Classification for $C_2$

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

Last update: 1 April 2013

Classification for $C_2$

The root system $R$ for $C_2$ has simple roots ${\alpha }_1$ and ${\alpha }_2,$ fundamental weights ${\omega }_1$ and ${\omega }_2,$ and

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

Irreducible representations. Table 5.1 lists the irreducible $\stackrel{\sim }{H}\text{-modules}$ by their central characters. We have listed only those central characters t for which the principal series module $M\left(t\right)$ is not irreducible (see Theorem 1.16). 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 5.1. 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 5.1 we have replaced the representation $𝒯$ by the weight $t\text{.}$ The notation for the nilpotent elements in the indexing triples is as in Table 5.2. 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}{cccccc}\text{Central}& P\left(t\right)& \text{Dim.}& \text{Langlands}& \text{Indexing}& \text{Calibration}\\ \text{char.}& Z\left(t\right)& & \text{parameters}& \text{triple}& \text{set}\hspace{0.17em}J\\ \\ t_a& \{{\alpha }_1,{\alpha }_2\}& 1& (s_1{s}_2{s}_1{s}_2{t}_a,\varnothing )& (t_a,0,1)& \varnothing \\ & \varnothing & 3& (s_1{t}_a,\left\{1\right\})& (t_a,e_{{\alpha }_1},1)& \left\{{\alpha }_1\right\}\\ & & 3& (s_2{t}_a,\left\{2\right\})& (t_a,e_{{\alpha }_2},1)& \left\{{\alpha }_2\right\}\\ & & 1& \text{tempered}& (t_a,e_{{\alpha }_1}+e_{{\alpha }_2},1)& \{{\alpha }_1,{\alpha }_2\}\\ \\ t_b& \{{\alpha }_1,{\alpha }_1+{\alpha }_2,& 3& (t_b,\left\{2\right\})& (t_b,0,1)& \text{nc}\\ & {\alpha }_1+2{\alpha }_2\}& 1& (s_1{t}_b,\left\{1\right\})& (t_b,e_{{\alpha }_1},1)& \left\{{\alpha }_1\right\}\\ & \left\{{\alpha }_2\right\}& 1& \text{tempered}& (t_b,e_{{\alpha }_1+{\alpha }_2},-1)& \{{\alpha }_1,{\alpha }_1+{\alpha }_2\}\\ & & 3& \text{tempered}& (t_b,e_{{\alpha }_1+{\alpha }_2},1)& \text{nc}\\ \\ t_c& \{{\alpha }_1,{\alpha }_1+2{\alpha }_2\}& 2& (t_c,\left\{2\right\})& (t_c,0,1)& \varnothing \\ & \varnothing & 2& (s_1{t}_c,\left\{1\right\})& (t_c,e_{{\alpha }_1},1)& \left\{{\alpha }_1\right\}\\ & & 2& (s_1{s}_2{t}_c,\left\{1\right\})& (t_c,e_{{\alpha }_1+2{\alpha }_2},1)& \{{\alpha }_1+2{\alpha }_2\}\\ & & 2& \text{tempered}& (t_c,e_{{\alpha }_1}+e_{{\alpha }_1+2{\alpha }_2},1)& \{{\alpha }_1,{\alpha }_1+2{\alpha }_2\}\\ \\ t_d& \{{\alpha }_2,{\alpha }_1+{\alpha }_2\}& 4& (t_d,\left\{1\right\})& (t_d,0,1)& \text{nc}\\ & \left\{{\alpha }_1\right\}& 4& (s_2{t}_d,\left\{2\right\})& (t_d,e_{{\alpha }_2},1)& \text{nc}\\ \\ t_e& \left\{{\alpha }_1\right\}& 4& (s_2{t}_e,\left\{1\right\})& (t_e,0,1)& \text{nc}\\ & \{{\alpha }_1+2{\alpha }_2\}& 4& \text{tempered}& (t_e,e_{{\alpha }_1},1)& \text{nc}\\ \\ t_f& \left\{{\alpha }_1\right\}& 4& (t_f,\varnothing )& (t_f,0,1)& \varnothing \\ & \varnothing & 4& (s_1{t}_f,\left\{1\right\})& (t_f,e_{{\alpha }_1},1)& \left\{{\alpha }_1\right\}\\ \\ t_g& \left\{{\alpha }_2\right\}& 4& (t_g,\varnothing )& (t_g,0,1)& \varnothing \\ & \varnothing & 4& (s_2{t}_g,\left\{2\right\})& (t_g,e_{{\alpha }_2},1)& \left\{{\alpha }_2\right\}\\ \\ \multicolumn{6}{c}{\text{Table 5.1.}\hspace{0.17em}\text{Irreducible (non principal series) representations}}\end{array}$$

