Classification for $A_2$

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 $A_2$

The root system $R$ for $A_2$ has simple roots ${\alpha }_1$ and ${\alpha }_2,$ fundamental weights ${\omega }_1$ and ${\omega }_2,$ and

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

Irreducible representations. Table 4.1 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 4.1. The Langlands parameters usually consist of a pair $(𝒯,I)$ where $I$ is a subset of $\{1,2\}$ 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 4.1 we have replaced the representation $𝒯$ by the weight $t\text{.}$ The nilpotent elements $e_{{\alpha }_1}$ and $e_{{\alpha }_2}$ are representatives of the root spaces ${𝔤}_{{\alpha }_1}$ and ${𝔤}_{{\alpha }_2},$ respectively, where $𝔤$ is the Lie algebra $𝔤={𝔰𝔩}_3\text{.}$ 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.

$$\begin{array}{ccccccc}\text{Central}& P\left(t\right)& Z\left(t\right)& \text{Dimension}& \text{Langlands}& \text{Indexing}& \text{Calibration}\\ \text{character}& & & & \text{parameters}& \text{triple}& \text{set}\hspace{0.17em}J\\ \\ t_a& \{{\alpha }_1,{\alpha }_2\}& \varnothing & 1& (t_a,\varnothing )& (t_a,0,1)& \varnothing \\ & & & 2& (s_1{t}_a,\left\{2\right\})& (t_a,e_{{\alpha }_2},1)& \left\{{\alpha }_2\right\}\\ & & & 2& (s_2{t}_a,\left\{1\right\})& (t_a,e_{{\alpha }_1},1)& \left\{{\alpha }_1\right\}\\ & & & 1& \text{tempered}& (t_a,e_{{\alpha }_1}+e_{{\alpha }_2},1)& \{{\alpha }_1,{\alpha }_2\}\\ \\ t_b& \left\{{\alpha }_2\right\}& \varnothing & 3& (t_b,\varnothing )& (t_b,0,1)& \varnothing \\ & & & 3& {(s_2{t}_b,\left\{2\right\})}^{†}& (t_b,e_{{\alpha }_2},1)& \left\{{\alpha }_2\right\}\\ \\ t_c& \{{\alpha }_2,{\alpha }_1+{\alpha }_2\}& \left\{{\alpha }_1\right\}& 3& (t_c,\left\{1\right\})& (t_c,0,1)& \text{nc}\\ & & & 3& (s_2{t}_c,\left\{2\right\})& (t_c,e_{{\alpha }_2},1)& \text{nc}\\ \\ t_d& \{{\alpha }_1,{\alpha }_1+{\alpha }_2\}& \left\{{\alpha }_2\right\}& 3& (t_d,\left\{2\right\})& (t_d,0,1)& \text{nc}\\ & & & 3& (s_1{t}_d,\left\{1\right\})& (t_d,e_{{\alpha }_1},1)& \text{nc}\\ \\ t_e& \varnothing & \left\{{\alpha }_1\right\}& 6& (t_e,\left\{1\right\})& (t_e,0,1)& \text{nc}\\ \\ t_f& \varnothing & \left\{{\alpha }_2\right\}& 6& (t_f,\left\{2\right\})& (t_f,0,1)& \text{nc}\\ \\ t_g& \varnothing & \varnothing & 6& (t_g,\varnothing )& (t_g,0,1)& \varnothing \\ \\ t_o& \varnothing & \{{\alpha }_1+{\alpha }_2\}& 6& \text{tempered}& (t_o,0,1)& \text{nc}\\ \\ \multicolumn{7}{c}{\text{Table 4.1.}\hspace{0.17em}\text{Irreducible representations}}\end{array}$$

${}^{†}$ There is one case when this representation is tempered, see Table 4.2.

