Classification for $A_1$

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

Last update: 31 March 2013

Classification for $A_1$

The root system $R$ for $A_1$ has one simple root ${\alpha }_1$ and fundamental weight ${\omega }_1=\frac{1}{2}{\alpha }_1\text{.}$

Irreducible representations. Table 2.1 lists the irreducible $\stackrel{\sim }{H}\text{-modules}$ by their central characters. The sets $P\left(t\right)$ and $Z\left(t\right)$ are as given in (1.7) and correspond to the choice of representative for the central character displayed in Figure 2.1. The Langlands parameters usually consist of a pair $(𝒯,I)$ where $I$ is a subset of $\left\{1\right\}$ and $𝒯$ is a tempered representation for the parabolic subalgebra ${\stackrel{\sim }{H}}_I\text{.}$ In our cases the tempered representation $𝒯$ of ${\stackrel{\sim }{H}}_I$ is completely determined by a character $t\in T\text{.}$ Specifically, $𝒯$ is the only tempered representation of ${\stackrel{\sim }{H}}_I$ which has $t$ as a weight. In the labeling in Table 2.1 we have replaced the representation $𝒯$ by the weight $t\text{.}$ The nilpotent element $e_{{\alpha }_1}$ is a representative of the root space ${𝔤}_{{\alpha }_1}$ for the Lie algebra $𝔤={𝔰𝔩}_2\text{.}$ For each calibrated module with central character $t$ we have listed the subset $J\subseteq P\left(t\right)$ such that $(t,J)$ is the corresponding placed skew shape (see Theorem 1.11). The abbreviation ‘nc’ indicates modules that are not calibrated.

$$\begin{array}{ccccccc}\text{Central}& P\left(t\right)& Z\left(t\right)& \text{Dimension}& \text{Langlands}& \text{Indexing}& \text{Calibration}\\ \text{character}& & & & \text{parameters}& \text{triple}& \text{set}\hspace{0.17em}J\\ \\ t_a& \left\{{\alpha }_1\right\}& \varnothing & 1& (t_a,\varnothing )& (t_a,0,1)& \varnothing \\ & & & 1& \text{tempered}& (t_a,e_{{\alpha }_1},1)& \left\{{\alpha }_1\right\}\\ \\ t_b& \varnothing & \varnothing & 2& (t_b,\varnothing )& (t_b,0,1)& \varnothing \\ \\ t_o& \varnothing & \left\{{\alpha }_1\right\}& 2& \text{tempered}& (t_o,0,1)& \text{nc}\\ \multicolumn{7}{c}{\text{Table 2.1.}\hspace{0.17em}\text{Irreducible representations}}\end{array}$$

Figure 2.1 displays the real parts of the central characters in Table 2.1. If $t\in T$ then the polar decomposition $t=t_r{t}_c$ determines an element $\mu \in {ℝ}^n$ such that $t_r\left(X^{\lambda }\right)=e^{⟨\lambda ,\mu ⟩}$ (see (1.3)). For each central character $t_p$ the point labeled by $p$ in Figure 2.1 is the graph of the corresponding ${\mu }_p\in {ℝ}^n\text{.}$ Assume (for pictorial convenience) that $q$ is a positive real number and let

$$H_{{\alpha }_1}=\{x\in ℝ\hspace{0.17em}\mid \hspace{0.17em}⟨{\alpha }_1,x⟩=0\},\text{and}{H}_{{\alpha }_1\pm \delta }=\{x\in ℝ\hspace{0.17em}\mid \hspace{0.17em}⟨{\alpha }_1,x⟩=\text{ln}\left(q^{\pm 2}\right)\}\text{.}$$

The $\mid$ marks indicate the (affine) hyperplanes $H_{{\alpha }_1\pm \delta }\text{.}$

$$\begin{array}{c} to ta tb Hα1-δ Hα1 Hα1+δ \\ \text{Figure 2.1.}\hspace{0.17em}\text{Real parts of central characters in Table 2.1}\end{array}$$

Tempered and square integrable representations. The tempered (resp. square integrable) $\stackrel{\sim }{H}\text{-modules}$ are the ones which have all their weight spaces in the closure (resp. interior) of the dotted region of Figure 2.2.

