Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 42 (2001), No. 2, 419-430.

Quantum Co-Adjoint Orbits of the Group of Affine Transformations of the Complex Line^{1}

Do Ngoc Diep Nguyen Viet Hai

Institute of Mathematics, National Center for Natural Sciences and Technology, P. O. Box 631, Bo Ho, 10.000, Hanoi, Vietnam e-mail: dndiep@hn.vnn.vn

Haiphong Teacher’s Training College Haiphong City, Vietnam e-mail: hainviet@yahoo.com

Abstract. We construct star-products on the co-adjoint orbit of the Lie group Aff(ℂ) of affine transformations of the complex line and apply them to obtain the irreducible unitary representations of this group. These results show the effectiveness of the Fedosov quantization even for groups which are neither nilpotent nor exponential. Together with the result for the group Aff(ℝ) (see [5]), we thus have a description of quantum \overline{MD} co-adjoint orbits.

1. Introduction

The notion of ⋆-products was a few years ago introduced and played a fundamental role in the basic problem of quantization, see e.g. references [1, 2, 6, 7, …], as a new approach to quantization on arbitrary symplectic manifolds. In [5] we have constructed star-products on upper half-plane, obtained the operator ℓ̂_{Z}, Z ∈ aff(ℝ) = Lie Aff(ℝ) and proved that the representation

exp(ℓ̂_{Z}) = exp(α\frac{∂}{∂s} + iβe^{s})

1This work was supported in part by the National Foundation for Fundamental Science research of Vietnam and the Alexander von Humboldt Foundation, Germany 0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2001 Heldermann Verlag

420 Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-Adjoint Orbits …

of the group Aff_{0}(ℝ) coincides with the representation T_{Ω_{+}} obtained from the orbit method or Mackey small subgroup method. One of the advantages of this group, with which the computation is rather accessible is the fact that its connected component Aff_{0}(ℝ) is exponential. We could use therefore the canonical coordinates for Kirillov form on the orbits. It is natural to consider the same problem for the group Aff(ℂ). We can expect that the calculations and final expressions could be similar to the corresponding ones in real line case, but this group Aff(ℂ) is no more the exponential, i.e. the exponential map

exp : aff(ℂ) → { \atop Aff(ℂ)}

is no longer a global diffeomorphism and the general theory of D. Arnal and J. Cortet (see [1], [2]) could not be directly applicable. We overcame these difficulties by a rather different way which could give new ideas for more general non-exponential groups: To overcome the main difficulty in applying the deformation quantization to this group, we replace the global diffeomorphism in Arnal-Cortet’s setting by a local diffeomorphism. With this replacement, we need to pay attention on the complexity of the symplectic Kirillov form in new coordinates. We then computed the inverse image of the Kirillov form on appropriate local charts. The question raised here is how to choose a good local chart in order to make the calculation as simple as possible. The calculation we propose is realized by using complex analysis on very simple complex domain.

Our main result consists of an explicit star-product formula (Proposition 3.5) on the local charts. This means that the functional algebras on co-adjoint orbits admit a suitable deformation, or in other words, we obtained the quantum co-adjoint orbits of this group as exact models of new quantum objects, called “quantum punctured complex planes” (ℂ^{2} \L)_{q}. Then, by using the Fedosov deformation quantization, it is not hard to obtain the list of all irreducible unitary representations (Theorem 4.2) of the group Aff(ℂ), although the computation in this case, using the Mackey small subgroup method or the modern orbit method, is rather delicate. The infinitesimal generators of those exact models of infinite dimensional irreducible unitary representations, nevertheless, are given by rather simple formulae. We introduce some preliminary result in §2. The operators ℓ̂_{A} which define the representation of the Lie algebra aff(ℂ) are found in §3. In particular, we obtain the unitary representations of the Lie group { \atop Aff(ℂ)}, the universal covering group of the Aff(ℂ), in Theorem 4.3 of §4.

2. Preliminary results

Recall that the Lie algebra 𝔤 = aff(ℂ) of affine transformations of the complex line is described as follows (see [4]). It is well-known that the group Aff(ℂ) is a real 4-dimensional Lie group which is isomorph to the group of matrices:

Aff(ℂ) ≅ \{( \begin{matrix}a & b \\ 0 & 1 \end{matrix} ) |a, b ∈ ℂ, a ≠ 0\}

The most easy method is to consider X,Y as complex generators, X = X_{1}+iX_{2} and Y = Y_{1}+ iY_{2}. Then from the relation [X, Y] = Y, we get [X_{1}, Y_{1}]−[X_{2}, Y_{2}]+i([X_{1}Y_{2}]+[X_{2}, Y_{1}]) = Y_{1} + iY_{2}. This means that the Lie algebra aff(ℂ) is a real 4-dimensional Lie algebra, having four generators with the only nonzero Lie brackets: [X_{1}, Y_{1}]−[X_{2}, Y_{2}] = Y_{1}; [X_{2}, Y_{1}]+[X_{1}, Y_{2}] = Y_{2} and we can choose another basis denoted again by the same letters such that:

