Type $A_1$

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

Last update: 9 March 2013

Type $A_1$

The Affine Hecke Algebra

The type $A_1$ affine Hecke algebra is built on the root data of ${\text{SL}}_2\text{.}$ This is the only reduced rank one root system (up to isometry.) Though our main goal is the rank two affine Hecke algebras, the rank one case is fundamental in the classifcation of the higher-rank cases.

Let

$$P=\mathbb{Z}{\omega }_1\text{and}X=\{X^{k{\omega }_1}\hspace{0.17em}\mid \hspace{0.17em}k\in \mathbb{Z}\}$$

so that $X$ is the group generated by $X^{{\omega }_1}$ and is isomorphic to $P\text{.}$ The Weyl group is

$$W_0=\{1,s_1\}\text{with}{s}_1^2=1,\text{and}{s}_1{X}^{{\omega }_1}=X^{-{\omega }_1}$$

defines an action of $W_0$ on $X\text{.}$

Let $q\in {ℂ}^{\times }\text{.}$ The affine Hecke algebra of type $A_1$ is

$$\stackrel{\sim }{H}=ℂ\text{-span}\{X^{k{\omega }_1},T_1{X}^{k{\omega }_1}\hspace{0.17em}\mid \hspace{0.17em}k\in \mathbb{Z}\},$$

with relations

$$\begin{array}{cccc}{T}_1^2& =& (q-q^{-1})T_1+1& \text{(2.1)}\\ X^{k{\omega }_1}{X}^{m{\omega }_1}& =& X^{(k+m){\omega }_1}& \text{(2.2)}\\ X^{{\omega }_1}{T}_1& =& T_1{X}^{-{\omega }_1}+(q-q^{-1})X^{{\omega }_1}\text{.}& \text{(2.3)}\end{array}$$

Then

$$ℂ\left[X\right]=\text{span}\{X^{k{\omega }_1}\hspace{0.17em}\mid \hspace{0.17em}k\in \mathbb{Z}\},\text{with}{X}^{k{\omega }_1}{X}^{\ell {\omega }_1}=X^{(k+\ell ){\omega }_1}$$

is a subalgebra of $\stackrel{\sim }{H}\text{.}$ A weight $t$ is an element of

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

A weight $t:ℂ\left[X\right]\to ℂ$ is completely determined by the value $t\left(X^{{\omega }_1}\right),$ which must be invertible, so that $T\cong {ℂ}^{\times }\text{.}$

This definition is a special case of the definitions given in sections 1.2.1 and 1.2.2, using the root system

$$R=\{\pm {\alpha }_1\},$$

with ${\alpha }_1=2{\omega }_1\text{.}$

Proposition 2.1. There are four 1-dimensional $\stackrel{\sim }{H}$ representations

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

If $q^2=-1,$ then $q=-q^{-1},$ so that $L_q=L_{-q^{-1}}$ and $L_{-q}=L_{q^{-1}}\text{.}$

		Proof.
	A straightforward check of the relations (2.1) − (2.3) shows that the maps $L_{\pm q^{\pm 1}}$ are homomorphisms. Assume $M$ is a 1-dimensional $\stackrel{\sim }{H}\text{-module,}$ with $M=\text{span}\left\{v\right\}\text{.}$ By (2.1), $$T_{1}v=qv\text{or}{T}_{1}v=-q^{-1}v\text{.}$$ By (2.3), if $T_{1}v=qv,$ then $$X^{2{\omega }_1}v=q^{2}v\text{and either}M\cong L_q\text{or}M\cong L_{-q}\text{.}$$ Similarly, if $T_{1}v=-q^{-1}v,$ then $$X^{2{\omega }_1}v=q^{-2}v\text{and either}M\cong L_{q^{-1}}\text{or}M\cong L_{-q^{-1}}\text{.}$$ $\square$

Let $t\in T$ and let ${ℂ}_t=\text{span}\left\{v_t\right\}$ be the one-dimensional $ℂ\left[X\right]\text{-module}$ given by

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

Then the principal series module is

$$M\left(T\right)={\text{Ind}}_{ℂ\left[X\right]}^{\stackrel{\sim }{H}}{ℂ}_t=\stackrel{\sim }{H}{\otimes }_{ℂ\left[X\right]}{ℂ}_t=\text{span}\{v_t,T_1{v}_t\},$$

so that $\{v_t,T_1{v}_t\}$ is a basis for $M\left(t\right)\text{.}$

Proposition 2.2.

