## The double affine group

The double affine group is $DGLℓ(ℂ) = { g∈GLℓ+2 (ℂ) 12 | 12 g= ( 1 λ z ⋮ 0 g‾ μ ⋮ 0 ⋯ 0 ⋯ 1 ) }$ Let $Xμ = ( 1 0 0 ⋮ 0 1 μ ⋮ 0 ⋯ 0 ⋯ 1 ) , Yλ = ( 1 λ 0 ⋮ 0 1 0 ⋮ 0 ⋯ 0 ⋯ 1 ) ,$ $qz = ( 1 0 z ⋮ 0 1 0 ⋮ 0 ⋯ 0 ⋯ 1 ) , g‾ = ( 1 0 0 ⋮ 0 g‾ 0 ⋮ 0 ⋯ 0 ⋯ 1 ) ,$ for $\mu ,\lambda \in {ℂ}^{\ell }$, $z\in ℂ$ and $\stackrel{‾}{g}\in {\mathrm{GL}}_{\ell }\left(ℂ\right)$.

The double affine group ${\mathrm{DGL}}_{\ell }\left(ℂ\right)$ is presented by generators $Xμ, Yλ, qz, g‾ ( μ,λ∈ℂℓ , z∈ℂ, g‾, ∈ GLℓ(ℂ) )$ with relations $qz∈ Z(DGLℓ (ℂ)), Xμ Xν = Xμ+ν, Yλ Yσ = Yλ+σ,$ $g‾ Xμ g‾-1 = Xg‾ μ , g‾ Yμ g‾-1 = Yλ g‾-1 , and Yλ Xμ = q(λ|μ) Xμ Yλ .$ where $(λ|μ) = λ1μ1+⋯+ λℓμℓ, if λ=(λ1, …,λℓ) and μ=(μ1, …,μℓ).$

Let $𝔥∘ℂ* = { ( 0 T μ T 0 ) 12 | 12 μ∈ℂℓ } , Λ0 = ( 0 ⋮ 0 ⋮ 1 ) , δ = ( 1 ⋮ 0 ⋮ 0 ) ,$ so that $𝔥ℂ* =ℂδ⊕ 𝔥∘ℂ* ⊕ ℂΛ0 ≃ℂℓ+2.$ Let $m\in ℂ$. The level $m$ action of ${\mathrm{DGL}}_{\ell }\left(ℂ\right)$ is the

 $DGLℓ(ℂ) -action on 𝔥m*= { ( a T ν T m ) 12 | 12 ν∈ℂℓ , a∈ℂ }$ (lvlm)
given by matrix multiplication: $( 1 λ z ⋮ 0 g‾ μ ⋮ 0 ⋯ 0 ⋯ 1 ) ( a T ν T m ) = ( a+(λ|ν) +mz T g‾ν+mμ T m ) .$ Note that ${𝔥}_{m}^{*}$ is not a ${\mathrm{DGL}}_{\ell }\left(ℂ\right)$-module, but a set with a ${\mathrm{DGL}}_{\ell }\left(ℂ\right)$-action.

The subspace $ℂ\delta$ is a trivial ${\mathrm{DGL}}_{\ell }\left(ℂ\right)$-module of ${𝔥}_{ℂ}^{*}$, and the affine linear group (see [Bou, Alg. Ch. II §9.4] $AGLℓ(ℂ) = { g∈GLℓ+1 (ℂ) 12 | 12 g= ( g‾ μ ⋯ 0 ⋯ 1 ) }$ acts on ${𝔥}_{ℂ}^{*}/ℂ\delta$ by matrix multiplication.