[X_{1}, Y_{1}] = Y_{1}; [X_{1}, Y_{2}] = Y_{2}; [X_{2}, Y_{1}] = Y_{2}; [X_{2}, Y_{2}] = −Y_{1}

Remark 2.1. The exponential map

exp : ℂ ⟶ ℂ^{∗} := ℂ\{0}

given by z ↦ e^{z} is just the covering map and therefore the universal covering of ℂ^{∗}: ℂ̃^{∗} ≅ ℂ. As a consequence one deduces that

{ \atop Aff(ℂ)} ≅ ℂ ⋉ ℂ ≅ \{(z, w)|z, w ∈ ℂ\}

with the following multiplication law:

(z, w)(z′, w′) := (z + z′, w + e^{z}w′)

Remark 2.2. The co-adjoint orbit of { \atop Aff(ℂ)} in 𝔤^{∗} passing through F ∈ 𝔤^{∗} is denoted by

Ω := K({ \atop Aff(ℂ)})F = \{K(g)F |g ∈ { \atop Aff(ℂ)}\}.

Then, (see [4]):

1. Each point (α, 0, 0, δ) is 0-dimensional co-adjoint orbit Ω_{(α,0,0,δ)}.

2. The open set β^{2} + γ^{2} ≠ 0 is the single 4-dimensional co-adjoint orbit Ω = Ω_{β^{2}+γ^{2}≠0}. We shall use Ω in the form Ω ≅ ℂ × ℂ^{∗}.

Remark 2.3. Set

ℍ_{k} = {w = q_{1} + iq_{2} ∈ ℂ| − ∞ < q_{1} < +∞; 2kπ < q_{2} < 2kπ + 2π}; k = 0, ±1, …
L = {ρe^{iφ} ∈ ℂ | 0 < ρ < +∞; φ = 0}

and ℂ_{k} = ℂ\L is a univalent sheet of the Riemann surface of the multi-valued complex analytic function Ln(w), (k = 0, ±1, …). Then there is a natural diffeomorphism w ∈ ℍ_{k} ⟼ e^{w} ∈ ℂ_{k} with each k = 0, ±1, …. Now consider the map:

ℂ × ℂ ⟶ Ω = ℂ × ℂ^{∗}
(z, w) ⟼ (z, e^{w}),

with a fixed k ∈ ℤ. We have a local diffeomorphism

φ_{k} : ℂ × ℍ_{k} ⟶ ℂ × ℂ_{k}
(z, w) ⟼ (z, e^{w}).

This diffeomorphism φ_{k} will be needed in the sequel.

On ℂ × ℍ_{k} we have the natural symplectic form

ω_{o} = \frac{1}{2}[dz ∧ dw + d\overline{z} ∧ d\overline{w}],  (1)

induced from ℂ^{2}. Put z = p_{1} + ip_{2}, w = q_{1} + iq_{2} and (x^{1}, x^{2}, x^{3}, x^{4}) = (p_{1}, q_{1}, p_{2}, q_{2}) ∈ ℝ^{4}, then

ω_{o} = dp_{1} ∧ dq_{1} − dp_{2} ∧ dq_{2}.

The corresponding symplectic matrix of ω_{o} is

⋀ = ( \begin{matrix}0 & −1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & −1 & 0 \end{matrix} ) and ⋀^{−1} = ( \begin{matrix}0 & 1 & 0 & 0 \\ −1 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 \\ 0 & 0 & 1 & 0 \end{matrix} )

We have therefore the Poisson brackets of functions as follows. With each f, g ∈ C^{∞}(Ω),

\{f, g\} = ∧^{ij} \frac{∂f}{∂x^{i}} \frac{∂g}{∂x^{j}} = \frac{∂f}{∂p_{1}} \frac{∂g}{∂q_{1}} − \frac{∂f}{∂q_{1}} \frac{∂g}{∂p_{1}} − \frac{∂f}{∂p_{2}} \frac{∂g}{∂q_{2}} + \frac{∂f}{∂q_{2}} \frac{∂g}{∂p_{2}} =

= 2[\frac{∂f}{∂z} \frac{∂g}{∂w} − \frac{∂f}{∂w} \frac{∂g}{∂z} + \frac{∂f}{∂\overline{z}} \frac{∂g}{∂\overline{w}} − \frac{∂f}{∂\overline{w}} \frac{∂g}{∂\overline{z}} ].

Proposition 2.4. Fixing the local diffeomorphism φ_{k}(k ∈ ℤ), we have:

1. For any element A ∈ aff(ℂ), the corresponding Hamiltonian function Ã in local coordinates (z, w) of the orbit Ω is of the form

Ã ∘ φ_{k}(z, w) = \frac{1}{2}[αz + βe^{w} + \overline{α}\overline{z} + \overline{β}e\overline{w}]

2. In local coordinates (z, w) of the orbit Ω, the Kirillov form Ω is just the standard form (1).

Proof. 1. Each element F ∈ Ω ⊂ (Aff(ℂ))^{∗} is of the form

F = zX^{∗} + e^{w}Y^{∗} = (\begin{matrix}z & 0 \\ e^{w} & 0 \end{matrix})

in local Darboux coordinates (z, w). From this it follows that

Ã(F) = ⟨F, A⟩ = ℜ tr(F.A) =

= ℜ tr (\begin{matrix}αz & βz \\ αe^{w} & βe^{w} \end{matrix}) = \frac{1}{2}[αz + βe^{w} + \overline{α}\overline{z} + \overline{β}e\overline{w}]

