## Classification for ${C}_{2}$

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

$⟨α1,α2∨⟩ = -2 ⟨α2,α1∨⟩ = -1, ω1 = α1+α2 ω2 = 12α1+α2, and α1 = 2ω1-2ω2 α2 = -ω1+2ω2.$

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 $\left(𝒯,I\right)$ where $I$ is a subset of $\left\{1,2\right\}$ 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 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 $\left(t,J\right)$ is the corresponding placed skew shape (see Theorem 1.11). The abbreviation ‘nc’ indicates modules that are not calibrated.

$Central P(t) Dim. Langlands Indexing Calibration char. Z(t) parameters triple set J ta {α1,α2} 1 (s1s2s1s2ta,∅) (ta,0,1) ∅ ∅ 3 (s1ta,{1}) (ta,eα1,1) {α1} 3 (s2ta,{2}) (ta,eα2,1) {α2} 1 tempered (ta,eα1+eα2,1) {α1,α2} tb {α1,α1+α2, 3 (tb,{2}) (tb,0,1) nc α1+2α2} 1 (s1tb,{1}) (tb,eα1,1) {α1} {α2} 1 tempered (tb,eα1+α2,-1) {α1,α1+α2} 3 tempered (tb,eα1+α2,1) nc tc {α1,α1+2α2} 2 (tc,{2}) (tc,0,1) ∅ ∅ 2 (s1tc,{1}) (tc,eα1,1) {α1} 2 (s1s2tc,{1}) (tc,eα1+2α2,1) {α1+2α2} 2 tempered (tc,eα1+eα1+2α2,1) {α1,α1+2α2} td {α2,α1+α2} 4 (td,{1}) (td,0,1) nc {α1} 4 (s2td,{2}) (td,eα2,1) nc te {α1} 4 (s2te,{1}) (te,0,1) nc {α1+2α2} 4 tempered (te,eα1,1) nc tf {α1} 4 (tf,∅) (tf,0,1) ∅ ∅ 4 (s1tf,{1}) (tf,eα1,1) {α1} tg {α2} 4 (tg,∅) (tg,0,1) ∅ ∅ 4 (s2tg,{2}) (tg,eα2,1) {α2} Table 5.1. Irreducible (non principal series) representations$

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

$Hβ= {x∈ℝn ∣ ⟨β,x⟩=0} ,andHβ±δ= { x∈ℝn ∣ ⟨β,x⟩= ln(q±2) } ,$

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

$Hα1 Hα2 Hα1+α2 Hα1+2α2 a b,c d e f g Figure 5.1. Real parts of central characters in Table 5.1$

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.

$Hα1 Hα2 Hα1+α2 Hα1+2α2 to s1e s1b s2s1e s1s2s1s2a s1s2s1b,s1s2s1s2c Figure 5.2. Real parts of weights of tempered representations$

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 $\left(n,\rho \right)$ 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.

$Nilpotent orbit ZG(n)/ZG(n)∘ Indexing triple Square integrable W representation regular 1 (ta,eα1+eα2,1) yes (∅,11) subregular ℤ/2ℤ (tb,eα1+α2,1) yes (1,1) (tb,eα1+α2,-1) yes (∅,2) minimal 1 (te,eα1,1) no (11,∅) 0 1 (to,0,1) no (2,∅) Table 5.2. Tempered representations and the Springer correspondence$

The only other tempered representation is the square integrable representation $\left({t}_{c},{e}_{{\alpha }_{1}}+{e}_{{\alpha }_{1}+2{\alpha }_{2}},1\right)\text{.}$ 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 $𝒢\prime \oplus 𝒢\prime \prime \text{.)}$

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

The analysis. A general calculation with the defining relations of $\stackrel{\sim }{H}$ shows that there are only four one dimensional $\stackrel{\sim }{H}\text{-modules}$ $ℂ{v}_{1},$ $ℂ{v}_{2},$ $ℂ{v}_{3},$ and $ℂ{v}_{4}$ 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

$T1v1=qv1, T2v1=qv1, Xα1v1=q2v1, Xα2v1=q2v1, T1v2=qv2, T2v2=-q-1v2, Xα2v2=q2v2, Xα2v1=q-2v2, T1v3=-q-1v3, T2v3=qv3, Xα1v3=q-2v3, Xα2v3=q2v3, T1v4=-q-1v4, T2v4=-q-1v4, Xα1v4=q-2v4, Xα2v4=q-2v4.$

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 $ℂ{v}_{b}$ and $ℂ{v}_{{s}_{1}b}$ be the one dimensional representations of ${\stackrel{\sim }{H}}_{\left\{1\right\}}$ given by

$T1vb=qvb, Xα1vb=q2vb, Xα2vb=vb, T1vs1b=-q-1vs1b, Xα1vs1b=q-2vs1b, Xα2vs1b=q2vs1b.$

Let

$M1=IndH∼{1}H∼ (ℂvb)and M2=IndH∼{1}H∼ (ℂvs1b).$

By Lemma 1.17 these modules have weights $\text{supp}\left({M}_{1}\right)=\left\{{t}_{b},{s}_{1}{t}_{b},{s}_{2}{s}_{1}{t}_{b}\right\}$ and $\text{supp}\left({M}_{2}\right)=\left\{{s}_{1}{t}_{b},{s}_{2}{s}_{1}{t}_{b},{s}_{1}{s}_{2}{s}_{1}{t}_{b}\right\},$ 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 $\left\{{t}_{b},{s}_{1}{t}_{b}\right\}\text{.}$ By Lemma 1.19, any module $P$ with weights $\left\{{t}_{b},{s}_{1}{t}_{b}\right\}$ 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 $ℂ{v}_{d}$ and $ℂ{v}_{{s}_{2}d}$ be the one dimensional representations of ${\stackrel{\sim }{H}}_{\left\{2\right\}}$ given by

$T2vd=qvd, Xα1vd=vd, Xα2vd=q2vd, T2vs2d=-q-1vs2d, Xα1vs2d=q4vs2d, Xα2vs2d=q-2vs2d.$

Let

$M1= IndH∼{2}H∼ (ℂvd) and M2= IndH∼{2}H∼ (ℂvs2d).$

By Lemma 1.17 these modules have weights $\text{supp}\left({M}_{1}\right)=\left\{{t}_{d},{s}_{2}{t}_{d},{s}_{1}{s}_{2}{t}_{d}\right\}$ and $\text{supp}\left({M}_{2}\right)=\left\{{s}_{2}{t}_{d},{s}_{1}{s}_{2}{t}_{d},{s}_{2}{s}_{1}{s}_{2}{t}_{d}\right\},$ 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 $ℂ{v}_{e}$ and $ℂ{v}_{{s}_{1}e}$ be the one dimensional representations of ${\stackrel{\sim }{H}}_{\left\{1\right\}}$ given by

$T1ve=qve, Xα1ve=q2ve, Xα2ve=q-1ve, T1vs1e=-q-1vs1e, Xα1vs1e=q-2vs1e, Xα2vs1e=qvs1e.$

Let

$M1= IndH∼{1}H∼ (ℂve) and M2= IndH∼{1}H∼ (ℂvs1e).$

By Lemma 1.17 these modules have weights $\text{supp}\left({M}_{1}\right)=\left\{{t}_{e},{s}_{2}{t}_{e}\right\}$ and $\text{supp}\left({M}_{2}\right)=\left\{{s}_{1}{t}_{e},{s}_{2}{s}_{1}{t}_{e}\right\}$ 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.