Figure 5.1 displays the real parts of the central characters in Table 5.1. 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 5.1 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

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

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

$$\begin{array}{c} Hα1 Hα2 Hα1+α2 Hα1+2α2 a b,c d e f g \\ \\ \text{Figure 5.1.}\hspace{0.17em}\text{Real parts of central characters in Table 5.1}\end{array}$$

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 5.2.

$$\begin{array}{c} Hα1 Hα2 Hα1+α2 Hα1+2α2 to s1e s1b s2s1e s1s2s1s2a s1s2s1b,s1s2s1s2c \\ \\ \text{Figure 5.2.}\hspace{0.17em}\text{Real parts of weights of tempered representations}\end{array}$$

The irreducible tempered representations with real central character can be indexed by the irreducible representations of the Weyl group $W$ of type $C_2$ (see [BMo1989, p. 34]). Equivalently, these representations 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 $𝔤=\text{Lie}\left(G\right),$ $G$ is the complex simple group over $ℂ$ of type $C_2$ and $\rho$ is an irreducible representation of the component group $Z_G\left(n\right)/Z_G{\left(n\right)}^{\circ }$ (see [Car1985]). For each root $\beta \in R$ let $e_{\beta }$ be an element of the root space ${𝔤}_{\beta }\text{.}$ The four nilpotent orbits in $𝔤$ and the corresponding tempered representations of $\stackrel{\sim }{H}$ are as in Table 5.2. We have used the notation of Carter [Car1985, p.424] to label the irreducible representations of the Weyl group.

$$\begin{array}{ccccc}\text{Nilpotent orbit}& Z_G\left(n\right)/Z_G{\left(n\right)}^{\circ }& \text{Indexing triple}& \text{Square integrable}& W\hspace{0.17em}\text{representation}\\ \text{regular}& 1& (t_a,e_{{\alpha }_1}+e_{{\alpha }_2},1)& \text{yes}& (\varnothing ,11)\\ \text{subregular}& \mathbb{Z}/2\mathbb{Z}& (t_b,e_{{\alpha }_1+{\alpha }_2},1)& \text{yes}& (1,1)\\ & & (t_b,e_{{\alpha }_1+{\alpha }_2},-1)& \text{yes}& (\varnothing ,2)\\ \text{minimal}& 1& (t_e,e_{{\alpha }_1},1)& \text{no}& (11,\varnothing )\\ 0& 1& (t_o,0,1)& \text{no}& (2,\varnothing )\\ \\ \multicolumn{5}{c}{\text{Table 5.2.}\hspace{0.17em}\text{Tempered representations and the Springer correspondence}}\end{array}$$