2. Using the definition of the Poisson brackets{, }, associated to a symplectic form ω, we have

\{Ã, f\} = α\frac{∂f}{∂w} − βe^{w} \frac{∂f}{∂z} − \overline{β}e\overline{w} \frac{∂f}{∂\overline{z}} + \overline{α}\frac{∂f}{∂\overline{w}}.  (2)

Let us from now on denote by ξ_{A} the Hamiltonian vector field (symplectic gradient) corresponding to the Hamiltonian function Ã, A ∈ aff(ℂ). Now we consider two vector fields:

ξ_{A} = α_{1} \frac{∂}{∂w} − β_{1}e^{w} \frac{∂}{∂z} − \overline{β_{1}}e\overline{w} \frac{∂}{∂\overline{z}} + \overline{α_{1}} \frac{∂}{∂\overline{w}}; ξ_{B} = α_{2} \frac{∂}{∂w} − β_{2}e^{w} \frac{∂}{∂z} − \overline{β_{2}}e\overline{w} \frac{∂}{∂\overline{z}} + \overline{α_{2}} \frac{∂}{∂\overline{w}},

where A = (\begin{matrix}α_{1} & β_{1} \\ 0 & 0 \end{matrix}); B = (\begin{matrix}α_{2} & β_{2} \\ 0 & 0 \end{matrix}) ∈ aff(ℂ). It is easy to check that

ξ_{A} ⊗ ξ_{B} = β_{1}β_{2}e^{2w} \frac{∂}{∂z} ⊗ \frac{∂}{∂z} + α_{1}α_{2} \frac{∂}{∂w} ⊗ \frac{∂}{∂w} + \overline{β_{1}}\overline{β_{2}}e^{2\overline{w}} \frac{∂}{∂\overline{z}} ⊗ \frac{∂}{∂\overline{z}} + \overline{α_{1}}\overline{α_{2}} \frac{∂}{∂\overline{w}} ⊗ \frac{∂}{∂\overline{w}}+

+(α_{1}β_{2} − α_{2}β_{1})e^{w} \frac{∂}{∂z} ⊗ \frac{∂}{∂w} + (\overline{α_{1}}\overline{β_{2}} − \overline{α_{2}}\overline{β_{1}})e\overline{w} \frac{∂}{∂\overline{z}} ⊗ \frac{∂}{∂\overline{w}} + (α_{1}\overline{β_{2}} − α_{2}\overline{β_{1}})e\overline{w} \frac{∂}{∂\overline{z}} ⊗ \frac{∂}{∂w}+

+(\overline{α_{1}}β_{2} − \overline{α_{2}}β_{1})e^{w} \frac{∂}{∂z} ⊗ \frac{∂}{∂\overline{w}} + (β_{1}\overline{β_{2}} − \overline{β_{1}}β_{2})e^{w+\overline{w}} \frac{∂}{∂z} ⊗ \frac{∂}{∂\overline{z}} + (α_{1}\overline{α_{2}} − \overline{α_{1}}α_{2})\frac{∂}{∂w} ⊗ \frac{∂}{∂\overline{w}}.

Thus,

⟨ω, ξ_{A} ⊗ ξ_{B}⟩ = \frac{1}{2}[(α_{1}β_{2} − α_{2}β_{1})e^{w} + (\overline{α_{1}}\overline{β_{2}} − \overline{α_{2}}\overline{β_{1}})e\overline{w}] = ℜ tr(F.[A, B]) = ⟨F, [A, B]⟩.

Thus the proposition is proved. □

3. Computation of operators ℓ̂_{A}^{(k)}

Proposition 3.1. With A, B ∈ aff(ℂ), the Moyal ⋆-product satisfies the relation:

iÃ ⋆ iB̃ − iB̃ ⋆ iÃ = i{ \atop [A, B]}.  (3)

Proof. Consider two arbitrary elements A = α_{1}X + β_{1}Y; B = α_{2}X + β_{2}Y. Then the corresponding Hamiltonian functions are:

Ã = \frac{1}{2}[α_{1}z + β_{1}e^{w} + \overline{α_{1}}\overline{z} + \overline{β_{1}}e\overline{w}]; B̃ = \frac{1}{2}[α_{2}z + β_{2}e^{w} + \overline{α_{2}}\overline{z} + \overline{β_{2}}e\overline{w}].

It is easy, then, to see that:

P^{0}(Ã, B̃) = Ã.B̃

P^{1}(Ã, B̃) = \{Ã.B̃\} = 2[\frac{∂Ã}{∂z} \frac{∂B̃}{∂w} − \frac{∂Ã}{∂w} \frac{∂B̃}{∂z} + \frac{∂Ã}{∂\overline{z}} \frac{∂B̃}{∂\overline{w}} − \frac{∂Ã}{∂\overline{w}} \frac{∂B̃}{∂\overline{z}} ]

= \frac{1}{2}[(α_{1}β_{2} − α_{2}β_{1})e^{w} + (\overline{α_{1}}\overline{β_{2}} − \overline{α_{2}}\overline{β_{1}})e\overline{w}]

and P^{r}(Ã, B̃) = 0, ∀r ≥ 2.

