Double affine braid groups and Hecke algebras of classical type

Last update: 12 March 2012

Introduction

I thank M. Noumi for providing me with a copy of his article [N95].

Type $\left({C}_{n}^{\vee },{C}_{n}\right),$ for $n\ge 2$

Use a graphical notation for relations so that

The double affine braid group

The double affine braid group is the group ${\stackrel{˜}{𝔅}}_{n}$ is generated by ${q}^{\frac{1}{2}},{T}_{1},...,{T}_{n}$ and ${T}_{0},{T}_{0}^{\prime },{T}_{0}^{\vee }$ with relations $T0 T1 T2 Tn-2 Tn-1 Tn T0∨ T1 T2 Tn-2 Tn-1 Tn T0′ T1 T2 Tn-2 Tn-1 Tn (Daff 2.2)$ There are redundant relations in this presentation of ${\stackrel{˜}{𝔅}}_{n}.$ In particular, the second relation in (Daff 2.1) implies ${T}_{1}{T}_{0}^{\vee }{T}_{1}^{-1}{T}_{0}={T}_{0}{T}_{1}{T}_{0}^{-1}{T}_{1},$ giving that ${T}_{0}$ commutes with ${T}_{1}{T}_{0}^{\vee }{T}_{1}^{-1}$ and thus that ${T}_{0}^{-1}$ commutes with ${T}_{1}{\left({T}_{0}^{\vee }\right)}^{-1}{T}_{1}^{-1}.$ But this is the last relation in (Daff 2.1).

The elements of the braid group on $n+3$ strands given by $T0 = and Tn = T0∨ = and Ti = i i+1 ,$ satisfy the relations in (Daff 2.1) and (Daff 2.2).

For $j= 1,...,n$ define ${Y}^{{\epsilon }_{j}^{\vee }}$ and ${X}^{{\epsilon }_{j}}$ in ${\stackrel{˜}{𝔅}}_{n}$ by $Yε1∨ = T0T1⋯Tn⋯T1 and Yεj+1∨ = Tj-1 Yεj∨ Tj-1, Xε1 = (T0∨)-1T1-1⋯Tn-1⋯T1-1 and Xεj+1 = TjXεjTj, (Daff 2.3)$ for $j= 1,...,n-1 .$ Pictorially, $Yεj∨ = j and Xεj = j .$ The pictorial viewpoint makes it straightforward to verify that the subgroups $X= ⟨Xε1,...,Xεn⟩ and Y= ⟨Yε1∨,...,Yεn∨⟩ are abelian.$

In ${\stackrel{˜}{𝔅}}_{n}$ let ${T}_{{s}_{{\epsilon }_{1}}}={T}_{1}\cdots {T}_{n}\cdots {T}_{1}$ and $T0′ = q-12 Xε1 T0-1 = q-12 (T0∨)-1 T1-1 ⋯ Tn-1 ⋯ T1-1 T0-1 = q-12 (T0∨)-1 Y-ε1∨ (Daff 2.4)$ DRAW A PICTURE OF THIS ELEMENT? Then, with ${T}_{{s}_{{\epsilon }_{1}}}={T}_{1}\cdots {T}_{n}\cdots {T}_{1},$ $T0′ T1 T2 Tn-2 Tn-1 Tn and T0T0′T0∨Tsε1 = q-12,$ since

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The affine braid groups ${𝔅}_{n},{𝔅}_{n}^{\prime }$ and ${𝔅}_{n}^{\vee }$ are the subgroups of ${\stackrel{˜}{𝔅}}_{n}$ given by $𝔅n = ⟨T0,T1,...,Tn⟩, 𝔅n′ = ⟨T0′,T1,...,Tn⟩ and 𝔅n∨ = ⟨T0∨,T1,...,Tn∨⟩.$

1. The double affine braid group ${\stackrel{˜}{𝔅}}_{n}$ is generated by ${q}^{\frac{1}{2}},$ the affine braid group ${𝔅}_{n}^{\vee },$ and the abelian group $Y$ with additional relations and
2. The double affine braid group ${\stackrel{˜}{𝔅}}_{n}$ is generated by ${q}^{\frac{1}{2}},$ the affine braid group ${𝔅}_{n},$ and the abelian group $X$ with additional relations ${q}^{\frac{1}{2}}\in Z\left({\stackrel{˜}{𝔅}}_{n}\right),$ and

