## Root system type ${C}_{2}$

Last update: 26 June 2012

## Root system type ${C}_{2}$

An example is when the lattice $P=ℤ{\epsilon }_{1}+ℤ{\epsilon }_{2}$ with $\left\{{\epsilon }_{1},{\mathrm{&epsilon}}_{2}\right\}$ an orthonormal basis of ${𝔥}_{ℝ}^{*}\cong {ℝ}^{2}$ and $W=\left\{1,{s}_{1},{s}_{2},{s}_{1}{s}_{2},{s}_{2}{s}_{1},{s}_{1}{s}_{2}{s}_{1},{s}_{2}{s}_{1}{s}_{2},{s}_{1}{s}_{2}{s}_{1}{s}_{2}\right\}$ is the dihedral group of order 8 generated by the reflections ${s}_{1}$ and ${s}_{2}$ in the hyperplanes ${H}_{{\alpha }_{1}}$ and ${H}_{{\alpha }_{2}},$ respectively, where $Hα1 = { x∈𝔥ℝ* | ⟨x,ε1⟩=0 }, and Hα2 = { x∈𝔥ℝ* | ⟨x,ε2-ε1⟩ = 0 }.$

This is type ${C}_{2}.$

Define $P+ = P∩C_ and P++ = P∩C$ so that ${P}^{+}$ is a set of representatives of the orbits of the action of $W$ on $P.$ The fundamental weights are the generators ${\omega }_{1},...,{\omega }_{n}$ of the ${ℤ}_{\ge 0}-$module ${P}^{+}$ so that $C = ∑i=1n ℝ≥0 ωi, P+ = ∑i=1n ℤ≥0 ωi, and P++ = ∑i=1n ℤ>0 ωi. (TC2 1)$ The lattice $P$ has $ℤ-$basis ${\omega }_{1},...,{\omega }_{n}$ and the map $P+ → P++ λ ↦ ρ+λ where ρ = ω1+⋯+ωn (TC2 2)$ is a bijection.

In the case of type ${C}_{2}$ the picture is $Hα1 Hα2 C s1C s2C s1s2C s2s1C s1s2s1C s2s1s2C s1s2s1s2C ε2 = ω2 ω1 ε1 0 Hα1 Hα2 C s1C s2C s1s2C s2s1C s1s2s1C s2s1s2C s1s2s1s2C ε2 ε1 ρ The set P+ The set P++$ with $ω1 = ε1+ε2, α1 = 2ε1, α1∨ = ε1, ω2 = ε2, α2 = ε2-ε1, α2∨ = ε2,$ and $R = { ±α1, ±α2, ±(α1+α2), ±(α1+2α2) }.$

Let $⟨\phantom{A},\phantom{A}⟩:{𝔥}_{ℝ}^{*}×{𝔥}_{ℝ}^{*}\to ℝ$ be a nondegenerate $W-$invariant symmetric bilinear form on ${𝔥}_{ℝ}^{*}.$ Any symmetric bilinear form $\left(\phantom{A},\phantom{A}\right):{𝔥}_{ℝ}^{*}×{𝔥}_{ℝ}^{*}\to ℝ$ can be made into a $W-$invariant form $⟨\phantom{A},\phantom{A}⟩$ by defining The simple coroots are ${\alpha }_{1}^{\vee },...,{\alpha }_{n}^{\vee }$ the dual basis to the fundamental weights, $⟨ ωi, αj∨ ⟩ = δij. (TC2 3)$ Define $C∨_ = ∑i=1n ℝ≤0 αi∨ and C∨ = ∑i=1n ℝ<0 αi∨. (TC2 4)$ The dominance order is the partial order on ${𝔥}_{ℝ}^{*}$ given by $μ≤λ if μ∈λ+C∨_. (TC2 5)$

PICTURE

## Notes and References

An alternate reference for this material is [Bou] (Bourbaki Chpt. IV-VI).

## References

[RR] A. Ram and J. Ramagge, Affine Hecke Algebras, http://researchers.ms.unimelb.edu.au/~aram@unimelb/publications.html.