Thus,

iÃ ⋆ iB̃ − iB̃ ⋆ iÃ = \frac{1}{2i}[P^{1}(iÃ, iB̃) − P^{1}(iB̃, iÃ)] =

= \frac{i}{2}[(α_{1}β_{2} − α_{2}β_{1})e^{w} + (\overline{α_{1}}\overline{β_{2}} − \overline{α_{2}}\overline{β_{1}})e\overline{w}]

on one hand. On the other hand, because of [A, B] = (α_{1}β_{2} − α_{2}β_{1})Y we have

i[{ \atop A, B}] = i⟨F, [A, B]⟩ = \frac{i}{2}[(α_{1}β_{2} − α_{2}β_{1})e^{w} + (\overline{α_{1}}\overline{β_{2}} − \overline{α_{2}}\overline{β_{1}})e\overline{w}]

The proposition is hence proved. □

For each A ∈ aff(ℂ), the corresponding Hamiltonian function is

Ã = \frac{1}{2}[αz + βe^{w} + \overline{α}\overline{z} + \overline{β}e\overline{w}]

and we can consider the operator ℓ_{A}^{(k)} acting on the dense subspace of smooth functions L^{2}(ℝ^{2} × (ℝ^{2})^{∗}, dp_{1}dq_{1}dp_{2}dq_{2}/(2π)^{2})^{∞} by left ⋆-multiplication by iÃ, i.e. ℓ_{A}^{(k)}(f) = iÃ ⋆ f. Because of the relation in Proposition 3.1, we have

Corollary 3.2.

ℓ_{[A,B]}^{(k)} = ℓ_{A}^{(k)} ∘ ℓ_{B}^{(k)} − ℓ_{B}^{(k)} ∘ ℓ_{A}^{(k)} := [ℓ_{A}^{(k)}, ℓ_{B}^{(k)}]  (4)

From this it is easy to see that the correspondence A ∈ aff(ℂ) ⟼ ℓ_{A}^{(k)} = iÃ ⋆ . is a representation of the Lie algebra aff(ℂ) on the space C^{∞}(Ω)[[\frac{i}{2}]] of formal power series, see [7] for more detail.

Now, let us denote F_{z}(f) the partial Fourier transform of the function f from the variable z = p_{1} + ip_{2} to the variable ξ = ξ_{1} + iξ_{2}, i.e.

F_{z}(f)(ξ, w) = \frac{1}{2π} ∫∫ _{ℝ^{2}} e^{−iℜ(ξ\overline{z})}f(z, w)dp_{1}dp_{2}.

Let us denote by

F_{z}^{−1}(f)(z, w) = \frac{1}{2π} ∫∫ _{ℝ^{2}} e^{iℜ(ξ\overline{z})}f(ξ, w)dξ_{1}dξ_{2}

the inverse Fourier transform.

Lemma 3.3. Putting g = g(z, w) = F_{z}^{−1}(f)(z, w) we obtain:

1. ∂_{z}g = \frac{i}{2} \overline{ξ}g ; ∂_{z}^{r}g = (\frac{i}{2} \overline{ξ})^{r}g, r = 2, 3, …

2. ∂\overline{z}g = \frac{i}{2}ξg ; ∂r ; \overline{z}g = (\frac{i}{2}ξ)^{r}g, r = 2, 3, …

3. F_{z}(zg) = 2i∂\overline{ξ}F_{z}(g) = 2i∂\overline{ξ}f ; F_{z}(\overline{z}g) = 2i∂_{ξ}F_{z}(g) = 2i∂_{ξ}f

4. ∂_{w}g = ∂_{w}(F_{z}^{−1}(f)) = F_{z}^{−1}(∂_{w}f); ∂\overline{w}g = ∂\overline{w}(F_{z}^{−1}(f) = F_{z}^{−1}(∂\overline{w}f)

Proof. As ∂_{z} = \frac{1}{2}(∂_{p_{1}} − i∂_{p_{2}}); ∂\overline{z} = \frac{1}{2}(∂_{p_{1}} + i∂_{p_{2}}) we obtain 1. and 2.

3. We have F_{z}(zg) =

= \frac{1}{2π} ∫∫ e^{−i(p_{1}ξ_{1}+p_{2}ξ_{2})}p_{1}g(z, w)dp_{1}dp_{2} + i\frac{1}{2π} ∫∫ e^{−i(p_{1}ξ_{1}+p_{2}ξ_{2})}p_{2}g(z, w)dp_{1}dp_{2} =

= i∂_{ξ_{1}}F_{z}(g) + i^{2}∂_{ξ_{2}}F_{z}(g) = (i∂_{ξ_{1}} − ∂_{ξ_{2}})F_{z}(g) = 2i∂\overline{ξ}F_{z}(g) = 2i∂\overline{ξ}f

Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-Adjoint Orbits … 425

and F_{z}(\overline{z}g) =

= \frac{1}{2π} ∫∫ e^{−i(p_{1}ξ_{1}+p_{2}ξ_{2})}p_{1}g(z, w)dp_{1}dp_{2} − i\frac{1}{2π} ∫∫ e^{−i(p_{1}ξ_{1}+p_{2}ξ_{2})}p_{2}g(z, w)dp_{1}dp_{2} = 2i∂_{ξ}f.

