Double Weyl groups, braid groups and Hecke algebras

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

Last update: 18 February 2013

Double Weyl groups, braid groups and Hecke algebras

In this section we review the basic definitions and notations for affine Weyl groups and double affine Hecke algebras following the expositions in [Ram0601343], [Che1992], [Mac1976581] and [Hai2275709]. Following the definitions we prove Theorem 2.2, a formula for the expansion of products of intertwining operators in the DAHA. This formula is a “lift into the DAHA” of the expansions of Macdonald polynomials given in Section 3.

Double affine Weyl groups

Let ${𝔥}_{\mathbb{Z}}$ be a $\mathbb{Z}\text{-lattice}$ with an action of a finite subgroup $W_0$ of $GL\left({𝔥}_{\mathbb{Z}}\right)$ generated by reflections. Then $W_0$ acts on ${𝔥}_{\mathbb{Z}}^{*}$ by

$$\begin{array}{cc}⟨w\mu ,{\lambda }^{\vee }⟩=⟨\mu ,w^{-1}{\lambda }^{\vee }⟩,\text{where}⟨{\lambda }^{\vee },\mu ⟩=\mu \left({\lambda }^{\vee }\right)\text{for}\hspace{0.17em}{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}},\mu \in {𝔥}_{\mathbb{Z}}^{*}\text{.}& \text{(2.1)}\end{array}$$

Let $R^{+}\subseteq {𝔥}_{\mathbb{Z}}^{*}$ and ${\left(R^{\vee }\right)}^{+}\subseteq {𝔥}_{\mathbb{Z}}$ denote fixed choices of the positive roots and the positive coroots so that the reflections $s_{\alpha }$ in $W_0$ act on ${𝔥}_{\mathbb{Z}}$ and on ${𝔥}_{\mathbb{Z}}^{*}$ by

$$\begin{array}{cc}{s}_{\alpha }\lambda =\lambda -⟨\lambda \text{.}{\alpha }^{\vee }⟩\alpha \text{and}{s}_{\alpha }{\lambda }^{\vee }={\lambda }^{\vee }-⟨{\lambda }^{\vee },\alpha ⟩{\alpha }^{\vee },\text{respectively.}& \text{(2.2)}\end{array}$$

The groups

$$\begin{array}{cc}X=\{X^{\mu }\hspace{0.17em}\mid \hspace{0.17em}\mu \in {𝔥}_{\mathbb{Z}}^{*}\}\text{and}Y=\{Y^{{\lambda }^{\vee }}\hspace{0.17em}\mid \hspace{0.17em}{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}\}& \text{(2.3)}\end{array}$$

with

$$\begin{array}{cc}{X}^{\mu }{X}^{\nu }=X^{\mu +\nu }\text{and}{Y}^{{\lambda }^{\vee }}{Y}^{{\sigma }^{\vee }}=Y^{{\lambda }^{\vee }+{\sigma }^{\vee }}& \text{(2.4)}\end{array}$$

are the groups ${𝔥}_{\mathbb{Z}}^{*}$ and ${𝔥}_{\mathbb{Z}}$ respectively, except written multiplicatively, and the semidirect product

$$\begin{array}{cc}{W}_0⋉(X\times Y)=\{X^{\mu }w{Y}^{{\lambda }^{\vee }}\hspace{0.17em}\mid \hspace{0.17em}w\in W_0,\mu \in {𝔥}_{\mathbb{Z}}^{*},{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}\}& \text{(2.5)}\end{array}$$

has additional relations

$$\begin{array}{cc}w{X}^{\mu }=X^{w\mu }w\text{and}w{Y}^{{\lambda }^{\vee }}=Y^{w{\lambda }^{\vee }}w,& \text{(2.6)}\end{array}$$

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

Assume that the action of $W_0$ on ${𝔥}_{ℂ}=ℂ{\otimes }_{\mathbb{Z}}{𝔥}_{\mathbb{Z}}$ is irreducible. The double affine Weyl group $\stackrel{\sim }{W}$ is the universal central extension of $W_0⋉(X\times Y)\text{.}$ If $e$ is the smallest integer such that $⟨{\lambda }^{\vee },\mu ⟩\in \frac{1}{e}\mathbb{Z}$ for all ${\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}$ and $\mu \in {𝔥}_{\mathbb{Z}}^{*}$ then $\stackrel{\sim }{W}$ is presented by

$$\stackrel{\sim }{W}=\{q^k{X}^{\mu }w{Y}^{{\lambda }^{\vee }}\hspace{0.17em}\mid \hspace{0.17em}k\in \frac{1}{e}\mathbb{Z},\mu \in {𝔥}_{\mathbb{Z}}^{*},{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}},w\in W_0\}$$

with (2.4), (2.6) and

$$\begin{array}{cc}{X}^{\mu }{Y}^{{\lambda }^{\vee }}=q^{⟨{\lambda }^{\vee },\mu ⟩}{Y}^{{\lambda }^{\vee }}{X}^{\mu },\text{for}\hspace{0.17em}\mu \in {𝔥}_{\mathbb{Z}}^{*},{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}\text{.}& \text{(2.7)}\end{array}$$