$$\begin{array}{c} s1ta to \\ \text{Figure 2.2.}\hspace{0.17em}\text{Real parts of weights of tempered representations}\end{array}$$

The irreducible tempered representations with real central character can be indexed by the irreducible representations of the symmetric group $S_2$ (see [BMo1989]). These representations are indexed by the partitions (2), $\left(1^2\right)$ of 2. Let $e_{{\alpha }_1}$ be an element of the root space ${𝔤}_{{\alpha }_1}$ for the Lie algebra $𝔤={𝔰𝔩}_2\text{.}$ The two nilpotent orbits in $𝔤$ and the corresponding tempered representations of $\stackrel{\sim }{H}$ are as in Table 2.2.

$$\begin{array}{cccc}\text{Nilpotent orbit}& \text{Indexing triple}& \text{Square integrable}& W\hspace{0.17em}\text{representation}\\ \text{regular}& (t_a,e_{{\alpha }_1},1)& \text{yes}& \left(1^2\right)\\ 0& (t_o,0,1)& \text{no}& \left(2\right)\\ \multicolumn{4}{c}{\text{Table 2.2.}\hspace{0.17em}\text{Tempered representations and the Springer correspondence}}\end{array}$$ $$\begin{array}{ccc} s1ta ta & s1tb tb & to \\ \multicolumn{3}{c}{\text{Figure 2.3.}\hspace{0.17em}\text{Calibration graphs for central characters in Table 2.1}}\end{array}$$

The analysis.

Central character $t_a\text{:}$ There are two one-dimensional representations $ℂv_a$ and $ℂv_{s_{1}a}$ with central character $t_a\text{.}$ These representations are given explicitly by

$$\begin{array}{ccc}{X}^{\lambda }{v}_a& =& t_a\left(X^{\lambda }\right)v_a,\\ T_1{v}_a& =& q{v}_a,\end{array}\text{and}\begin{array}{ccc}{X}^{\lambda }{v}_{s_{1}a}& =& \left(s_1{t}_a\right)\left(X^{\lambda }\right)v_{s_{1}a},\\ T_1{v}_{s_{1}a}& =& -q^{-1}{v}_{s_{1}a},\end{array}$$

respectively. One uses Theorem 1.15 and the fact that the principal series module $M\left(t_a\right)$ is two dimensional to conclude that $ℂv_a$ and $ℂv_{s_{1}a}$ are the only irreducible representations of $\stackrel{\sim }{H}$ with central character $t_a\text{.}$

Central character $t_b\text{:}$ By Theorem 1.15 and Kato’s irreducibility criterion, Theorem 1.16, the only irreducible representation with central character $t_b$ is the principal series module $M\left(t_b\right)\text{.}$ Alternatively, this module can be constructed by applying Theorem 1.11 to the placed skew shape $(t_b,\varnothing )\text{.}$

Central character $t_o\text{:}$ The weights given by $t_o\left(X^{{\omega }_1}\right)=\pm 1$ are the two central characters $t_o\in T$ which satisfy $P\left(t\right)=\varnothing ,$ $Z\left(t\right)=\left\{{\alpha }_1\right\}\text{.}$ In either case Kato’s irreducibility criterion (Theorem 1.16) tells us that the principal series module $M\left(t_o\right)$ is irreducible. This module has basis $\{v_t,T_1{v}_t\}$ and action given by

$$\varphi \left(X^{\lambda }\right)=t\left(X^{\lambda }\right)\left(\begin{array}{cc}1& (q-q^{-1})⟨\lambda ,{\alpha }_1^{\vee }⟩\\ 0& 1\end{array}\right)\text{and}\varphi \left(T_1\right)=\left(\begin{array}{cc}0& 1\\ 1& q-q^{-1}\end{array}\right)\text{.}$$

Notes and References

This is an excerpt of a preprint entitled Representations of rank two affine Hecke Algebras, written by Arun Ram, Department of Mathematics, Princeton University, August 5, 1989.

Research supported in part by National Science Foundation grant DMS-9622985, and a Postdoctoral Fellowship at Mathematical Sciences Research Institute.

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