1. If $t\left(X^{{\omega }_1}\right)\ne \pm q^{\pm 1},$ then $M\left(t\right)$ is irreducible.
2. If $t\left(X^{{\omega }_1}\right)=\pm q^{\pm 1},$ $M\left(t\right)$ has composition series $M\left(t\right)\supseteq M_1\supseteq 0\text{.}$
   If $t\left(X^{{\omega }_1}\right)=\pm q,$ then $$\begin{array}{ccc}{M}_1& =& \text{span}\{T_1{v}_t-q{v}_t\}\cong L_{\pm q^{-1}}\text{and}\\ M\left(t\right)/M_1& \cong & L_{\pm q},\text{and}\end{array}$$
   If $t\left(X^{{\omega }_1}\right)=\pm q^{-1},$ $$\begin{array}{ccc}{M}_1& =& \text{span}\{T_1{v}_t-q^{-1}{v}_t\}\cong L_{\pm q}\text{and}\\ M\left(t\right)/M_1& \cong & L_{\pm q^{-1}}\text{.}\end{array}$$
3. If $q^2=1$ and $t\left(X^{{\omega }_1}\right)=\pm q,$ then $M\left(t\right)=M^{+}\oplus M^{-},$ where $$\begin{array}{ccc}{M}^{+}& =& \text{span}\{T_1{v}_t-q{v}_t\},\text{and}\\ M^{-}& =& \text{span}\{T_1{v}_t+q^{-1}{v}_t\}\text{.}\end{array}$$

Note that if $q$ is a primitive fourth root of unity, the two cases in (b) coincide.

		Proof.
	(a) Any non-trivial proper submodule of $M\left(t\right)$ must be 1-dimensional. Such a submodule can exist only if $t\left(X^{{\omega }_1}\right)=\pm q^{\pm 1},$ by proposition 2.1. (b) If $t\left(X^{{\omega }_1}\right)=\pm q,$ let $n=T_1{v}_t-q{v}_t\text{.}$ Then $$T_{1}n=T_1^2{v}_t-q{T}_1{v}_t=-q^{-1}{T}_1{v}_t+v_t=-q^{-1}n$$ and $$\begin{array}{ccc}{X}^{{\omega }_1}n& =& T_1\left(X^{-{\omega }_1}\right)v_t+(q-q^{-1})X^{{\omega }_1}{v}_t-q{X}^{{\omega }_1}{v}_t\\ & =& t\left(X^{-{\omega }_1}\right)(T_1{v}_t-q^{-1}t\left(X^{2{\omega }_1}\right)v_t)\\ & =& t\left(X^{-{\omega }_1}\right)(T_1{v}_t-q{v}_t)\\ & =& t\left(X^{-{\omega }_1}\right)n,\end{array}$$ so that $N=ℂn$ is a 1-dimensional submodule of $M\left(t\right)\text{.}$ Since $n=T_1{v}_t-q{v}_t,$ if $\overline{v_t}$ is the image of $v_t$ in $M\left(t\right)/N,$ then $T_1\overline{v_t}=q\overline{v_t}$ and $X^{{\omega }_1}\overline{v_t}=t\left(X^{{\omega }_1}\right)\overline{v_t}\text{.}$ If $t\left(X^{{\omega }_1}\right)=\pm q^{-1},$ then $n=T_1{v}_t+q^{-1}{v}_t$ spans a 1-dimensional submodule $N\subseteq M\left(t\right),$ with $T_1\overline{v_t}=-q^{-1}\overline{v_t}$ and $X^{{\omega }_1}\overline{v_t}=t\left(X^{{\omega }_1}\right)\overline{v_t}$ in $M\left(t\right)/N\text{.}$ If $t\left(X^{{\omega }_1}\right)\ne t\left(X^{-{\omega }_1}\right),$ then $M\left(t\right)$ has two distinct weight spaces: $M{\left(t\right)}_t=ℂv_t$ and $M{\left(t\right)}_{s_{1}t}=ℂn\text{.}$ If $M\left(t\right)$ were the direct sum of two 1-dimensional submodules, it would have to be the direct sum of the two weight spaces. However, $\stackrel{\sim }{H}{v}_t$ is all of $M\left(t\right),$ so that $M\left(t\right)$ is indecomposable. (c) If $q^2=1$ and $t\left(X^{{\omega }_1}\right)=\pm q=\pm q^{-1},$ then the weight space $M{\left(t\right)}_t$ is all of $M\text{.}$ Both $T_1{v}_t-q{v}_t$ and $T_1{v}_t+q^{-1}{v}_t$ span 1-dimensional submodules of $M\left(t\right)$ which are disjoint. $\square$

