Journal of Lie Theory Volume 7 (1997) 147–164 ◯^{C}1997 Heldermann Verlag

Contraction of an Adapted Functional Calculus

P. Cotton and A. H. Dooley

Communicated by J. Ludwig

Abstract. We aim to show, using the example of a Riemannian symmetric pair (G, K) = (SL_{2}(ℝ), SO(2)), how contraction ideas may be applied to functional calculi constructed on coadjoint orbits of Lie groups. We construct such calculi on principal series orbits and generic orbits of the Cartan motion group V ⋊ K, and show how the two are related. Since the calculi are adapted to the representations traditionally attached to the orbits, we recover at the Lie algebra level the contraction results of Dooley and Rice [5].

1. Introduction

Let G be a real semisimple Lie group with finite centre and K a closed subgroup making (G, K) a Riemannian symmetric pair ([8] p. 209). We have the Cartan decomposition 𝔤 = 𝔨 + V where V is an Ad(K)–invariant subspace and 𝔨 is the Lie algebra of K. The Cartan motion group associated with the pair (G, K) is the semidirect product V ⋊ K formed with respect to the adjoint action of K on V .

The motion group is related to G by a family of contraction mappings π_{λ} : V ⋊ K → G defined by

π_{λ}(v, k) = exp_{G}(λv) · k,

for v ∈ V and k ∈ K, and indexed by λ ∈ ℝ^{+}. These maps were introduced by Dooley and Rice in [5] and are approximate group homomorphisms. In [5] the unitary irreducible representations of V ⋊ K are obtained as limits of sequences of principal series representations of G composed with contraction mappings.

The derivatives dπ_{λ} : V + 𝔨 → 𝔤 have in a sense been around for longer, albeit in a less desirable form. The idea of obtaining a new Lie algebra by deformation of the structure constants first appears in Inonu and Wigner [9]. It is clear however that Dooley’s contraction mappings

dπ_{λ} (v, A) = λv + A

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag for v ∈ V ,A ∈ 𝔨 and λ ∈ ℝ^{+} represent a more coordinate free approach.

The present work is stimulated by the above considerations and a recent PhD thesis by Benjamin Cahen of the University of Metz [3]. Cahen has shown that on certain coadjoint orbits of G we may construct a symbol calculus (a mapping from a class of functions or distributions on the orbit to operators in a Hilbert space) which is adapted to the orbit in the following sense : Let O be some such orbit and dσ : 𝔤 → H the representation of 𝔤 associated with O by the traditional orbit methods [10]. For each X ∈ 𝔤, define the function X̃ on O by

X̃(ξ) = ⟨ξ, X⟩

for ξ ∈ O ⊂ 𝔤^{∗}. Then the symbol X̃ is assigned to an operator A_{X} acting on H, and

A_{X} = \frac{1}{i}dσ(X)

holds on a dense subspace of H.

Cahen’s procedure is analogous to that of Arnal and Cortet [1] and Wildberger [17], establishing first suitable parametrisations of the coadjoint orbits of a nilpotent (connected simply connected) group and using the same parametrisations given by Duflo in [6]. If O is a principal series orbit then we have a diffeomorphism

Ψ : \overline{N} × \overline{𝔫} × O_{M} → Õ

where \overline{N} is a nilpotent Lie group with Lie algebra \overline{𝔫}, O_{M} is a coadjoint orbit of a compact subgroup M ⊂ K and Õ is an open dense subset of O. A combination of the Berezin calculus (Berezin [2], Rawnsley [12]) on O_{M} and a generalisation of the Weyl calculus (Weyl [16], Voros [14]) on \overline{N} × \overline{𝔫} yields the desired calculus on O.

Cahen also provides a construction for a similarly adapted calculus on V ⋊ K –orbits which has prompted us to ask whether this second calculus may be obtained as a limit is some sense of the calculus on G–orbits. This paper is an attempt to answer this question, in the special case (G, K) = (SL_{2}(ℝ), SO(2)). Here the analysis is simplified by degeneracy of the orbits O_{M} mentioned in the previous paragraph. Operators in the Berezin calculus on O_{M} become complex numbers, and symbols in the calculus on O are precisely the symbols on the calculus on \overline{N} × \overline{𝔫} in Cahen [3] Paragraph 7.2, Chapter 1.

In Section 3 we find explicit realisations of the principal series representations of G in L^{2}(ℝ) and L^{2}(𝕋), where 𝕋 is the torus with underlying set (−π, π]. In Sections 4 and 5 we define calculi adapted to these realisations. The first is Cahen’s construction, defined by the Bruhat decomposition and referred to here as the Ψ–calculus. It maps symbols on a given orbit to operators in C_{c}^{∞}(ℝ). The second, which we call the Υ–calculus, uses instead the Iwasawa decomposition. It maps the same symbols to operators in C^{∞}(𝕋) and is defined on each orbit O using a diffeomorphism Υ : 𝕋 × ℝ → O. The Υ–calculus is better suited to our purposes.

In Section 7 we find irreducible unitary representations of the motion group V ⋊ K indexed in the same way as the principal series representations of G. We then establish the Γ–calculus using, for a given coadjoint orbit O, a diffeomorphism Γ : 𝕋 × ℝ → O. It is a slight modification of Cahen [3], and is adapted to the representations of V ⋊ K.