 Proof. (Daff 2.1) $⇒$ (a): The relations (Daff 2.5) for $j\ne 0$ follow easily from the pictorial expression for ${Y}^{{\epsilon }_{j}^{\vee }}$ and the additional relations (Daff 2.5) for $j=0$ follow from the definition of ${Y}^{{\epsilon }_{j}^{\vee }}$ and the relation ${T}_{0}^{\vee }{T}_{1}^{-1}{T}_{0}{T}_{1}={T}_{1}^{-1}{T}_{0}{T}_{1}{T}_{0}^{\vee }.$ (a) $⇒$ (Daff 2.1): Since $T0 = Yε1∨T1-1 ⋯ Tn-1-1Tn-1Tn-1-1 ⋯ T1-1,$ the generators in (Daff 2.1) can be written in terms of the generators in (a). We need to show that the relations (Daff 2.5) imply Pictorially, $Tsφ = = T1⋯Tn⋯T1,$ so that ${T}_{i}{T}_{{s}_{\phi }}^{-1}={T}_{{s}_{\phi }}^{-1}{T}_{i},$ for $i=2,3,...,n.$ Thus, Using and ${T}_{0}{T}_{i}={T}_{i}{T}_{0}$ for $i=2,...,n,$ $T0∨ T1-1 T0T1 = T0∨ T1-1 Yε1∨ T1-1 ⋯ Tn-1 ⋯ T1-1 T1 = T0∨ Yε2∨ T2-1 ⋯ Tn-1 ⋯ T2-1 = Yε2∨ T0∨ T2-1 ⋯ Tn-1 ⋯ T2-1 = T1-1 Yε1∨ T1-1 T2-1 ⋯ Tn-1 ⋯ T2-1 T0∨ = T1-1 T0T1 T0∨.$ Since ${Y}^{{\epsilon }_{1}^{\vee }}={T}_{0}{T}_{1}\cdots {T}_{n}\cdots {T}_{1}$ and ${Y}^{{\epsilon }_{2}^{\vee }}={T}_{1}^{-1}{Y}^{{\epsilon }_{1}^{\vee }}{T}_{1}^{-1}={T}_{1}^{-1}{T}_{0}{T}_{1}\cdots {T}_{n}\cdots {T}_{2}$ we have $Yε1∨+ε2∨ = T0T1 ⋯ Tn ⋯ T2T0T1 ⋯ Tn ⋯ T2 = T0T1T0T2 ⋯ Tn ⋯ T2T1 ⋯ Tn ⋯ T2.$ Then ${T}_{1}^{-1}{Y}^{{\epsilon }_{1}^{\vee }+{\epsilon }_{2}^{\vee }}={T}_{1}^{-1}{Y}^{{\epsilon }_{1}^{\vee }}{Y}^{{\epsilon }_{2}^{\vee }}={T}_{1}^{-1}{Y}^{{\epsilon }_{1}^{\vee }}{T}_{1}^{-1}{Y}^{{\epsilon }_{1}^{\vee }}{T}_{1}^{-1}={Y}^{{\epsilon }_{2}^{\vee }}{Y}^{{\epsilon }_{1}^{\vee }}{T}_{1}^{-1}={Y}^{{\epsilon }_{2}^{\vee }+{\epsilon }_{1}^{\vee }}{T}_{1}^{-1}$ gives $T1 Yε1∨+ε2∨ = T1T0T1T0T2 ⋯ Tn ⋯ T2T1 ⋯ Tn ⋯ T2 = Yε1∨+ε2∨ T1 = T0T1T0T2 ⋯ Tn ⋯ T2Tsφ = T0T1T0TsφT2 ⋯ Tn ⋯ T2 = T0T1T0T1T2 ⋯ Tn ⋯ T2T1 ⋯ Tn ⋯ T2,$ since ${T}_{{s}_{\phi }}{T}_{i}={T}_{i}{T}_{{s}_{\phi }}$ for $i=2,...,n.$ thus ${T}_{1}{T}_{0}{T}_{1}{T}_{0}={T}_{0}{T}_{1}{T}_{0}{T}_{1}.$ (b) Since $\iota \left({Y}^{{\epsilon }_{i}^{\vee }}\right)={X}^{{\epsilon }_{i}},$ applying the automorphism $\iota$ to the presentation in (a) gives the presentation in (b). $\square$

Letting (see [M03, (1.4.3)]) the relations (Daff 2.5) are equivalent to $Y$ being abelian and The last relation is vacuous for $i=n$ since $⟨{\lambda }^{\vee },2{\epsilon }_{n}⟩\in 2ℤ$ for all ${\lambda }^{\vee }\in {𝔥}_{ℤ}.$

The presentation of the double affine braid group given in (???) is implicit in [H06, ???] and quite explicit in [?, ???].

Automorphisms of the double affine braid group

(Duality) [M03, (3.5.1)] There is an involutive automorphism $\iota :{\stackrel{˜}{𝔅}}_{n}\to {\stackrel{˜}{𝔅}}_{n}$ given by

 Proof. The involution $\iota$ fixes the first relation in (Daff 2.1), switches the second and third relations in (Daff 2.1), and switches the relations in (Daff 2.2). $\square$

