim }{H}\text{.}$ If ${\varphi }_M:\stackrel{\sim }{H}\to {\text{GL}}_3$ and ${\varphi }_N:\stackrel{\sim }{H}\to {\text{GL}}_3$ are the maps describing the action of $\stackrel{\sim }{H}$ on $M$ and $N$ respectively, then ${\varphi }_M\circ \theta ={\varphi }_N$ and ${\varphi }_N\circ \theta ={\varphi }_M\text{.}$ Thus if one of $M$ and $N$ is irreducible, the other is as well. However, this would account for all the composition factors of $M\left(t\right),$ a contradiction since there is a 1-dimensional module with central character $t\text{.}$ Thus $M$ and $N$ must both be reducible, so $M$ has a 1-dimensional submodule or quotient. It must have weight $s_{1}t$ since that is the only weight of $M$ that supports a 1-dimensional module. Then the 2-dimensional composition factor is $M_t^{\text{gen}},$ by Lemma 1.11. Then since $${\text{Hom}}_{\stackrel{\sim }{H}}(M,M_t^{\text{gen}})={\text{Hom}}_{{\stackrel{\sim }{H}}_{\left\{1\right\}}}({ℂ}_{v_t},M_t^{\text{gen}})\ne 0,$$ $M_{s_{1}t}$ is a submodule of $M\text{.}$ Similarly, $N_{s_{1}t}$ is a submodule of $N$ since $N_{s_2{s}_{1}t}^{\text{gen}}$ must be a quotient of $N\text{.}$ $\square$

$$\begin{array}{cc} (M(t))t (M(t))s2t (M(t))s1s2t & (M(t))t (M(t))s2s1t (M(t))s1t \\ t_{1,q^2},q^4\ne 1& t_{q^2,1},q^4\ne 1\end{array}$$

To construct the irreducibles with $t{\mid }_Q=t_{1,q^2},$ composing with

$$\begin{array}{cccc}\varphi :& \stackrel{\sim }{H}& ⟶& \stackrel{\sim }{H}\\ & T_1& ⟼& T_2\\ & T_2& ⟼& T_1\\ & X^{{\omega }_1}& ⟼& X^{{\omega }_2}\\ & X^{{\omega }_2}& ⟼& X^{{\omega }_1},\end{array}$$

an automorphism of $\stackrel{\sim }{H},$ gives a bijection between representations with central character $t_{q^2,1}$ and those with central character $t_{1,q^2}\text{.}$

$$\begin{array}{c} (M(t))t (M(t))s2t (M(t))s1s2t \\ t_{1,q^2},q^2=-1\end{array}$$

If $q^2=1,$ then the results of section 1.2.9 suffice to classify the representations of $\stackrel{\sim }{H}$ with central characters $t_{1,1}$ and $t_{1,z}$ for $z\ne q^{\pm 2}\text{.}$ Specifically, if $t{\mid }_Q=t_{1,1},$ then $W_t=W_0$ and a $\stackrel{\sim }{H}\text{-module}$ is merely a $W_0\text{-module}$ (via the isomorphism $H\cong ℂ\left[W_0\right]\text{)}$ on which $X^{\lambda }\in ℂ\left[X\right]$ acts by the scalar $t\left(X^{\lambda }\right)\text{.}$ In fact, $M\left(t\right)$ considered as a $W_0$ module is the regular $W_0$ module. If $t{\mid }_Q=t_{1,z}$ for $z\ne 1,$ then $W_t=\{1,s_1\}\text{.}$ Since $W_t$ has two 1-dimensional irreducible representations, $\stackrel{\sim }{H}$ has two irreducible 3-dimensional representations obtained by inducing up from ${\stackrel{\sim }{H}}_{\left\{1\right\}}\text{.}$

$$\begin{array}{ccc} (M(t))t=M(t) & (M(t)) t (M(t)) s2t (M(t)) s1s2t & (M(t))t (M(t))s2t (M(t))s1s2t (M(t))s2s1s2t (M(t))s2s1t (M(t))s1t \\ t_{1,1},q^2=1& t_{1,z},q^2=1& t_{z,w},q^2=1\end{array}$$

Note that in all cases, the local region picture is a picture of a small neighborhood around the point corresponding to $t$ in the picture of $T_Q\text{.}$ The necessary information to understand the composition factors of $M\left(t\right)$ is contained in the local region picture around $t\text{.}$