The subgroup $\{q^k{X}^{\mu }{Y}^{{\lambda }^{\vee }}\hspace{0.17em}\mid \hspace{0.17em}k\in \frac{1}{e}\mathbb{Z},\mu \in {𝔥}_{\mathbb{Z}},{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}\}$ is a Heisenberg group and

$$\begin{array}{cc}W=\{X^{\mu }w\hspace{0.17em}\mid \hspace{0.17em}\mu \in {𝔥}_{\mathbb{Z}},w\in W_0\}\text{and}{W}^{\vee }=\{w{Y}^{{\lambda }^{\vee }}\hspace{0.17em}\mid \hspace{0.17em}{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}},w\in W_0\}& \text{(2.8)}\end{array}$$

are affine Weyl groups inside $\stackrel{\sim }{W}\text{.}$ Letting

$$\begin{array}{cc}q=X^{\delta }=Y^{-d}& \text{(2.9)}\end{array}$$

and extending the notation of (2.6) gives actions of $W^{\vee }$ on ${𝔥}_{\mathbb{Z}}^{*}+\mathbb{Z}\delta$ and $W$ on ${𝔥}_{\mathbb{Z}}\oplus \mathbb{Z}d$ with

$$\begin{array}{cc}{Y}^{{\lambda }^{\vee }}\mu =\mu -⟨\mu ,{\lambda }^{\vee }⟩\delta \text{and}{X}^{\mu }{\lambda }^{\vee }={\lambda }^{\vee }-⟨{\lambda }^{\vee },\mu ⟩d\text{.}& \text{(2.10)}\end{array}$$

Let $\phi \in R$ be the highest root and ${\phi }^{\vee }\in R^{\vee }$ the highest coroot and let

$$\begin{array}{cc}{s}_0=Y^{{\phi }^{\vee }}{s}_{\phi }\text{and}{s}_0^{\vee }=X^{\phi }{s}_{{\phi }^{\vee }}\text{.}& \text{(2.11)}\end{array}$$

Let

$$\begin{array}{cc}{\alpha }_0=-\phi +\delta ,{\alpha }_0^{\vee }=-{\phi }^{\vee }+d,⟨d,\mu ⟩=0,⟨{\lambda }^{\vee },\delta ⟩=0,⟨d,\delta ⟩=0,& \text{(2.12)}\end{array}$$

so that

$$\begin{array}{cc}{s}_0\mu =\mu -⟨\mu ,{\alpha }_0^{\vee }⟩{\alpha }_0\text{and}{s}_0^{\vee }{\lambda }^{\vee }={\lambda }^{\vee }-⟨{\lambda }^{\vee },{\alpha }_0⟩{\alpha }_0^{\vee }\text{.}& \text{(2.13)}\end{array}$$

The alcoves of ${𝔥}_{ℝ}^{*}=ℝ{\otimes }_{\mathbb{Z}}{𝔥}_{\mathbb{Z}}^{*}$ are the connected components of

$$\begin{array}{cc}{𝔥}_{ℝ}^{*}\backslash \left(\bigcup _{{\alpha }^{\vee }\in {\left(R^{\vee }\right)}^{+},j\in \mathbb{Z}}{𝔥}^{{\alpha }^{\vee }+jd}\right)\text{where}{𝔥}^{{\alpha }^{\vee }+jd}=\{x\in {𝔥}_{ℝ}^{*}\hspace{0.17em}\mid \hspace{0.17em}⟨x,{\alpha }^{\vee }⟩=-j\}\text{.}& \text{(2.14)}\end{array}$$

The action of $W=\{X^{\mu }w\hspace{0.17em}\mid \hspace{0.17em}\mu \in {𝔥}_{\mathbb{Z}}^{*},w\in W_0\}$ on ${𝔥}_{ℝ}^{*}$ given by

$$\begin{array}{cc}{X}^{\mu }·\nu =\nu +\mu \text{and}w·\nu =w\nu ,\text{for}\hspace{0.17em}w\in W_0,\mu \in {𝔥}_{\mathbb{Z}}^{*}\hspace{0.17em}\text{and}\hspace{0.17em}\nu \in {𝔥}_{ℝ}^{*},& \text{(2.15)}\end{array}$$

sends alcoves to alcoves; $s_0^{\vee },\dots ,s_n^{\vee }$ are the reclections in the walls ${𝔥}^{{\alpha }_0^{\vee }},\dots ,{𝔥}^{{\alpha }_n^{\vee }}$ of the fundamental alcove