4. The proof is straightforward. The Lemma 3.3 is therefore proved. □ We also need another lemma which will be used in the sequel. Lemma 3.4. With g = F_{z}^{−1}(f)(z, w), we have:

1. F_{z}(P^{0}(Ã, g)) = i(α∂\overline{ξ} + \overline{α}∂_{ξ})f + \frac{1}{2}βe^{w}f + \frac{1}{2} \overline{β}e\overline{w}f.

2. F_{z}(P^{1}(Ã, g)) = \overline{α}∂\overline{w}f + α∂_{w}f − \overline{β}e\overline{w}(\frac{i}{2}ξ)f − βe^{w}(\frac{i}{2} \overline{ξ})f.

3. F_{z}(P^{r}(Ã, g)) = (−1)^{r}.2^{r−1}[\overline{β}e\overline{w}(\frac{i}{2}ξ)^{r} + βe^{w}(\frac{i}{2} \overline{ξ})^{r}]f ∀r ≥ 2. Proof. 1. Applying Lemma 3.3 we obtain P^{0}(Ã, g) = Ã.g = \frac{1}{2}[αzg + βe^{w}g + \overline{α}\overline{z}g + \overline{β}e\overline{w}g]. Thus,

F_{z}(P^{0}(Ã, g)) = \frac{1}{2}[αF_{z}(zg) + βe^{w}F_{z}(g) + \overline{α}F_{z}(\overline{z}g) + \overline{β}e\overline{w}F_{z}(g)] =

\frac{1}{2}[2iα∂\overline{ξ}F_{z}(g) + 2i\overline{α}∂_{ξ}F_{z}(g) + βe^{w}F_{z}(g) + \overline{β}e\overline{w}F_{z}(g)] = i(α∂\overline{ξ} + \overline{α}∂_{ξ})f + \frac{1}{2}βe^{w}f + \frac{1}{2}\overline{β}e\overline{w}f.

2. (P^{1}(Ã, g)) = ∧^{12}∂_{p_{1}} Ã∂_{q_{1}}g + ∧^{21}∂_{q_{1}} Ã∂_{p_{1}}g + ∧^{34}∂_{p_{2}} Ã∂_{q_{2}}g + ∧^{43}∂_{q_{2}} Ã∂_{p_{2}}g = = \overline{α}∂\overline{w}g + α∂_{w}g − \overline{β}e\overline{w}∂\overline{z}g − βe^{w}∂_{z}g.

This implies that: F_{z}(P^{1}(Ã, g)) =

= \overline{α}∂\overline{w}F_{z}(g) + α∂_{w}F_{z}(g) − \overline{β}e\overline{w}∂\overline{z}F_{z}(g) − βe^{w}∂_{z}F_{z}(g) =

= \overline{α}∂\overline{w}f + α∂_{w}f − \overline{β}e\overline{w}(\frac{i}{2}ξ)f − βe^{w}(\frac{i}{2}\overline{ξ})f.

3. P^{2}(Ã, g) = ∧^{21} ∧^{21} ∂_{q_{1}q_{1}} Ã∂_{p_{1}p_{1}}g + ∧^{21} ∧^{43} ∂_{q_{1}q_{2}} Ã∂_{p_{1}p_{2}}g + ∧^{43} ∧^{21} ∂_{q_{2}q_{1}} Ã∂_{p_{2}p_{1}}g +

+ ∧^{43} ∧^{43}∂_{q_{2}q_{2}} Ã∂_{p_{2}p_{2}}g = \frac{1}{2}[(\overline{β}e\overline{w} + βe^{w} − βe^{w} + \overline{β}e\overline{w} + \overline{β}e\overline{w} − βe^{w} + βe^{w} + \overline{β}e\overline{w})∂2 ; \overline{z}g +

+ (\overline{β}e\overline{w} + βe^{w} + βe^{w} − \overline{β}e\overline{w} − \overline{β}e\overline{w} + βe^{w} + βe^{w} + \overline{β}e\overline{w})∂_{z}^{2}g] = 2\overline{β}e\overline{w}∂2 ; \overline{z}g + 2βe^{w}∂_{z}^{2}g.

This implies also that:

F_{z}(P^{2}(Ã, g)) = 2\overline{β}e\overline{w}F_{z}(∂2 ; \overline{z}g) + 2βe^{w}F_{z}(∂_{z}^{2}g) = 2\overline{β}e\overline{w}(\frac{i}{2}ξ)^{2}f + 2βe^{w}(\frac{i}{2}\overline{ξ})^{2}f.