The following tables summarize the classification. It should be noted that for any value of $q$ with $q^2\ne \pm 1$ and $q^2$ not a primitive third root of unity, the representation theory of $\stackrel{\sim }{H}$ can be described in terms of $q$ only. If $q^2$ is a primitive root of unity of order 3 or less, then the representation theory of $\stackrel{\sim }{H}$ does not fit that same description. This fact can be seen through a number of different lenses. It is a reflection of the fact that the sets $P\left(t\right)$ and $Z\left(t\right)$ for all possible central characters $t$ can be described solely in terms of $q\text{.}$ In the local region pictures, this is reflected in the fact that the hyperplanes $H_{\alpha }$ and $H_{\alpha \pm \delta }$ are distinct unless $q^2$ is a root of unity of order 3 or less. When these hyperplanes coincide, the sets $P\left(t\right)$ and $Z\left(t\right)$ change for characters on those hyperplanes.

$$\begin{array}{c}\begin{array}{|cccc|}\hline t{\mid }_Q& Z\left(t\right)& P\left(t\right)& \text{Dimensions of Irreds.}\\ t_{1,1}& R^{+}& \varnothing & 6\\ t_{1,z}& \left\{{\alpha }_1\right\}& \varnothing & 6\\ t_{1,q^2}& \left\{{\alpha }_1\right\}& \{{\alpha }_2,{\alpha }_1+{\alpha }_2\}& 3,3\\ t_{q^2,1}& \left\{{\alpha }_2\right\}& \{{\alpha }_1,{\alpha }_1+{\alpha }_2\}& 3,3\\ t_{q^2,q^2}& \varnothing & \{{\alpha }_1,{\alpha }_2\}& 1,1,2,2\\ t_{q^2,z}& \varnothing & \left\{{\alpha }_1\right\}& 3,3\\ t_{z,w}& \varnothing & \varnothing & 6\\ \hline\end{array}\\ \text{Table 4: Table of possible central characters in Type}\hspace{0.17em}{A}_2\text{, with}\hspace{0.17em}q\hspace{0.17em}\text{generic.}\end{array}$$

$$\begin{array}{c}\begin{array}{|cccc|}\hline t{\mid }_Q& Z\left(t\right)& P\left(t\right)& \text{Dimensions of Irreds.}\\ t_{1,1}& R^{+}& \varnothing & 6\\ t_{1,z}& \left\{{\alpha }_1\right\}& \varnothing & 6\\ t_{1,q^2}& \left\{{\alpha }_1\right\}& \{{\alpha }_2,{\alpha }_1+{\alpha }_2\}& 3,3\\ t_{q^2,1}& \left\{{\alpha }_2\right\}& \{{\alpha }_1,{\alpha }_1+{\alpha }_2\}& 3,3\\ t_{q^2,q^2}& \varnothing & R^{+}& 1,1,1,1,1,1\\ t_{q^2,z}& \varnothing & \left\{{\alpha }_1\right\}& 3,3\\ t_{z,w}& \varnothing & \varnothing & 6\\ \hline\end{array}\\ \text{Table 5: Table of possible central characters in Type}\hspace{0.17em}{A}_2\text{, with}\hspace{0.17em}{q}^6=1\text{.}\end{array}$$

$$\begin{array}{c}\begin{array}{|cccc|}\hline t{\mid }_Q& Z\left(t\right)& P\left(t\right)& \text{Dimensions of Irreds.}\\ t_{1,1}& R^{+}& \varnothing & 6\\ t_{1,z}& \left\{{\alpha }_1\right\}& \varnothing & 6\\ t_{1,q^2}& \left\{{\alpha }_1\right\}& \{{\alpha }_2,{\alpha }_1+{\alpha }_2\}& 1,2,2\\ t_{q^2,z}& \varnothing & \left\{{\alpha }_1\right\}& 3,3\\ t_{z,w}& \varnothing & \varnothing & 6\\ \hline\end{array}\\ \text{Table 6: Table of possible central characters in Type}\hspace{0.17em}{A}_2\text{, with}\hspace{0.17em}{q}^2=-1\text{.}\end{array}$$

$$\begin{array}{c}\begin{array}{|cccc|}\hline t{\mid }_Q& Z\left(t\right)& P\left(t\right)& \text{Dimensions of Irreds.}\\ t_{1,1}& R^{+}& R^{+}& 1,1,2\\ t_{1,z}& \left\{{\alpha }_1\right\}& \left\{{\alpha }_1\right\}& 3,3\\ t_{z,w}& \varnothing & \varnothing & 6\\ \hline\end{array}\\ \text{Table 7: Table of possible central characters in Type}\hspace{0.17em}{A}_2\text{, with}\hspace{0.17em}q=-1\text{.}\end{array}$$

Notes and References

This is an excerpt from Matt Davis' Ph.D Thesis entitled Representations of Rank Two Affine Hecke Algebras at Roots of Unity, University of Wisconsin, 2010.

page history