Note that

The double affine Weyl group

Type $\left({C}_{n}^{\vee },{C}_{n}\right)$

Let $𝔥ℤ = ∑i=1n ℤεi∨ and 𝔥ℤ* = ∑i=1n ℤεi, with ⟨εi∨,εj⟩ = δij.$ The Weyl group ${W}_{0}$ is generated by ${s}_{1},...,{s}_{n}$ with $s1 s2 sn-2 sn-1 sn and si2=1, (Daff 3.1)$ for $i=1,...,n.$ The group ${W}_{0}$ acts on ${𝔥}_{ℤ}$ by Then ${W}_{0}$ acts on ${𝔥}_{ℤ}^{*}$ by setting

The double affine Weyl group is with $q12∈ Z(W˜), wXμ = Xwμw, wYλ∨ = Ywλ∨w, (Daff 3.5)$ for $\mu ,\nu \in {𝔥}_{ℤ}^{*},{\lambda }^{\vee },{\sigma }^{\vee }\in {𝔥}_{ℤ},w\in {W}_{0}.$

Noting that define $s0 = sε1 Y-ε1∨ , s0∨ = Xε1 sε1 , s0′ = q-12 Xε1 sε1 Y-ε1∨ . (Daff 3.8)$ The following proposition shows that $\stackrel{˜}{W}$ is a quotient of that double affine braid group ${\stackrel{˜}{𝔅}}_{n}.$

The group $\stackrel{˜}{W}$ is presented by generators ${q}^{\frac{1}{2}},{s}_{0}^{\vee },{s}_{0},{s}_{1},...,{s}_{n}$ and relations

 Proof. With definitions of ${s}_{0},{s}_{0}^{\vee }$ and ${s}_{0}^{\prime }$ as in (Daff 3.8) it is straightforward to check that the relations of Proposition 3.1 hold in $\stackrel{˜}{W}.$ Since $Xε1 = s0∨ s1⋯sn⋯s1, and Xεj+1 = sj Xεj sj, and Yε1∨ = s0 s1⋯sn⋯s1, and Yεj+1∨ = sj Yεj∨ sj, (Daff 3.10)$ the generators of $\stackrel{˜}{W}$ can be written in terms of ${q}^{\frac{1}{2}},{s}_{0}^{\vee },{s}_{0},{s}_{1},...,{s}_{n} .$ Theorem 2.1(a) shows that the group presented in Proposition 3.1 satisfies the relations Similarly Theorem 2.1(b) shows that the group presented in Proposition 3.1 satisfies the relations The relations in (Daff 3.11) and (Daff 3.14) imply the first relation in (Daff 3.5) and the first two relations of (Daff 3.6). The relation ${s}_{0}^{2}=1$ gives ${s}_{{\epsilon }_{1}}{Y}^{-{\epsilon }_{1}^{\vee }}={Y}^{{\epsilon }_{1}^{\vee }}{s}_{{\epsilon }_{1}}$ which implies follows from ${\left({s}_{0}^{\vee }\right)}^{2}=1.$ In combination with the relations in (Daff 3.12), (Daff 3.13), (Daff 3.15), (Daff 3.16) these imply the last two relations in (Daff 3.5). The relation ${s}_{0}^{\vee }{Y}^{{\epsilon }_{j}^{\vee }}={Y}^{{\epsilon }_{j}^{\vee }}{s}_{0}^{\vee }$ gives ${X}^{{\epsilon }_{1}}{s}_{{\epsilon }_{1}}{Y}^{{\epsilon }_{j}^{\vee }}={Y}^{{\epsilon }_{j}^{\vee }}{X}^{{\epsilon }_{1}}{s}_{{\epsilon }_{1}}$ so that, follows from ${s}_{0}{X}^{{\epsilon }_{j}}={X}^{{\epsilon }_{j}}{s}_{0},$ for $j=2,...,n.$ Finally, ${\left({s}_{0}^{\prime }\right)}^{2}=1$ gives ${X}^{{\epsilon }_{1}}{Y}^{{\epsilon }_{1}^{\vee }}=q{Y}^{{\epsilon }_{1}^{\vee }}{X}^{{\epsilon }_{1}},$ which establishes the third relation of (Daff 3.6). $\square$

The group algebra of $\stackrel{˜}{W}$ contains two Laurent polynomial rings This is analogous to the Weyl algebra generated by the two polynomial rings CHECK AGAIN?

Affine root systems

The notation for affine root systems is where

Define and ${O}_{5}={O}_{5}^{-}\bigsqcup {O}_{5}^{+}$ so that the affine root systems of classical type are missing Note that and

Where the left notation is the notation in [M03, §1.3] and the right notation is that of [BT]. The Weyl groups of these are where $WCn = W0⋉𝔥ℤ ⊇ WBn = W0⋉𝔥_ℤ ⊇ WDn = W0_⋉𝔥_ℤ$ with $W0=G2,1,n, W0_=G2,2,n$ and SAY SOMETHING ABOUT ${W}_{S}$ orbits on $S.$ HOW DO THESE COMPARE TO KAC AND SAHI-ION???

When $n=2$ define Then the "classical" affine root systems of rank 2 are

When $n=1$ define Then the "classical" affine root systems of rank 1 are

Notes and References

Where are these from?

References?