The double affine Weyl group

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

Last update: 16 September 2012

The double affine Weyl group

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

$$⟨,⟩:{𝔥}_{\mathbb{Z}}^{*}\times {𝔥}_{\mathbb{Z}}⟶\frac{1}{e}\mathbb{Z},\text{where}e\in {\mathbb{Z}}_{>0}\text{.}$$

Let $W_0$ be a finite subgroup of $\text{GL}{𝔥}_{\mathbb{Z}}$ generated by reflections and assume that $W_0$ acts on ${𝔥}_{\mathbb{Z}}^{*}$ so that

$$\begin{array}{cc}⟨w\mu ,{\lambda }^{\vee }⟩=⟨\mu ,w^{-1}{\lambda }^{\vee }⟩,\text{where}\text{for}{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}},\mu \in {𝔥}_{\mathbb{Z}}^{*}\text{.}& \text{(1)}\end{array}$$

The double affine Weyl group is

$$\stackrel{\sim }{W}=\{q^k{X}^{\mu }w{Y}^{{\lambda }^{\vee }}\mid k\in \frac{1}{e}\mathbb{Z},\mu \in {𝔥}_{\mathbb{Z}}^{*},w\in W_0\}$$

with

$$\begin{array}{cc}{X}^{\mu }{X}^{\nu }=X^{\mu +\nu }\text{and}{Y}^{{\lambda }^{\vee }}{Y}^{{\sigma }^{\vee }}=Y^{{\lambda }^{\vee }+{\sigma }^{\vee }}& \text{(2)}\\ X^{\mu }{Y}^{{\lambda }^{\mu }}=q^{⟨{\lambda }^{\vee },\mu ⟩}{Y}^{{\lambda }^{\vee }}{X}^{\mu },\text{for}\mu \in {𝔥}_{\mathbb{Z}}^{*},{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}\text{.}& \text{(3)}\\ w{X}^{\mu }=X^{w\mu }w\text{and}w{Y}^{{\lambda }^{\vee }}=Y^{w{\lambda }^{\vee }}w,& \text{(4)}\end{array}$$

for $w\in W_0,$ $\mu \in {𝔥}_{\mathbb{Z}}^{*}$ and ${\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}\text{.}$

DO WE NEED TO SAY THAT we often put

$$q=X^{\delta }=Y^{-d}\text{.}$$

Let

$$s_0=Y^{{\phi }^{\vee }}{s}_{\phi },s_0^{\vee }=X^{\phi }{s}_{{\phi }^{\vee }},s_0^{\prime }=q^{-1}{T}_0^{-1}{X}^{\phi },$$

OR SOMETHING LIKE THAT.

For the moment assume that there is a $W_0$–module isomorphism ${}^{\vee }:{𝔥}_{\mathbb{Z}}\to {𝔥}_{\mathbb{Z}}^{*}$ which we use to identify ${𝔥}_{\mathbb{Z}}$ and ${𝔥}_{\mathbb{Z}}^{*}$ and assume that ${𝔥}_{\mathbb{Z}}={𝔥}_{\mathbb{Z}}^{*}=Q,$ 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

$$W=⟨s_0,s_1,\dots ,s_n⟩,W^{\vee }=⟨s_0^{\vee },s_1,\dots ,s_n⟩,W^{\prime }=⟨s_0^{\prime },s_1,\dots ,s_n⟩$$

affine Weyl subgroups,

$$s_0{s}_0^{\prime }{s}_0^{\vee }{s}_{\phi }\in Z\left(\stackrel{\sim }{W}\right),s_0^{\vee }{s}_1{s}_0{s}_1=s_1{s}_0{s}_1{s}_0^{\vee }\text{if}-⟨{\alpha }_0,{\alpha }_1^{\vee }⟩>1$$

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.

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