The only other tempered representation is the square integrable representation $(t_c,e_{{\alpha }_1}+e_{{\alpha }_1+2{\alpha }_2},1)\text{.}$ This representation does not have real central character. It is the representation constructed in [Lus1983-2, 4.14, 4.23]. (In Lusztig’s notation it is the star of the representation corresponding to the graph $𝒢\prime \oplus 𝒢\prime \prime \text{.)}$

$$\begin{array}{ccc} ta s2ta s1s2ta s2s1s2ta s1s2s1s2ta s1s2s1ta s2s1ta s1ta & tb s2tb s1s2tb s2s1s2tb s1s2s1s2tb s1s2s1tb s2s1tb s1tb & tc s2tc s1s2tc s2s1s2tc s1s2s1s2tc s1s2s1tc s2s1tc s1tc \\ td s2td s1s2td s2s1s2td s1s2s1s2td s1s2s1td s2s1td s1td & te s2te s1s2te s2s1s2te s1s2s1s2te s1s2s1te s2s1te s1te & tf s2tf s1s2tf s2s1s2tf s1s2s1s2tf s1s2s1tf s2s1tf s1tf \\ & tg s2tg s1s2tg s2s1s2tg s1s2s1s2tg s1s2s1tg s2s1tg s1tg & \\ \\ \multicolumn{3}{c}{\text{Figure 5.3.}\hspace{0.17em}\text{Calibration graphs for central characters in Table 5.1}}\end{array}$$

The analysis. A general calculation with the defining relations of $\stackrel{\sim }{H}$ shows that there are only four one dimensional $\stackrel{\sim }{H}\text{-modules}$ $ℂv_1,$ $ℂv_2,$ $ℂv_3,$ and $ℂv_4$ which have weights $t_a,$ $s_2{s}_1{t}_b,$ $s_1{t}_b$ and $s_1{s}_2{s}_1{s}_2{t}_a,$ respectively. These modules are given explicitly by

$$\begin{array}{cccc}{T}_1{v}_1=q{v}_1,& T_2{v}_1=q{v}_1,& X^{{\alpha }_1}{v}_1=q^2{v}_1,& X^{{\alpha }_2}{v}_1=q^2{v}_1,\\ T_1{v}_2=q{v}_2,& T_2{v}_2=-q^{-1}{v}_2,& X^{{\alpha }_2}{v}_2=q^2{v}_2,& X^{{\alpha }_2}{v}_1=q^{-2}{v}_2,\\ T_1{v}_3=-q^{-1}{v}_3,& T_2{v}_3=q{v}_3,& X^{{\alpha }_1}{v}_3=q^{-2}{v}_3,& X^{{\alpha }_2}{v}_3=q^2{v}_3,\\ T_1{v}_4=-q^{-1}{v}_4,& T_2{v}_4=-q^{-1}{v}_4,& X^{{\alpha }_1}{v}_4=q^{-2}{v}_4,& X^{{\alpha }_2}{v}_4=q^{-2}{v}_4\text{.}\end{array}$$

Central character $t_a:$ Since $Z\left(t_a\right)=\varnothing ,$ $t_a$ is regular and thus all irreducible representations with central character $t_a$ are calibrated. They are in one to one correspondence with the connected components of the calibration graph and can be constructed with the use of Theorem 1.11. 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-Lusztig (see the discussion in [BMo1989, p.34]).

Central character $t_b:$ We already know from our general computation above, that there are two one-dimensional $\stackrel{\sim }{H}\text{-modules}$ with central character $t_b\text{.}$ One has weight $s_1{t}_b$ and the other has weight $s_2{s}_1{t}_b\text{.}$ Let $ℂv_b$ and $ℂv_{s_{1}b}$ be the one dimensional representations of ${\stackrel{\sim }{H}}_{\left\{1\right\}}$ given by

$$\begin{array}{ccc}{T}_1{v}_b=q{v}_b,& X^{{\alpha }_1}{v}_b=q^2{v}_b,& X^{{\alpha }_2}{v}_b=v_b,\\ T_1{v}_{s_{1}b}=-q^{-1}{v}_{s_{1}b},& X^{{\alpha }_1}{v}_{s_{1}b}=q^{-2}{v}_{s_{1}b},& X^{{\alpha }_2}{v}_{s_{1}b}=q^2{v}_{s_{1}b}\text{.}\end{array}$$