By analogy,

P^{3}(Ã, g) = (−1)^{3}[4\overline{β}e\overline{w}∂3 ; \overline{z}g + 4βe^{w}∂_{z}^{3}g],

F_{z}(P^{3}(Ã, g)) = (−1)^{3}.2^{2}[\overline{β}e\overline{w}(\frac{i}{2}ξ)^{3}f + βe^{w}(\frac{i}{2}\overline{ξ})^{3}f],

426 Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-Adjoint Orbits …

and with r ≥ 4

P^{r}(Ã, g) = (−1)^{r}.2^{r−1}[\overline{β}e\overline{w}∂r ; \overline{z}g + βe^{w}∂_{z}^{r}g],

F_{z}(P^{r}(Ã, g)) = (−1)^{r}.2^{r−1}[\overline{β}e\overline{w}(\frac{i}{2}ξ)^{r} + βe^{w}(\frac{i}{2}\overline{ξ})^{r}]f.

Lemma 3.4 is therefore proved. □ Proposition 3.5. For each A = (\begin{matrix}α & β \\ 0 & 0 \end{matrix}) ∈ aff(ℂ) and for each compactly supported C^{∞}- function f ∈ C_{c}^{∞}(ℂ × ℍ_{k}), we have: ℓ_{A}^{(k)}f := F_{z} ∘ ℓ_{A}^{(k)} ∘ F_{z}^{−1}(f) =

= [α(\frac{1}{2}∂_{w} − ∂\overline{ξ})f + \overline{α}(\frac{1}{2}∂\overline{w} − ∂_{ξ})f + \frac{i}{2}(βe^{w− \frac{1}{2} \overline{ξ}} + \overline{β}e^{\overline{w}− \frac{1}{2} ξ})f].  (5)

Proof. Applying Lemma 3.4, we have:

ℓ_{A}^{(k)}(f) := F_{Z}(iÃ ⋆ F_{z}^{−1}(f)) = i∑_{r≥0} \frac{1}{r!}(\frac{1}{2i})^{r}F_{z}(P^{r}(Ã, F_{z}^{−1}(f))) =

= i\{[i(α∂\overline{ξ} + \overline{α}∂_{ξ})f + \frac{1}{2}βe^{w}f + \frac{1}{2}\overline{β}e\overline{w}f] + \frac{1}{1!}(\frac{1}{2i})[\overline{α}∂\overline{w}f + α∂_{w}f − \overline{β}e\overline{w}(\frac{i}{2}ξ)f−

−βe^{w}(\frac{i}{2}\overline{ξ})f] + \frac{1}{2!}(\frac{−1}{2i})^{2}2[\overline{β}e\overline{w}(\frac{i}{2}ξ)^{2}f + βe^{w}(\frac{i}{2}\overline{ξ})^{2}f] + ⋯ +

+\frac{1}{r!}(\frac{−1}{2i})^{r}2^{r−1}[\overline{β}e\overline{w}(\frac{i}{2}ξ)^{r}f + βe^{w}(\frac{i}{2}\overline{ξ})^{r}f] + … \}

= −(α∂\overline{ξ} − \overline{α}∂_{ξ})f + \frac{1}{2}(\overline{α}∂\overline{w} + α∂_{w})f + i\{[\frac{1}{2}βe^{w} + \frac{1}{2}\overline{β}e\overline{w} − \frac{1}{2}\overline{β}e\overline{w}(\frac{1}{2}ξ) − \frac{1}{2}βe^{w}(\frac{1}{2}\overline{ξ})]f+

+\frac{1}{2}.\frac{1}{2!}[\overline{β}e\overline{w}(\frac{−1}{2}\overline{ξ})^{2} + βe^{w}(\frac{−1}{2}ξ)^{2}]f + ⋯ + \frac{1}{2} \frac{1}{k!}[\overline{β}e\overline{w}(\frac{−1}{2}ξ)^{r} + βe^{w}(\frac{−1}{2}\overline{ξ})^{r}]f + … \}

= [α(\frac{1}{2}∂_{w} − ∂\overline{ξ}) + \overline{α}(\frac{1}{2}∂\overline{w} − ∂_{ξ}) + \frac{i}{2}βe^{w}e^{− \frac{1}{2} \overline{ξ}} + \frac{i}{2}\overline{β}e\overline{w}e^{− \frac{1}{2} ξ}]f

= [α(\frac{1}{2}∂_{w} − ∂\overline{ξ}) + \overline{α}(\frac{1}{2}∂\overline{w} − ∂_{ξ}) + \frac{i}{2}(βe^{w− \frac{1}{2} \overline{ξ}} + \overline{β}e^{\overline{w}− \frac{1}{2} ξ})]f

The proposition is therefore proved. □ Remark 3.6. Set u=w − \frac{1}{2} \overline{ξ};v = w + \frac{1}{2} \overline{ξ} we obtain

ℓ̂_{A}^{(k)}(f) = α\frac{∂f}{∂u} + \overline{α}\frac{∂f}{∂\overline{u}} + \frac{i}{2}(βe^{u} + \overline{β}e\overline{u})f|_{(u,v)}, i.e.  (6)

ℓ̂_{A}^{(k)} = α\frac{∂}{∂u} + \overline{α}\frac{∂}{∂\overline{u}} + \frac{i}{2}(βe^{u} + \overline{β}e\overline{u}),

which provides a (local) representation of the Lie algebra aff(ℂ).

