The double affine linear groups, affine linear groups and Heisenberg groups

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

Last updates: 29 November 2011

The double affine group

The double affine group is $${\mathrm{DGL}}_{\ell }\left(ℂ\right)=\{g\in {\mathrm{GL}}_{\ell +2}\left(ℂ\right)\phantom{\frac{1}{2}}|\phantom{\frac{1}{2}}g=\left(\begin{array}{ccccc}1& & \lambda & & z\\ ⋮& & & & \\ 0& & \bar{g}& & \mu \\ ⋮& & & & \\ 0& \cdots & 0& \cdots & 1\end{array}\right)\}$$ Let $$X^{\mu }=\left(\begin{array}{ccccc}1& & 0& & 0\\ ⋮& & & & \\ 0& & 1& & \mu \\ ⋮& & & & \\ 0& \cdots & 0& \cdots & 1\end{array}\right),Y^{\lambda }=\left(\begin{array}{ccccc}1& & \lambda & & 0\\ ⋮& & & & \\ 0& & 1& & 0\\ ⋮& & & & \\ 0& \cdots & 0& \cdots & 1\end{array}\right),$$ $$q^z=\left(\begin{array}{ccccc}1& & 0& & z\\ ⋮& & & & \\ 0& & 1& & 0\\ ⋮& & & & \\ 0& \cdots & 0& \cdots & 1\end{array}\right),\bar{g}=\left(\begin{array}{ccccc}1& & 0& & 0\\ ⋮& & & & \\ 0& & \bar{g}& & 0\\ ⋮& & & & \\ 0& \cdots & 0& \cdots & 1\end{array}\right),$$ for $\mu ,\lambda \in {ℂ}^{\ell }$, $z\in ℂ$ and $\bar{g}\in {\mathrm{GL}}_{\ell }\left(ℂ\right)$.

The double affine group ${\mathrm{DGL}}_{\ell }\left(ℂ\right)$ is presented by generators $$X^{\mu },Y^{\lambda },q^z,\bar{g}(\mu ,\lambda \in {ℂ}^{\ell },z\in ℂ,\bar{g},\in {\mathrm{GL}}_{\ell }\left(ℂ\right))$$ with relations $$q^z\in Z\left({\mathrm{DGL}}_{\ell }\right(ℂ\left)\right),X^{\mu }{X}^{\nu }=X^{\mu +\nu },Y^{\lambda }{Y}^{\sigma }=Y^{\lambda +\sigma },$$ $$\bar{g}{X}^{\mu }{\bar{g}}^{-1}=X^{\bar{g}\mu },\bar{g}{Y}^{\mu }{\bar{g}}^{-1}=Y^{\lambda {\bar{g}}^{-1}},\text{and}{Y}^{\lambda }{X}^{\mu }=q^{\left(\lambda \right|\mu )}{X}^{\mu }{Y}^{\lambda }.$$ where $$\left(\lambda \right|\mu )={\lambda }_1{\mu }_1+\cdots +{\lambda }_{\ell }{\mu }_{\ell },\text{if}\lambda =({\lambda }_1,\dots ,{\lambda }_{\ell })\text{and}\mu =({\mu }_1,\dots ,{\mu }_{\ell }).$$

Let $${\stackrel{\circ }{𝔥}}_{ℂ}^{*}=\{\left(\begin{array}{c}0\\ \phantom{T}\\ \mu \\ \phantom{T}\\ 0\end{array}\right)\phantom{\frac{1}{2}}|\phantom{\frac{1}{2}}\mu \in {ℂ}^{\ell }\},{\Lambda }_0=\left(\begin{array}{c}0\\ ⋮\\ 0\\ ⋮\\ 1\end{array}\right),\delta =\left(\begin{array}{c}1\\ ⋮\\ 0\\ ⋮\\ 0\end{array}\right),$$ so that $${𝔥}_{ℂ}^{*}=ℂ\delta \oplus {\stackrel{\circ }{𝔥}}_{ℂ}^{*}\oplus ℂ{\Lambda }_0\simeq {ℂ}^{\ell +2}.$$ Let $m\in ℂ$. The level $m$ action of ${\mathrm{DGL}}_{\ell }\left(ℂ\right)$ is the

$${\mathrm{DGL}}_{\ell }\left(ℂ\right)\text{-action on}{𝔥}_m^{*}=\{\left(\begin{array}{c}a\\ \phantom{T}\\ \nu \\ \phantom{T}\\ m\end{array}\right)\phantom{\frac{1}{2}}|\phantom{\frac{1}{2}}\nu \in {ℂ}^{\ell },a\in ℂ\}$$

(lvlm)

given by matrix multiplication: $$\left(\begin{array}{ccccc}1& & \lambda & & z\\ ⋮& & & & \\ 0& & \bar{g}& & \mu \\ ⋮& & & & \\ 0& \cdots & 0& \cdots & 1\end{array}\right)\left(\begin{array}{c}a\\ \phantom{T}\\ \nu \\ \phantom{T}\\ m\end{array}\right)=\left(\begin{array}{c}a+\left(\lambda \right|\nu )+mz\\ \phantom{T}\\ \bar{g}\nu +m\mu \\ \phantom{T}\\ m\end{array}\right).$$ 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] $${\mathrm{AGL}}_{\ell }\left(ℂ\right)=\{g\in {\mathrm{GL}}_{\ell +1}\left(ℂ\right)\phantom{\frac{1}{2}}|\phantom{\frac{1}{2}}g=\left(\begin{array}{cccc}& & & \\ & \bar{g}& & \mu \\ & & & \\ \cdots & 0& \cdots & 1\end{array}\right)\}$$ acts on ${𝔥}_{ℂ}^{*}/ℂ\delta$ by matrix multiplication.

