## The double affine Weyl group

Last update: 16 September 2012

## The double affine Weyl group

Let ${𝔥}_{ℤ}$ and ${𝔥}_{ℤ}^{*}$ be $ℤ$–lattices with a $ℤ$–bilinear map

$⟨,⟩:𝔥ℤ* ×𝔥ℤ⟶1eℤ, wheree∈ℤ>0.$

Let ${W}_{0}$ be a finite subgroup of $\text{GL}\phantom{\rule{0.2em}{0ex}}{𝔥}_{ℤ}$ generated by reflections and assume that ${W}_{0}$ acts on ${𝔥}_{ℤ}^{*}$ so that

$⟨wμ,λ∨⟩= ⟨μ,w-1λ∨⟩ ,whereforλ∨ ∈𝔥ℤ,μ∈𝔥ℤ*. (1)$

The double affine Weyl group is

$W∼= { qkXμwYλ∨ ∣k∈1eℤ, μ∈𝔥ℤ*,w∈W0 }$

with

$XμXν=Xμ+ν andYλ∨ Yσ∨= Yλ∨+σ∨ (2) XμYλμ= q⟨λ∨,μ⟩ Yλ∨Xμ, forμ∈𝔥ℤ*, λ∨∈𝔥ℤ. (3) wXμ=Xwμw andwYλ∨= Ywλ∨w, (4)$

for $w\in {W}_{0},$ $\mu \in {𝔥}_{ℤ}^{*}$ and ${\lambda }^{\vee }\in {𝔥}_{ℤ}\text{.}$

DO WE NEED TO SAY THAT we often put

$q=Xδ=Y-d.$

Let

$s0=Yφ∨sφ, s0∨=Xφ sφ∨,s0′ =q-1T0-1 Xφ,$

OR SOMETHING LIKE THAT.

For the moment assume that there is a ${W}_{0}$–module isomorphism ${}^{\vee }:\phantom{\rule{0.2em}{0ex}}{𝔥}_{ℤ}\to {𝔥}_{ℤ}^{*}$ which we use to identify ${𝔥}_{ℤ}$ and ${𝔥}_{ℤ}^{*}$ and assume that ${𝔥}_{ℤ}={𝔥}_{ℤ}^{*}=Q,$ the root lattice. Then $\stackrel{\sim }{W}$ is presented by generators ${s}_{0}^{\prime },{s}_{0},{s}_{0}^{\vee },{s}_{1},\dots ,{s}_{n}$ with

$W=⟨s0,s1,…,sn⟩ ,W∨= ⟨ s0∨,s1, …,sn ⟩ ,W′= ⟨ s0′,s1, …,sn ⟩$

affine Weyl subgroups,

$s0s0′s0∨ sφ∈Z(W∼), s0∨s1s0s1= s1s0s1s0∨ if- ⟨α0,α1∨⟩ >1$

STATE THIS LAST CONDITION BETTER.

## 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.