Remark: The machinery of section 1.2 also suffices to obtain the information in this theorem. Part a is exactly Kato’s criterion (Theorem 1.8). Part b follows from the properties of the $\tau$ operators, since ${\tau }_1\left(v_t\right)\ne 0,$ but ${\tau }_1{\tau }_1\left(v_t\right)=0$ exactly when $t\left(X^{{\alpha }_1}\right)=q^{\pm 2}\text{.}$ In that case, $T_1·{\tau }_1\left(m\right)$ is a multiple of ${\tau }_1\left(m\right),$ so ${\tau }_1\left(m\right)$ spans a 1-dimensional submodule of $M\left(t\right)\text{.}$ We have given an explicit proof to keep the type $A_1$ example as clear as possible.

For $z\in {ℂ}^{\times },$ let $t_z\in T$ be the weight given by

$$t_z\left(X^{{\omega }_1}\right)=z\text{.}$$

Pictorially, identify $\{t_{q^x}\hspace{0.17em}\mid \hspace{0.17em}x\in ℝ\}$ with the real line. In this picture, the set

$$H_{\alpha }=\{x\hspace{0.17em}\mid \hspace{0.17em}{t}_{q^x}\left(X^{{\alpha }_1}\right)=1\}$$

is marked with a solid line, while

$$H_{\alpha +\delta }=\{x\hspace{0.17em}\mid \hspace{0.17em}{t}_{q^x}\left(X^{{\alpha }_1}\right)=q^2\}\text{and}{H}_{\alpha -\delta }=\{x\hspace{0.17em}\mid \hspace{0.17em}{t}_{q^x}\left(X^{{\alpha }_1}\right)=q^{-2}\}$$

are denoted by dashed lines.

$$\begin{array}{c} Hα-δ Hα Hα+δ tq-1 tq0 tq tq2 tq3 \\ \text{Characters}\hspace{0.17em}{t}_{q^x}\text{, generic}\hspace{0.17em}q\text{.}\end{array}$$

If $q$ is a primitve $2\ell \text{th}$ root of unity then $\{t_{q^x}\hspace{0.17em}\mid \hspace{0.17em}x\in ℝ\}$ is identified with $ℝ/2\ell \mathbb{Z}$ and $H_{\alpha }=\{k\ell \hspace{0.17em}\mid \hspace{0.17em}k\in \mathbb{Z}\}=\{\bar{0},\bar{\ell }\}\text{.}$

The following is the specific case $\ell =2,$ so that $H_{\alpha }=\{0,\pm 2,\pm 4,\dots \}=\{\bar{0},\bar{2}\}$ The periodicity is evident in the picture.

$$\begin{array}{c} Hα Hα±δ Hα Hα±δ Hα Hα±δ Hα Hα±δ Hα tq0 tq1 tq2 tq3 tq0 tq tq2 tq3 tq0 \\ \text{Characters}\hspace{0.17em}{t}_{q^x},\hspace{0.17em}{q}^4=1\text{.}\end{array}$$

If $q^2=1,$ then $H_{\alpha \pm \delta }=H_{\alpha }=\{\bar{0},\bar{1}\}\text{.}$

$$\begin{array}{c} Hα Hα Hα Hα Hα Hα Hα Hα Hα tq0 tq tq0 tq tq0 tq tq0 tq tq0 \\ \text{Characters}\hspace{0.17em}{t}_{q^x},\hspace{0.17em}{q}^2=1\text{.}\end{array}$$

The structure of $M\left(t\right)$ can be seen using the τ operator described above (1.2.4). If $t\left(X^{{\alpha }_1}\right)=1,$ then ${\tau }_1$ is not defined. If $t\left(X^{{\alpha }_1}\right)=q^{\pm 2}\ne 1,$ then ${\tau }_1$ is non-zero on $M{\left(t\right)}_t,$ but is zero on $M{\left(t\right)}_{s_{1}t}\text{.}$ This accounts for the structure of $M\left(t\right)$ described above (Proposition 2.2 (b)). If $t\left(X^{{\omega }_1}\right)\ne q^2$ or 1, then ${\tau }_1$ is invertible on $M\left(t\right)\text{.}$ The following pictures show the “local regions” as described in (1.2.6), which describe $M\left(t\right)\text{.}$

