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 be a -lattice with an action of a finite subgroup W0 of GL(𝔥) generated by reflections. Then W0 acts on 𝔥* by

wμ,λ= μ,w-1λ, where λ,μ= μ(λ)for λ𝔥,μ 𝔥*. (2.1)

Let R+𝔥* and (R)+𝔥 denote fixed choices of the positive roots and the positive coroots so that the reflections sα in W0 act on 𝔥 and on 𝔥* by

sαλ=λ- λ.α αandsαλ =λ- λ,α α,respectively. (2.2)

The groups

X= { Xμμ 𝔥* } andY= { Yλ λ𝔥 } (2.3)

with

XμXν= Xμ+νand Yλ Yσ= Yλ+σ (2.4)

are the groups 𝔥* and 𝔥 respectively, except written multiplicatively, and the semidirect product

W0(X×Y)= { Xμw Yλ wW0,μ 𝔥*,λ 𝔥 } (2.5)

has additional relations

wXμ= Xwμwandw Yλ= Ywλw, (2.6)

for wW0, μ𝔥* and λ𝔥.

Assume that the action of W0 on 𝔥=𝔥 is irreducible. The double affine Weyl group W is the universal central extension of W0(X×Y). If e is the smallest integer such that λ,μ 1e for all λ𝔥 and μ𝔥* then W is presented by

W= { qkXμw Yλ k1e,μ 𝔥*,λ 𝔥,wW0 }

with (2.4), (2.6) and

XμYλ= qλ,μ YλXμ,for μ𝔥*, λ𝔥. (2.7)

The subgroup { qkXμYλ k1e ,μ𝔥,λ 𝔥 } is a Heisenberg group and

W= { Xμwμ 𝔥,wW0 } andW= { wYλ λ𝔥,w W0 } (2.8)

are affine Weyl groups inside W. Letting

q=Xδ=Y-d (2.9)

and extending the notation of (2.6) gives actions of W on 𝔥*+δ and W on 𝔥d with

Yλμ=μ- μ,λ δandXμ λ=λ- λ,μ d. (2.10)

Let φR be the highest root and φR the highest coroot and let

s0= Yφsφand s0= Xφsφ. (2.11)

Let

α0=-φ+δ, α0=-φ+d, d,μ=0 ,λ,δ =0,d,δ =0, (2.12)

so that

s0μ=μ- μ,α0 α0and s0λ=λ -λ,α0 α0. (2.13)

The alcoves of 𝔥*= 𝔥* are the connected components of

𝔥*\ ( α (R)+, j 𝔥α+jd ) where 𝔥α+jd= { x𝔥* x,α =-j } . (2.14)

The action of W= { Xμwμ 𝔥*,wW0 } on 𝔥* given by

Xμ·ν=ν+μ andw·ν=wν, forwW0,μ 𝔥*andν 𝔥*, (2.15)

sends alcoves to alcoves; s0,,sn are the reclections in the walls 𝔥α0,, 𝔥αn of the fundamental alcove

1 = { x𝔥* x,αi 0,fori=0, 1,,n } ;and (2.16) (v) = (number of hyperplanes between 1 andv) (2.17)

is the length of vW. Let Ω be the set of length zero elements of W. The affine Weyl group W has an alternate presentation by generators s0, s1,, sn and Ω with relations

(si)2=1, si sj mij = sj si mij ,and gsi (g)-1= sσ(i), forgΩ, (2.18)

where π/mij is the angle between 𝔥αi and 𝔥αj and σ denotes the permutation of the 𝔥αi induced by the action of g. If Ω×𝔥* is Ω copies of 𝔥* (sheets), with Ω acting by switching sheets then there is a bijection

W { alcoves inΩ ×𝔥* } (2.19)

and we will often identify vW with the corresponding alcove in Ω×𝔥*. the pictures illustrating this bijection in type SL3 are displayed in the appendix.

the periodic orientation is the orientation of the hyperplanes 𝔥α+kd such that

(a) 1 is on the positive side of 𝔥αfor α (R)+, (b) 𝔥α+kd and𝔥α have parallel orientations. (2.20)

The pictures in the appendix illustrate the periodic orientation for type SL3.

A similar "pictorial" viewpoint applies to the group W acting on Ω×𝔥 where 𝔥=𝔥 and Ω is the set of length zero elements of W. Then W has an alternate presentation by generators s0,s1,,sn and Ω with relations

si2=1, sisjmij= sjsimij, andgsig-1= sσ(i),for gΩ, (2.21)

where π/mij is the angle between 𝔥αi and 𝔥αj and σ denotes the permutation of the 𝔥αi induced by the action of g.

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.

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