## The Heisenberg group

The Heisenberg group is (see [KP, §3.1]) $Nℝ = 𝔥∘ℝ* × 𝔥∘ℝ* ×ℝ = { (α,β,t) | α∈ 𝔥∘ℝ* , β∈ 𝔥∘ℝ* , t∈ℝ }$ with product given by $(α,β,t) (α′,β′, t′) = (α+α′, β+β′, t+t′ +12 ((α′|β) -(α|β′) ) ) .$ The Heisenberg group is a subgroup of the group ${\mathrm{DGL}}_{\ell }\left(ℂ\right)$ by $(α,β,t) = ( 1 -α t-12 (α|β) ⋮ 0 1 β ⋮ 0 ⋯ 0 ⋯ 1 )$ Identify ${\stackrel{\circ }{𝔥}}_{ℤ}$ with a subgroup of ${N}_{ℝ}$ by setting $tβ= (β,β,0) , for β∈ 𝔥∘ℤ .$ (WHERE WAS ${\stackrel{\circ }{𝔥}}_{ℤ}$ DEFINED? WHAT ABOUT ${\stackrel{\circ }{𝔥}}_{ℝ}^{*}$? SHOULD ONE OF THE TWO ${\stackrel{\circ }{𝔥}}_{ℝ}^{*}$ IN THE DEFINITION OF THE HEISENBERG GROUP BE ${\stackrel{\circ }{𝔥}}_{ℝ}$ ) It follows from (lvlm) that if $\lambda =a\delta +\stackrel{‾}{\lambda }+m{\Lambda }_{0}$ then

 $tβ⋅λ = ( a-(β| λ‾) -m2 (β|β) )δ +λ‾+mβ +mΛ0 = λ+mβ- (λ‾ +12 mβ |β)δ.$ (lvlmtr)

## Affine linear functions

Let $V$ be a vector space over $𝔽.$

For $v\in V,$ the translation in $v$ is the function $tv: V → V x ↦ v+x.$ Define a hyperplane in the vector space $V\oplus 𝔽\delta .$

Let $V$ and $W$ be vector spaces over $𝔽.$

An affine linear function from $V$ to $W$ is a function $f:{V}_{1}\to {W}_{1}$ such that there exists a linear transformation $Df:V\to W$ with $ftv = tDf(v)f.$ If $f\left(0\right)$ denotes $f\left(0+\delta \right)$ then $f(v+δ) = Df(v) + f(0)+δ.$ Let Let and choose bases $\left\{{v}_{1},...,{v}_{m}\right\}$ and $\left\{{w}_{1},...,{w}_{n}\right\}$ of $V$ and $W$ respectively, to identify $Hom(V,W) →∼ Mm,n(𝔽).$ The function $F → Hom(V,W)⊕W → Mm+1,n+1(𝔽) f ↦ (Df,f(0)) ↦ Df 0 f(0) 1 (*)$ is an isomorphism of vector spaces such that $f(v+δ) = Df(v) + f(0)+δ = v 1 Df 0 f(0) 1$

## Affine reflections

Let $\beta :{V}_{1}\to {ℝ}_{1}$ be affine linear so that (i.e. $D\beta =\alpha$ and $\beta \left(0\right)=k$ and $\beta \left(\stackrel{_}{\lambda }+\delta \right)=\alpha \left(\stackrel{_}{\lambda }\right)+k+\delta$).

Define $sα+kδ: V1→V1 by sα+kδ(λ_+δ) = sα(λ_)-kα∨+δ$ so that $D{s}_{\alpha +k\delta }={s}_{\alpha }$ and ${s}_{\alpha +k\delta }\left(0\right)=-\beta \left(0\right){\left(D\beta \right)}^{\vee }=-k{\alpha }^{\vee }$ where $α∨ = 2 ⟨α,α⟩ α and sα(λ_) = λ_-⟨α∨,λ_⟩α.$ In terms of matrices via (*) $sα 0 -kα∨ 1$

## Notes and References

This realization of the double affine group provides a convenient formalism for working with isometries of Euclidean space, affine Weyl groups, and Heisenberg groups. The notation has been chosen to coincide with certain notations in [Kac], in order to help the reader make the connections to the theory of Kac-Moody Lie algebras. In particular, the formula (lvlmtr) is the, sometimes mysteriously introduced, formula for the level $m$ action of a translation.

Affine linear functions are treated in [Bou, Alg Ch.II §9.4].

## References

[Bou] N. Bourbaki, Algebra, Springer-Verlag, Berlin 1989. MR?????

[KP] V. Kac and D. Peterson, Infinite dimensional Lie algebras, theta functions and modular forms, Advances in Math. 53 (1984), 125-264, MR0750341

[Kac] V. Kac, Infinite dimensional Lie algebras, Third edition, Cambridge University Press, Cambridge 1990. xxii+400pp. ISBN: 0-521-37215-1; 0-521-46693-8, MR1104219