Each region is divided into two chambers, representing the two weights in the $W_0\text{-orbit}$ of the central character. Each chamber $wC$ contains a number of large dots equal to the dimension of $M{\left(t\right)}_{wt}^{\text{gen}},$ the generalized weight space with weight $wt,$ and lines connecting the dots show that the corresponding basis vectors are in the same composition factor. A solid line is drawn where ${\tau }_1$ is not defined, as in the following picture.

$$\begin{array}{ccc} & & \\ t_1,t_{-1},q^2\ne 1& & t_1,t_{-1},q^2=1\end{array}$$

These two pictures represent weights with $t\left(X^{{\alpha }_1}\right)=1,$ so that $s_{1}t=t\text{.}$ Hence the two dots representing the weight spaces are drawn in the same chamber. If $q^2\ne 1,$ $M\left(t\right)$ is irreducible, so the weight spaces are connected by arcs. The picture reflects the fact that there is only one local region, since $P\left(t\right)=\varnothing \text{.}$

$$\begin{array}{ccc} & & \\ t_q,t_{-q},q^2\ne 1& & t_z,z\ne q^2,1\end{array}$$

These pictures show $M\left(t\right)$ for the other possible weights $t\text{.}$ A dashed line denotes a weight $t$ with $t\left(X^{{\alpha }_1}\right)=q^{\pm 2},$ so that the operator ${\tau }_1$ on the weight spaces of $M\left(t\right)$ is well-defined, but not invertible in both directions. This means that the corresponding weight spaces will be in different composition factors of $M\left(t\right)\text{.}$ A dotted line denotes a weight $t$ with $t\left(X^{{\alpha }_1}\right)\ne q^2$ or 1. In this case ${\tau }_1$ is invertible, forcing the weight spaces to be in the same composition factor of $M\left(t\right)\text{.}$ Accordingly, the dots representing them are connected.

Notice also the connection to the drawings of central characters above. The pictures of the local regions at a weight $t_{q^x}$ are a picture of a small open neighborhood around the point $t_{q^x}$ in the picture of the characters. This correspondence is not as clear in this case as in others, since it is the smallest example of the affine Hecke algebra, but the essential ingredients are present. The weights of the $M\left(t\right)$ are all displayed, as are the actions of the $\tau$ operators that determine the composition factors of $M\left(t\right)\text{.}$

The complete classification of $\stackrel{\sim }{H}$ modules is summarized in the following tables.

$$\begin{array}{c}\begin{array}{|cc|}\hline t& \text{Dimensions of Irreds.}\\ t_1& 2\\ t_{-1}& 2\\ t_q& 1,1\\ t_{-q}& 1,1\\ t_z,z\ne \pm 1\hspace{0.17em}\text{or}\hspace{0.17em}\pm q& 2\\ \hline\end{array}\\ \text{Table 1: Table of possible central characters in Type}\hspace{0.17em}{A}_1\text{, with}\hspace{0.17em}q\hspace{0.17em}\text{generic.}\end{array}$$

$$\begin{array}{c}\begin{array}{|cc|}\hline t& \text{Dimensions of Irreds.}\\ t_1& 2\\ t_{-1}& 2\\ t_q& 1,1\\ t_z,z\ne \pm 1\hspace{0.17em}\text{or}\hspace{0.17em}\pm q& 2\\ \hline\end{array}\\ \text{Table 2: Table of possible central characters in Type}\hspace{0.17em}{A}_1\text{, with}\hspace{0.17em}{q}^2=-1\text{.}\end{array}$$

$$\begin{array}{c}\begin{array}{|cc|}\hline t& \text{Dimensions of Irreds.}\\ t_1& 1,1\\ t_{-1}& 1,1\\ t_z,z\ne \pm 1\hspace{0.17em}\text{or}\hspace{0.17em}\pm q& 2\\ \hline\end{array}\\ \text{Table 3: Table of possible central characters in Type}\hspace{0.17em}{A}_1\text{, with}\hspace{0.17em}q=-1\text{.}\end{array}$$

It should be noted that for any value of $q$ with $q^2\ne \pm 1,$ the representation theory of $\stackrel{\sim }{H}$ can be described in terms of $q$ only. If $q^2=\pm 1,$ 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 $q^2=\pm 1\text{.}$ When these hyperplanes coincide, the sets $P\left(t\right)$ and $Z\left(t\right)$ change for characters on those hyperplanes.

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