Let

$$M_1={\text{Ind}}_{{\stackrel{\sim }{H}}_{\left\{1\right\}}}^{\stackrel{\sim }{H}}\left(ℂv_b\right)\text{and}{M}_2={\text{Ind}}_{{\stackrel{\sim }{H}}_{\left\{1\right\}}}^{\stackrel{\sim }{H}}\left(ℂv_{s_{1}b}\right)\text{.}$$

By Lemma 1.17 these modules have weights $\text{supp}\left(M_1\right)=\{t_b,s_1{t}_b,s_2{s}_1{t}_b\}$ and $\text{supp}\left(M_2\right)=\{s_1{t}_b,s_2{s}_1{t}_b,s_1{s}_2{s}_1{t}_b\},$ respectively. Both $M_1$ and $M_2$ are 4 dimensional. By Proposition 1.18 (c) one of the two operators ${\tau }_2:{\left(M_1\right)}_{s_2{s}_1{t}_b}\to {\left(M_1\right)}_{s_1{t}_b}$ or ${\tau }_2:{\left(M_1\right)}_{s_1{t}_b}\to {\left(M_1\right)}_{s_2{s}_1{t}_b}$ must have nonzero kernel. This implies that $M_1$ has either a 3 dimensional submodule or a 3 dimensional quotient, call it $N_1,$ with weights $\{t_b,s_1{t}_b\}\text{.}$ By Lemma 1.19, any module $P$ with weights $\{t_b,s_1{t}_b\}$ must have $\text{dim}\left(P_{t_b}^{\text{gen}}\right)\ge 2$ and $\text{dim}\left(P_{s_1{t}_b}^{\text{gen}}\right)\ge 1\text{.}$ It follows that $N_1$ is irreducible. A similar argument can be used to show that $M_2$ has either a 3 dimensional submodule or a 3-dimensional quotient which must be irreducible.

The representation $N_1$ constructed in the previous paragraph and the 1 dimensional representation with weight $s_1{t}_b$ are both tempered. One obtains the corresponding indexing triples by comparing the Langlands parameters for these modules with the labelings of the corresponding representations of $W$ in the Springer correspondence. See [Car1985, p.424], [BMo1989, p. 34] and Table 5.2.

Central character $t_c:$ Since $Z\left(t_c\right)=\varnothing ,$ $t_c$ is regular and thus all irreducible representations with central character $t_c$ are calibrated. They are in one to one correspondence with the connected components of the calibration graph and can be constructed with the use of Theorem 1.11. The Langlands parameters for each module can be determined from its weight structure. The only representation for which the indexing triple cannot be determined from the Langlands parameters and the [KLu0862716] induction theorem (see the discussion in [BMo1989, p. 34]) is the tempered representation. This representation is constructed in [Lus1983-2, 4.14 and 4.23]. In Lusztig’s notation, it is the star (see [Lus1983-2,4.23]) of the representation corresponding to the graph $𝒢\prime \oplus 𝒢\prime \prime \text{.}$ The indexing triple for this representation is given in the discussion for $B_2$ in [Lus1983-2, 2.10].

Central character $t_d:$ Let $ℂv_d$ and $ℂv_{s_{2}d}$ be the one dimensional representations of ${\stackrel{\sim }{H}}_{\left\{2\right\}}$ given by

$$\begin{array}{ccc}{T}_2{v}_d=q{v}_d,& X^{{\alpha }_1}{v}_d=v_d,& X^{{\alpha }_2}{v}_d=q^2{v}_d,\\ T_2{v}_{s_{2}d}=-q^{-1}{v}_{s_{2}d},& X^{{\alpha }_1}{v}_{s_{2}d}=q^4{v}_{s_{2}d},& X^{{\alpha }_2}{v}_{s_{2}d}=q^{-2}{v}_{s_{2}d}\text{.}\end{array}$$