The Heisenberg group

The Heisenberg group is (see [KP, §3.1]) $$N_{ℝ}={\stackrel{\circ }{𝔥}}_{ℝ}^{*}\times {\stackrel{\circ }{𝔥}}_{ℝ}^{*}\times ℝ=\left\{\right(\alpha ,\beta ,t)|\alpha \in {\stackrel{\circ }{𝔥}}_{ℝ}^{*},\beta \in {\stackrel{\circ }{𝔥}}_{ℝ}^{*},t\in ℝ\}$$ with product given by $$(\alpha ,\beta ,t)(\alpha \prime ,\beta \prime ,t\prime )=(\alpha +\alpha \prime ,\beta +\beta \prime ,t+t\prime +\frac{1}{2}(\left(\alpha \prime \right|\beta )-(\alpha \left|\beta \prime \right)\left)\right).$$ The Heisenberg group is a subgroup of the group ${\mathrm{DGL}}_{\ell }\left(ℂ\right)$ by $$(\alpha ,\beta ,t)=\left(\begin{array}{ccccc}1& & -\alpha & & t-\frac{1}{2}\left(\alpha \right|\beta )\\ ⋮& & & & \\ 0& & 1& & \beta \\ ⋮& & & & \\ 0& \cdots & 0& \cdots & 1\end{array}\right)$$ Identify ${\stackrel{\circ }{𝔥}}_{\mathbb{Z}}$ with a subgroup of $N_{ℝ}$ by setting $$t_{\beta }=(\beta ,\beta ,0),\text{for}\beta \in {\stackrel{\circ }{𝔥}}_{\mathbb{Z}}.$$ (WHERE WAS ${\stackrel{\circ }{𝔥}}_{\mathbb{Z}}$ 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 $\lambda =a\delta +\bar{\lambda }+m{\Lambda }_0$ then

$$\begin{array}{rl}{t}_{\beta }\cdot \lambda & =(a-\left(\beta \right|\bar{\lambda })-\frac{m}{2}\left(\beta \right|\beta ))\delta +\bar{\lambda }+m\beta +m{\Lambda }_0\\ & =\lambda +m\beta -(\bar{\lambda }+\frac{1}{2}m\beta |\beta )\delta .\end{array}$$

(lvlmtr)

Affine linear functions

Let $V$ be a vector space over $𝔽.$

For $v\in V,$ the translation in $v$ is the function $$\begin{array}{rrcl}{t}_v:& V& \to & V\\ & x& \mapsto & v+x.\end{array}$$ Define $$V_1=\{v+\delta \in V\oplus 𝔽\delta | v\in V\},$$ 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 $$f{t}_v=t_{Df\left(v\right)}f.$$ If $f\left(0\right)$ denotes $f(0+\delta )$ then $$f(v+\delta )=Df\left(v\right)+f\left(0\right)+\delta .$$ Let $$F=\{f:V_1\to W_1 | f \; is\; affine\; linear\}.$$ Let $m=\dim \left(V\right), n=\dim \left(W\right)$ and choose bases $\{v_1,...,v_m\}$ and $\{w_1,...,w_n\}$ of $V$ and $W$ respectively, to identify $$\mathrm{Hom}(V,W)\stackrel{\sim }{\to }{M}_{m,n}\left(𝔽\right).$$ The function $$\begin{array}{ll}\begin{array}{ccccc}F& \to & \mathrm{Hom}(V,W)\oplus W& \to & M_{m+1,n+1}\left(𝔽\right)\\ f& \mapsto & (Df,f\left(0\right))& \mapsto & \left(\begin{array}{cccc}& & & \\ & Df& & 0\\ & & & \\ & f\left(0\right)& & 1\end{array}\right)\end{array}& (*)\end{array}$$ is an isomorphism of vector spaces such that $$f(v+\delta )=Df\left(v\right)+f\left(0\right)+\delta =\left(\begin{array}{cccc}& v& & 1\end{array}\right)\left(\begin{array}{cccc}& & & \\ & Df& & 0\\ & & & \\ & f\left(0\right)& & 1\end{array}\right)$$

Affine reflections

Let $\beta :V_1\to {ℝ}_1$ be affine linear so that $$\beta =\alpha +k\delta ,with\; \alpha \in V^{*} \; and\; k\in ℝ$$ (i.e. $D\beta =\alpha$ and $\beta \left(0\right)=k$ and $\beta (\stackrel{\_}{\lambda }+\delta )=\alpha \left(\stackrel{\_}{\lambda }\right)+k+\delta$).

Define $$s_{\alpha +k\delta }:V_1\to V_{1}by{s}_{\alpha +k\delta }(\stackrel{\_}{\lambda }+\delta )=s_{\alpha }\left(\stackrel{\_}{\lambda }\right)-k{\alpha }^{\vee }+\delta$$ 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 $${\alpha }^{\vee }=\frac{2}{⟨\alpha ,\alpha ⟩}\alpha and{s}_{\alpha }\left(\stackrel{\_}{\lambda }\right)=\stackrel{\_}{\lambda }-⟨{\alpha }^{\vee },\stackrel{\_}{\lambda }⟩\alpha .$$ In terms of matrices via (*) $$\left(\begin{array}{cccc}& & & \\ & s_{\alpha }& & 0\\ & & & \\ & -k{\alpha }^{\vee }& & 1\end{array}\right)$$

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

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