Figure 4.1 displays the real parts of the central characters in Table 4.1. If $t\in T$ then the polar decomposition $t=t_r{t}_c$ determines an element $\nu \in {ℝ}^n$ such that $t_r\left(X^{\lambda }\right)=e^{⟨\lambda ,\nu ⟩}$ (see (1.3)). For each central character $t_p$ the point labeled by $p$ in Figure 4.1 is the graph of the corresponding ${\nu }_p\in {ℝ}^n\text{.}$ Assume (for pictorial convenience) that $q$ is a positive real number and let

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

for each positive root $\beta \text{.}$ The dotted lines display the (affine) hyperplanes $H_{\beta \pm \delta }\text{.}$

$$\begin{array}{c} Hα1 Hα2 Hα1+α2 a b c d e f g o \\ \text{Figure 4.1.}\hspace{0.17em}\text{Real parts of central characters in Table 4.1}\end{array}$$

Tempered and square integrable representations. The tempered (resp. square integrable) $\stackrel{\sim }{H}\text{-modules}$ are the ones which have the real parts of all their weights in the closure (resp. interior) of the shaded region of Figure 4.2. Let $t\in T$ be given by $t\left(X^{-{\alpha }_1}\right)=\pm q,$ $t\left(X^{-{\alpha }_2}\right)=\pm q\text{.}$ This is a special case of the central character $t_b$ in Table 4.1. For this particular special case there is one tempered representation with central character $t\text{.}$

$$\begin{array}{c} Hα1 Hα2 Hα1+α2 to s1t t s2t s2s1s2ta \\ \text{Figure 4.2.}\hspace{0.17em}\text{Real parts of weights of tempered representations}\end{array}$$

The irreducible tempered representations with real central character are in one-to-one cor- respondence with the irreducible representations of the symmetric group $S_3$ (see [BMo1989]). These representations are indexed by the partitions (3), (21), $\left(1^3\right)$ of 3. Equivalently, they can be indexed by the pairs $(n,\rho )$ which appear in the Springer correspondence. The $n$ and $\rho$ will also be elements of the indexing triple for the corresponding tempered representation of $\stackrel{\sim }{H}\text{.}$ Here $n$ is a nilpotent element of the Lie algebra $𝔤={𝔰𝔩}_3$ and $\rho$ is an irreducible representation of the component group $Z_G\left(n\right)/Z_G{\left(n\right)}^{\circ }\text{.}$ In type A the component group is always trivial. For each root $\beta \in R$ let $e_{\beta }$ be an element of the root space ${𝔤}_{\beta }\text{.}$ The three nilpotent orbits in $𝔤$ and the corresponding tempered representations of $\stackrel{\sim }{H}$ are as in Table 4.2.

$$\begin{array}{ccccc}\text{Nilpotent orbit}& Z_G\left(n\right)/Z_G{\left(n\right)}^{\circ }& \text{Indexing triple}& \text{Square integrable}& W\hspace{0.17em}\text{representation}\\ \text{regular}& 1& (t_a,e_{{\alpha }_1}+e_{{\alpha }_2},1)& \text{yes}& \left(3\right)\\ \text{subregular}& 1& (s_2{s}_{1}t,e_{{\alpha }_2},1)& \text{no}& \left(21\right)\\ 0& 1& (t_o,0,1)& \text{no}& \left(1^3\right)\\ \\ \multicolumn{5}{c}{\text{Table 4.2.}\hspace{0.17em}\text{Tempered representations and the Springer correspondence}}\end{array}$$ $$\begin{array}{c}\begin{array}{ccc} s1ta ta s2ta s1s2ta s2s1s2ta s2s1ta & s1tb tb s2tb s1s2tb s2s1s2tb s2s1tb & s1tc tc s2tc s1s2tc s2s1s2tc s2s1tc \end{array}\\ \begin{array}{ccc} s1td td s2td s1s2td s2s1s2td s2s1td & s1te te s2te s1s2te s2s1s2te s2s1te & s1tf tf s2tf s1s2tf s2s1s2tf s2s1tf \end{array}\\ \begin{array}{cc} s1tg tg s2tg s1s2tg s2s1s2tg s2s1tg & s1to to s2to s1s2to s2s1s2to s2s1to \end{array}\\ \text{Figure 4.3.}\hspace{0.17em}\text{Calibration graphs for central characters in Table 4.1}\end{array}$$

The analysis.