Let

$$M_1={\text{Ind}}_{{\stackrel{\sim }{H}}_{\left\{2\right\}}}^{\stackrel{\sim }{H}}\left(ℂv_d\right)\text{and}{M}_2={\text{Ind}}_{{\stackrel{\sim }{H}}_{\left\{2\right\}}}^{\stackrel{\sim }{H}}\left(ℂv_{s_{2}d}\right)\text{.}$$

By Lemma 1.17 these modules have weights $\text{supp}\left(M_1\right)=\{t_d,s_2{t}_d,s_1{s}_2{t}_d\}$ and $\text{supp}\left(M_2\right)=\{s_2{t}_d,s_1{s}_2{t}_d,s_2{s}_1{s}_2{t}_d\},$ respectively. Both $M_1$ and $M_2$ are 4 dimensional.

Let $M$ be any $\stackrel{\sim }{H}\text{-module}$ such that $M_{t_d}\ne 0\text{.}$ By Lemma 1.19 (a), $\text{dim}\left(M_{t_d}^{\text{gen}}\right)\ge 2\text{.}$ Since $⟨{\alpha }_2{\alpha }_1^{\vee }⟩\ne 0$ it follows from Lemma 1.19 (b) that $\text{dim}\left(M_{s_2{t}_d}^{\text{gen}}\right)\ge 1\text{.}$ Then, by Proposition 1.6, $\text{dim}\left(M_{s_1{s}_2{t}_d}^{\text{gen}}\right)\ge 1\text{.}$ Adding these numbers up we see that $\text{dim}\left(M\right)\ge 4\text{.}$ It follows that $M_1$ is irreducible. An analogous argument can be applied to conclude that $M_2$ is irreducible.

Central character $t_e:$ Let $ℂv_e$ and $ℂv_{s_{1}e}$ be the one dimensional representations of ${\stackrel{\sim }{H}}_{\left\{1\right\}}$ given by

$$\begin{array}{ccc}{T}_1{v}_e=q{v}_e,& X^{{\alpha }_1}{v}_e=q^2{v}_e,& X^{{\alpha }_2}{v}_e=q^{-1}{v}_e,\\ T_1{v}_{s_{1}e}=-q^{-1}{v}_{s_{1}e},& X^{{\alpha }_1}{v}_{s_{1}e}=q^{-2}{v}_{s_{1}e},& X^{{\alpha }_2}{v}_{s_{1}e}=q{v}_{s_{1}e}\text{.}\end{array}$$

Let

$$M_1={\text{Ind}}_{{\stackrel{\sim }{H}}_{\left\{1\right\}}}^{\stackrel{\sim }{H}}\left(ℂv_e\right)\text{and}{M}_2={\text{Ind}}_{{\stackrel{\sim }{H}}_{\left\{1\right\}}}^{\stackrel{\sim }{H}}\left(ℂv_{s_{1}e}\right)\text{.}$$

By Lemma 1.17 these modules have weights $\text{supp}\left(M_1\right)=\{t_e,s_2{t}_e\}$ and $\text{supp}\left(M_2\right)=\{s_1{t}_e,s_2{s}_1{t}_e\}$ respectively. Both $M_1$ and $M_2$ are 4 dimensional.

Let $M$ be any $\stackrel{\sim }{H}\text{-module}$ such that $M_{t_e}\ne 0\text{.}$ By Lemma 1.19 (a) and Proposition 1.6, $\text{dim}\left(M_{t_e}^{\text{gen}}\right)=\text{dim}\left(M_{s_2{t}_e}^{\text{gen}}\right)\ge 2\text{.}$ Thus $\text{dim}\left(M\right)\ge 4\text{.}$ It follows that $M_1$ is irreducible. An analogous argument can be applied to conclude that $M_2$ is irreducible.

Central character $t_f$ and $t_g:$ These cases are handled in the same way as for the central character $t_a\text{.}$

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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