## The double affine braid group

Last update: 19 September 2012

## The double affine braid group

Following Ion-Sahi [IS] the double affine braid group $\stackrel{\sim }{ℬ}$ is generated by ${T}_{0},{T}_{0}^{\prime },{T}_{0}^{\vee },{T}_{1},\dots ,{T}_{n}$ with

$ℬ= ⟨ T0,T1, …,Tn ⟩ ,ℬ′ ⟨ T0′,T1 ,…,Tn ⟩ ,ℬ∨= ⟨ T0∨,T1 ,…,Tn ⟩ ,$

which are affine braid groups,

$q∈Z(ℬ∼), whereq=T0T0′ T0∨Tsφ,$

and

$T0T1-1T0∨ T1=T1-1 T0∨T1T0, if nodes 0 and 1 are connected by a double edge.$

The conversion between presentations is given by

$T0=Yφ∨ Tsφ-1, T0∨=Xφ Tsφ,q=T0 T0′T0∨ Tsφ$

We should extend this to include the effect of $\Omega$ by using the relations

$g=Yωg Tw0wg-1 andg∨= XωgTw0wg,$

or whatever the correct versions of these are.

The braid group ${𝒜}_{3}$ on 3 strands is generated by ${a}_{1},{a}_{2}$ with relation ${a}_{1}{a}_{2}{a}_{1}={a}_{2}{a}_{1}{a}_{2}\text{.}$ Using the automorphism of the Dynkin diagram PICTURE build

$𝒜3⋊ℤ/2ℤ= { be∣b∈ 𝒜3,e∈ℤ/2ℤ } ,withea1=a2 e,ea2=a1e, e2=1.$

The isomorphism

$𝒜3⋊ℤ/2ℤ ≅ GL2(ℤ) a1 ⟼ ( 1 1 0 1 ) a2 ⟼ ( 1 0 -1 1 ) e ⟼ ( 0 1 1 0 )$

giving exact sequences

${1}⟶ ⟨(a1a2a1)4⟩ ⟶𝒜3⟶SL2(ℤ) ⟶{1}$ ${1}⟶Z(𝒜3)= ⟨(a1a2a1)2⟩ ⟶𝒜3⟶PSL2(ℤ) ⟶{1}$

The group $G{L}_{2}\left(ℤ\right)$ acts on $\stackrel{\sim }{ℬ}$ by automorphisms via

$a1: ℬ∼ ⟶ ℬ∼ T0 ⟼ T0′ T0′ ⟼ (T0′)-1 T0T0′ T0∨ ⟼ T0∨ Ti ⟼ Ti a2: ℬ∼ ⟶ ℬ∼ T0 ⟼ T0 T0′ ⟼ T0∨ T0∨ ⟼ (T0′)-1 T0′T0∨ Ti ⟼ Ti$ $e ℬ∼ ⟶ ℬ∼ T0 ⟼ (T0∨)-1 T0′ ⟼ (T0′)-1 T0∨ ⟼ (T0)-1 Ti ⟼ Ti-1 Xμ ⟼ Yμ$

as the automorphism of $\stackrel{\sim }{cB}\text{.}$ The existence of the automorphism $e$ is sometimes called duality.

The affine braid group ${ℬ}^{\vee }$ is given by ${T}_{0}^{\vee },{T}_{0}^{\vee },\dots ,{T}_{n}^{\vee }$ and ${\Omega }^{\vee }$ with relations

$Ti∨Tj∨… ⏟mij∨ = Tj∨Ti∨… ⏟mij∨ ,and g∨Ti∨ (g∨)-1= Tg(i)∨, forg∨∈Ω∨. (1)$

The affine Weyl group

$W∨= { Xμw∣μ∈ 𝔥ℤ*,w∈W0 } acts onY∼= { qk/e Yλ∨∣ k∈ℤ,λ∨∈𝔥ℤ } , (2)$

by conjugation. Write

$Yvλ∨=v Yλ∨v-1, forv∈W∨, λ∨∈𝔥ℤ. (3)$

The double affine braid group $\stackrel{\sim }{B}$ is the group generated by ${B}^{\vee }$ and $\stackrel{\sim }{Y}$ with relations

$Ti∨Tj∨… ⏟mij∨ = Tj∨Ti∨… ⏟mij∨ , g∨Ti∨ (g∨)-1= Tg(i)∨, g∨Yλ∨= Yg∨λ∨ g∨, (4)$ $(Ti∨)-1 Yλ∨= { Ysi∨λ∨ (Ti∨)-1 , if ⟨λ∨,αi⟩ =0 , Ysi∨λ∨ Ti∨ , if ⟨λ∨,αi⟩ =1 , (5)$

for ${g}^{\vee }\in {\Omega }^{\vee },{\lambda }^{\vee }\in {𝔥}_{ℤ}$ and $i=0,1,\dots ,n\text{.}$

For $w\in {W}^{\vee },$ view a reduced word $w=g{s}_{{i}_{1}}\dots {s}_{{i}_{\ell }}$ as a minimal length path $p$ from the fundamental alcove to $w$ in ${𝔥}_{ℝ}$ and define

with respect to the periodic orientation (see (??) and the pictures in the appendix). For $v\in W,$ view a reduced word $v={g}^{\vee }{s}_{{i}_{\ell }}^{\vee }\dots {s}_{{i}_{\ell }}^{\vee }$ as a minimal length path ${p}^{\vee }$ from the fundamental alcove to $v$ in ${𝔥}_{ℝ}^{*}$ and define

Let ${T}_{i}^{\vee }={T}_{i},$ for $i=1,2,\dots ,n,$

$g∨=Xωg Twgw0∨, (T0∨)-1= XφTsφ∨, g=Yωg∨ Tw0wg-1, T0=Yφ∨ Tsφ-1. (8)$

where $\phi$ and ${\phi }^{\vee }$ are as in (??) and, using the action in (2.7), ${\omega }_{g}={g}^{\vee }·0$ and ${w}_{g}$ is the longest element of the stabilizer of ${\omega }_{g}$ in ${W}_{0}\text{.}$

The following theorem, discovered by Cherednik [1, Thm. 2.2], is proved in [15, 3.5-3.7], in [7], and in [6, 4.13-4.18].

(Duality) Let ${Y}^{d}={q}^{-1}\text{.}$ The double affine braid group $\stackrel{\sim }{ℬ}$ is generated by ${T}_{0}^{\vee },{T}_{1}^{\vee },\dots ,{T}_{n}^{\vee },{\Omega }^{\vee }$ and $Y$ with relations

$Ti∨Tj∨… ⏟mij∨ = Tj∨Ti∨… ⏟mij∨ , g∨Ti∨ (g∨)-1= Tσ∨(i)∨, g∨Yλ∨= Yg∨λ∨ g∨, (2.28)$

for ${g}^{\vee }\in {\Omega }^{\vee },$ and

$Ti∨= Ysi∨λ∨ Ti∨ , if ⟨λ∨,αi⟩ =0 , (Ti∨) -1 Yλ∨ (Ti∨) -1 =Ysi∨λ∨ , if ⟨λ∨,αi⟩ =1 , fori=0,1, …,n, (2.29)$

where the action of $W$ on ${𝔥}_{ℤ}\oplus ℤd$ is as in (2.10).

## Notes and References

This page is taken from a paper entitled Relating double affine Hecke algebras and Rational Cherednik algebras by Stephen Griffeth and Arun Ram, May 4, 2009. (2.7) is a reference to the section entitled The double affine Weyl group.