Central characters $t_a,$ $t_b,$ $t_g\text{:}$ Since $Z\left(t\right)=\varnothing$ these weights are regular. Thus the representations corresponding to these central characters are in one to one correspondence with the connected components of the calibration graph $\Gamma \left(t\right)$ and can be constructed explicitly with the use of Theorem 1.11. Up to isomorphism the principal series module $M\left(t_g\right)$ is the only irreducible $\stackrel{\sim }{H}\text{-module}$ with central character $t_g\text{.}$ The Langlands parameters for each module can be determined from its weight structure and the indexing triple is then determined from the Langlands data by using the induction theorem of Kazhdan and Lusztig (see the discusssion in [BMo1989, p.34]).

There is one special case of the central character $t_b$ when the irreducible module constructed by applying Theorem 1.11 to the placed skew shape $(t_b,\varnothing )$ is tempered. This happens when $t_b=s_2{s}_{1}t$ for the weight $t\in T$ given by $t\left(X^{-{\alpha }_1}\right)=\pm q,$ $t\left(X^{-{\alpha }_2}\right)=\pm q\text{.}$ The indexing triple and the calibration set for this case are still given by $(t_b,e_{{\alpha }_2},1)$ and $J=\left\{{\alpha }_2\right\},$ respectively.

Central characters $t_c$ and $t_d\text{:}$ One can use the defining relations of $\stackrel{\sim }{H}$ to check that the only 1-dimensional representations of $\stackrel{\sim }{H}$ are the ones with central character $t_a\text{.}$ Construct two 3- dimensional representations of $\stackrel{\sim }{H}$ by

$${\text{Ind}}_{{\stackrel{\sim }{H}}_{\left\{2\right\}}}^{\stackrel{\sim }{H}}\left(ℂv_c\right)\text{and}{\text{Ind}}_{{\stackrel{\sim }{H}}_{\left\{2\right\}}}^{\stackrel{\sim }{H}}\left(ℂv_{s_{2}c}\right),$$

where $ℂv_c$ and $ℂv_{s_{2}c}$ are the two one-dimensional representations of ${\stackrel{\sim }{H}}_{\left\{2\right\}}$ given by

$$\begin{array}{ccc}{T}_2{v}_c=q{v}_c,& X^{{\alpha }_1}{v}_c=v_c,& X^{{\alpha }_2}{v}_c=q^2{v}_c,\\ T_2{v}_{s_{2}c}=-q^{-1}{v}_{s_{2}c},& X^{{\alpha }_1}{v}_{s_{2}c}=q^2{v}_{s_{2}c},& X^{{\alpha }_2}{v}_{s_{2}c}=q^{-2}{v}_{s_{2}c}\text{.}\end{array}$$

These representations must be irreducible since, if not, they would either have a 1-dimensional submodule or a one dimensional quotient. But there are no 1-dimensional modules with central character $t_c\text{.}$

The central characters $t_c$ and $t_d$ are taken into each other under the automorphism of the Dynkin diagram of $A_2$ which switches the two nodes and thus these two central characters will produce modules which have the same structure (up to twisting by the automorphism which switches ${\alpha }_1$ and ${\alpha }_2\text{).}$ Thus the representations with central character $t_d$ can be obtained from the ones with central character $t_c$ by switching all 1’s and 2’s and changing all $c\text{’s}$ to $d\text{’s.}$

Central characters $t_e$ and $t_f\text{:}$ Since $P\left(t_e\right)=\varnothing ,$ Kato’s irreducibility criterion (Theorem 1.16) implies that the principal series module $M\left(t_e\right)$ is irreducible. By Theorem 1.15 this is the only irreducible with central character $t_e\text{.}$ As for the case of $t_c$ and $t_d,$ the central characters $t_e$ and $t_f$ are taken into each other under the automorphism of the Dynkin diagram of $A_2\text{.}$ Thus the irreducible representations with central character $t_f$ can be obtained from the one with central character $t_e$ by switching all 1’s and 2’s and changing all $e\text{’s}$ to $f\text{’s.}$

Central characters $t_o,$ Since $P\left(t_o\right)=\varnothing ,$ Kato’s irreducibility criterion (Theorem 1.16) implies that the principal series module $M\left(t_o\right)$ is irreducible. By Theorem 1.15 this is the only irreducible with central character $t_o\text{.}$

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.

page history