4. The irreducible representations of { \atop Aff(ℂ)}

Since ℓ̂_{A}^{(k)} is a representation of the Lie algebra aff(ℂ), we have:

exp(ℓ̂_{A}^{(k)}) = exp(α\frac{∂}{∂u} + \overline{α}\frac{∂}{∂\overline{u}} + \frac{i}{2}(βe^{u} + \overline{β}e\overline{u}))

is just the corresponding representation of the corresponding connected and simply connected Lie group { \atop Aff(ℂ)}. Let us first recall the well-known list of all the irreducible unitary representations of the group of affine transformations of the complex line, see [4] for more details.

Theorem 4.1. Up to unitary equivalence, every irreducible unitary representation of { \atop Aff(ℂ)} is unitarily equivalent to one of the following one-to-another nonequivalent irreducible unitary representations:

1. The unitary characters of the group, i.e. the one-dimensional unitary representation U_{λ}, λ ∈ ℂ, acting in ℂ following the formula

U_{λ}(z, w) = e^{iℜ(z\overline{λ})}, ∀(z, w) ∈ { \atop Aff(ℂ)}, λ ∈ ℂ.

2. The infinite dimensional irreducible representations T_{θ}, θ ∈ S^{1}, acting on the Hilbert space L^{2}(ℝ × S^{1}) following the formula:

[T_{θ}(z, w)f](x) = exp(i(ℜ(wx) + 2πθ[\frac{ℑ(x + z)}{2π}]))f(x ⊕ z),  (7)

where (z, w) ∈ { \atop Aff(ℂ)}; x ∈ ℝ × S^{1} = ℂ\\{0\}; f ∈ L^{2}(ℝ × S^{1});

x ⊕ z = ℜ(x + z) + 2πi\{\frac{ℑ(x + z)}{2π}\}.

In this section we will prove the following important theorem which is of interest for us both in theory and practice.

Theorem 4.2. The representation exp(ℓ̂_{A}^{(k)}) of the group { \atop Aff(ℂ)} is the irreducible unitary representation T_{θ} of the group { \atop Aff(ℂ)} associated to Ω by the orbit method, i.e.

exp(ℓ̂_{A}^{(k)})f(x) = [T_{θ}(exp A)f](x),

where f ∈ L^{2}(ℝ × S^{1}); A = (\begin{matrix}α & β \\ 0 & 0 \end{matrix}) ∈ aff(ℂ); θ ∈ S^{1}; k = 0, ±1, …

Proof. Putting x = e^{u} ∈ ℂ\{0} = ℝ × S^{1} and recall that

(\begin{matrix}a & b \\ 0 & 1 \end{matrix}) = exp(A) = exp (\begin{matrix}α & β \\ 0 & 0 \end{matrix}) ,

428 Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-Adjoint Orbits …

we can rewrite (7) as following:

[T_{θ}(exp A)f](e^{u}) = exp(i(ℜ(\frac{e^{α} − 1}{α}βe^{u}) + 2πθ[\frac{ℑe^{u+α}}{2π}]))f(e^{u⊕α}),

where

u ⊕ α = ℜ(u + α) + 2πi\{\frac{ℑ(u + α)}{2π}\} = u + α − 2πi[\frac{ℑ(u + α)}{2π}].

Therefore, for the one-parameter subgroup exp tA, t ∈ ℝ, we have the action formula:

[T_{θ}(exp tA)f](e^{u}) = exp(i(ℜ\frac{e^{tα} − 1}{α}βe^{u} + 2πθ[\frac{ℑe^{u+tα}}{2π}]))f(e^{u⊕tα}).

By a direct computation:

\frac{∂}{∂t}([T_{θ}(exp tA)f](e^{u})) =  (8)

= \frac{∂}{∂t}(exp(\frac{i}{2}(\frac{e^{tα} − 1}{α}βe^{u} + \frac{e^{t\overline{α}} − 1}{\overline{α}}\overline{β}e\overline{u}) + 2πθi[\frac{ℑe^{u+tα}}{2π}])) + f(e^{u+tα−2πi[ \frac{ℑ(u+tα)}{2π}]})

+ exp(\frac{i}{2}(\frac{e^{tα} − 1}{α}βe^{u} + \frac{e^{t\overline{α}} − 1}{\overline{α}}\overline{β}e\overline{u}) + 2πθi[\frac{ℑe^{u+tα}}{2π}])\frac{∂}{∂t}f(e^{u+tα−2πi[ \frac{ℑ(u+tα)}{2π}]}) =

= \frac{i}{2}(βe^{u+tα} + \overline{β}e^{\overline{u}+t\overline{α})}[T_{θ}(exp tA)f](e^{u}) + exp(i(ℜ(\frac{e^{tα} − 1}{α}βe^{u}) + 2πθi[\frac{ℑe^{u+tα}}{2π}])αe^{u⊕tα} \frac{∂f}{∂u}

on one hand. On the other hand, we have:

ℓ̂_{A}^{(k)}([T_{θ}(exp tA)f](e^{u}) =  (9)

= α\frac{∂}{∂u}([T_{θ}(exp tA)f](e^{u})) + \overline{α}\frac{∂}{∂\overline{u}}([T_{θ}(exp tA)f](e^{u})) + \frac{i}{2}(βe^{u} + \overline{β}e\overline{u})[T_{θ}(exp tA)f](e^{u})] =

= α\frac{i}{2}(\frac{e^{tα} − 1}{α}βe^{u}) exp(i(ℜ(\frac{e^{tα} − 1}{α}βe^{u}) + 2πθ[\frac{ℑe^{u+tα}}{2π}]))f(e^{u⊕tα})+

+α exp(i(ℜ(\frac{e^{tα} − 1}{α}βe^{u}) + 2πθ[\frac{ℑe^{u+tα}}{2π}]))e^{u⊕tα} \frac{∂f}{∂u}+

+\overline{α}\frac{i}{2}(\frac{e^{t\overline{α}} − 1}{\overline{α}}\overline{β}e\overline{u}) exp(i(ℜ(\frac{e^{tα} − 1}{α}βe^{u}) + 2πθ[\frac{ℑe^{u+tα}}{2π}]))f(e^{u⊕tα})+

+\frac{i}{2}(\overline{β}e\overline{u} + βe^{u})[T_{θ}(exp tA)f](e^{u}) = \frac{i}{2}(βe^{u+tα} + \overline{β}e^{\overline{u}+t\overline{α}})[T_{θ}(exp tA)f](e^{u})+

+ exp(i(ℜ(\frac{e^{tα} − 1}{α}βe^{u}) + 2πθ[\frac{ℑe^{u+tα}}{2π}]))αe^{u⊕αt} \frac{∂f}{∂u}

(8) and (9) imply that

\frac{∂}{∂t}[T_{θ}(exp tA)f](x) = ℓ̂_{A}^{(k)}([T_{θ}(exp tA)f](x)) ∀x ∈ ℝ × S^{1}.

Furthermore,

T_{θ}(exp tA)f](e^{u})|_{t=0} = exp(2πi[\frac{ℑe^{u}}{2π}]θ)f(e^{u−2πi[ \frac{ℑu}{2π}]}) = f(e^{u}).

This means that: exp(ℓ̂_{A}^{(k)})f(x) and [T_{θ}(exp tA)f](x) together are the solution of the Cauchy problem

\{ \begin{matrix}\frac{∂}{∂t}u(t, x) & = & ℓ̂_{A}^{(k)}u(t, x); \\ u(0, x) & = & id. \end{matrix}

The operator ℓ̂_{A}^{(k)} is sufficiently well-behaved, so that the Cauchy problem has a unique solution. From this uniqueness we deduce that

exp(ℓ̂_{A}^{(k)})f(x) ≡ [T_{θ}(exp tA)f](x) ∀x ∈ ℝ × S^{1}.

The theorem is hence proved. □

Remark 4.3. We say that a real Lie algebra 𝔤 is in the class \overline{MD} iff every K-orbit is of dimension 0 or dim 𝔤. Furthermore, it was proven in [4, Theorem 4.4] that, up to isomorphism, every Lie algebra of class \overline{MD} is one of the following:

1. commutative Lie algebras,

2. the Lie algebra aff(ℝ) of affine transformations of the real line,

3. the Lie algebra of affine transformations of the complex line. Thus, by calculation for the group of affine transformations of the real line Aff(ℝ)) in [5] and here for the group of affine transformations of the complex line Aff(ℂ) we obtained a description of the quantum \overline{MD} co-adjoint orbits.

References

[1] Arnal, D.; Cortet, J. C.: ⋆-product and representations of nilpotent Lie groups. J. Geom. Phys. 2 (1985)(2), 86–116.

[2] Arnal, D.; Cortet, J. C.: Représentations ⋆ des groupes exponentiels. J. Funct. Anal. 92 (1990), 103–135.

[3] Arnold, V. I.: Mathematical Methods of Classical Mechanics. Springer Verlag, Berlin - New York - Heidelberg 1984.

[4] Do Ngoc Diep: Noncommutative Geometry Methods for Group C*-Algebras. Chapman & Hall/CRC. Press #LM 2003, 1999.

[5] Do Ngoc Diep; Nguyen Viet Hai: Quantum half-plane via Deformation Quantization. Math. QA/9905002, 2 May 1999.

[6] Fedosov, B.: Deformation Quantization and Index Theory. Akademie Verlag, Berlin 1996.

[7] Gutt, S.: Deformation quantization. ICTP Workshop on Representation Theory of Lie groups. SMR 686/14, 1993.

[8] Gel’fand, I. M.; Naimark, M. A.: Unitary representations of the group of affine transformations of the straight line. Dokl. AN SSSR 55 (1947)(7), 571–574.

[9] Kirillov, A. A.: Elements of the Theory of Representations. Springer Verlag, Berlin - New York - Heidelberg 1976.

Received October 10, 1999; revised version December 12, 2000