$$\begin{array}{cccc}1& =& \{x\in {𝔥}_{ℝ}^{*}\hspace{0.17em}\mid \hspace{0.17em}⟨x,{\alpha }_i^{\vee }⟩\ge 0,\hspace{0.17em}\text{for}\hspace{0.17em}i=0,1,\dots ,n\};\text{and}& \text{(2.16)}\\ \ell \left(v\right)& =& \text{(number of hyperplanes between 1 and}\hspace{0.17em}v\text{)}& \text{(2.17)}\end{array}$$

is the length of $v\in W\text{.}$ Let ${\Omega }^{\vee }$ be the set of length zero elements of $W\text{.}$ The affine Weyl group $W$ has an alternate presentation by generators $s_0^{\vee },s_1^{\vee },\dots ,s_n^{\vee }$ and ${\Omega }^{\vee }$ with relations

$$\begin{array}{cc}{\left(s_i^{\vee }\right)}^2=1,\underset{\underset{m_{ij}^{\vee }}{⏟}}{s_i^{\vee }{s}_j^{\vee }\dots }=\underset{\underset{m_{ij}^{\vee }}{⏟}}{s_j^{\vee }{s}_i^{\vee }\dots },\text{and}{g}^{\vee }{s}_i^{\vee }{\left(g^{\vee }\right)}^{-1}=s_{{\sigma }^{\vee }\left(i\right)}^{\vee },\text{for}\hspace{0.17em}{g}^{\vee }\in {\Omega }^{\vee },& \text{(2.18)}\end{array}$$

where $\pi /m_{ij}^{\vee }$ is the angle between ${𝔥}^{{\alpha }_i^{\vee }}$ and ${𝔥}^{{\alpha }_j^{\vee }}$ and ${\sigma }^{\vee }$ denotes the permutation of the ${𝔥}^{{\alpha }_i^{\vee }}$ induced by the action of $g^{\vee }\text{.}$ If ${\Omega }^{\vee }\times {𝔥}_{ℝ}^{*}$ is $\mid {\Omega }^{\vee }\mid$ copies of ${𝔥}_{ℝ}^{*}$ (sheets), with ${\Omega }^{\vee }$ acting by switching sheets then there is a bijection

$$\begin{array}{cc}W⟷\{\text{alcoves in}\hspace{0.17em}{\Omega }^{\vee }\times {𝔥}_{ℝ}^{*}\}& \text{(2.19)}\end{array}$$

and we will often identify $v\in W$ with the corresponding alcove in ${\Omega }^{\vee }\times {𝔥}_{ℝ}^{*}\text{.}$ the pictures illustrating this bijection in type $S{L}_3$ are displayed in the appendix.

the periodic orientation is the orientation of the hyperplanes ${𝔥}^{{\alpha }^{\vee }+kd}$ such that

$$\begin{array}{cc}\begin{array}{c}\text{(a) 1 is on the positive side of}\hspace{0.17em}{𝔥}^{{\alpha }^{\vee }}\hspace{0.17em}\text{for}\hspace{0.17em}{\alpha }^{\vee }\in {\left(R^{\vee }\right)}^{+},\\ \text{(b)}\hspace{0.17em}{𝔥}^{{\alpha }^{\vee }+kd}\hspace{0.17em}\text{and}\hspace{0.17em}{𝔥}^{{\alpha }^{\vee }}\hspace{0.17em}\text{have parallel orientations.}\end{array}& \text{(2.20)}\end{array}$$

The pictures in the appendix illustrate the periodic orientation for type $S{L}_3\text{.}$

A similar "pictorial" viewpoint applies to the group $W^{\vee }$ acting on $\Omega \times {𝔥}_{ℝ}$ where ${𝔥}_{ℝ}=ℝ{\otimes }_{\mathbb{Z}}{𝔥}_{\mathbb{Z}}$ and $\Omega$ is the set of length zero elements of $W^{\vee }\text{.}$ Then $W^{\vee }$ has an alternate presentation by generators $s_0,s_1,\dots ,s_n$ and $\Omega$ with relations

$$\begin{array}{cc}{s}_i^2=1,\underset{\underset{m_{ij}}{⏟}}{s_i{s}_j\dots }=\underset{\underset{m_{ij}}{⏟}}{s_j{s}_i\dots },\text{and}g{s}_i{g}^{-1}=s_{\sigma \left(i\right)},\text{for}\hspace{0.17em}g\in \Omega ,& \text{(2.21)}\end{array}$$

where $\pi /m_{ij}$ is the angle between ${𝔥}^{{\alpha }_i}$ and ${𝔥}^{{\alpha }_j}$ and $\sigma$ denotes the permutation of the ${𝔥}^{{\alpha }_i}$ induced by the action of $g\text{.}$

Notes and References

This is an excerpt from a paper entitled A combinatorial formula for Macdonald polynomials authored by Arun Ram and Martha Yip. It was dedicated to Adriano Garsia.

page history