Section 9 summarises the results of Dooley and Rice [5] in the case (G, K) = (SL_{2}(ℝ), SO(2)). The representations of G are “contracted” onto the representations of V ⋊ K, thus introducing the point of this paper. We aim to contract the Υ–calculus on G–orbits onto the Γ–calculus on V ⋊ K –orbits.

We demonstrate two methods of doing this. The first scheme runs as follows. Suppose f is a symbol on a V ⋊ K –orbit C_{ψ} passing through ψ ∈ 𝔤^{∗} and let A_{f}^{Γ} denote its corresponding operator under the Γ–calculus, acting on C^{∞}(𝕋). We take a family of G–orbits O_{ψ/λ} passing through ψ/λ and a family of symbols f_{λ} on O_{ψ/λ} related to f in a simple way. The operators A_{f_{λ}}^{Υ} corresponding to the symbols f_{λ} then also act on C^{∞}(𝕋) and converge (in a sense defined later) to the original operator A_{f}^{Γ}. As the calculi are adapted, one recovers at the Lie algebra level the aforementioned results of Dooley and Rice [5] pertaining to the pair (G, K).

The second method is more direct. For a symbol f on a V ⋊ K –orbit with operator A_{f}^{Γ} we define f̃ = f ∘ Γ ∘ Υ^{−1}. Then f̃ is a symbol on a G–orbit O and if AΥ ; _{f} denotes the corresponding operator under the Υ–calculus we have A_{f}^{Γ} = AΥ ; _{f} on C^{∞}(𝕋). Thus the Υ–calculus contracts nicely onto the Γ–calculus and we have a contractive relationship between the Lie group G and its motion group V ⋊ K not involving limits. Interestingly there is an immediate converse. If f̃ is a symbol on a G–orbit O then f = f̃∘Υ∘Γ^{−1} is a symbol on a V ⋊K –orbit and AΥ ; _{f} = A_{f}^{Γ}.

2. SL_{2}(ℝ) and its motion group M̃(2)

Hereafter G = SL_{2}(ℝ) is the matrix Lie group consisting of two by two real matrices with determinant unity. Its Lie algebra 𝔤 = sl_{2}(ℝ) consists of traceless two by two real matrices. Fix a basis

e_{1} = ( \begin{matrix}1/2 & 0 \\ 0 & −1/2 \end{matrix} ) , e_{2} = ( \begin{matrix}0 & 1/2 \\ 1/2 & 0 \end{matrix} ) , e_{3} = ( \begin{matrix}0 & 1/2 \\ −1/2 & 0 \end{matrix} )

for 𝔤 = sl_{2}(ℝ). Letting β denote the Killing form we take {ê_{i} = β(e_{i}, ·)}, i = 1, .., 3, as a basis for 𝔤^{∗}. All subspaces of 𝔤 and their duals may thus be identified with subspaces of ℝ^{3}.

We have the Cartan decomposition 𝔤 = 𝔨 + V where k = sp {e_{3}} and V = sp {e_{1}, e_{2}} are the eigenspaces of Θ(X) = −X^{T}. The restriction of the adjoint representation

Ad(g)X = gXg^{−1}, g ∈ G, X ∈ 𝔤

to K = exp(𝔨) preserves both 𝔨 and V , allowing us to define the semi-direct product M̃(2) = V ⋊ K which is the Cartan motion group associated with the pair (G, K). That is, we equip V ⋊ K with the group multiplication

(v, k) · (v′, k′) = (kv′ + v, kk′)

for v, v′ ∈ V and k, k′ ∈ K. The Lie algebra m̃(2) of M̃(2) shares the same underlying vector space V + 𝔨 as 𝔤, but has Lie bracket

[(ω, A), (ω′, A′)]_{m(2)} = ([A, ω′]_{𝔤} − [A′, ω]_{𝔤} , 0)

for ω, ω′ ∈ V and A, A′ ∈ 𝔨. The Lie subalgebra 𝔞 = sp {e_{1}} is maximal abelian in V ⊂ 𝔤. Let A = exp(𝔞), 𝔞^{+} = {a e_{1} : a ∈ ℝ^{+}}, and A^{+} = exp(𝔞^{+}). Let M be the centraliser of A in K, and 𝔪 its Lie algebra. Put Z = K/M. Define also 𝔫 = sp \{e_{2} + e_{3}\}, \overline{𝔫} = sp \{e_{2} − e_{3}\}, N = exp(𝔫) and \overline{N} = exp(\overline{𝔫}). Take bases for 𝔫 and \overline{𝔫} which are orthonormal with respect to the inner product (X, Y) = β ((X, Θ(Y)) - namely {(e_{2} + e_{3})/2} and {(e_{2} − e_{3})/2}. This fixes parametrisations by ℝ of 𝔫 and \overline{𝔫}, and also, in a manner consistent with the exponential map, N and \overline{N}.

As G = SL_{2}(ℝ) we find M = {±1} and 𝔪 = {0}. Letting 𝕋 and 2𝕋 denote the tori with respective underlying sets (−π, π] and (−2π, 2π] we parametrise K by

θ ∈ 2𝕋 ↦ ( \begin{matrix}cos(θ/2) & sin(θ/2) \\ − sin(θ/2) & cos(θ/2) \end{matrix} ) ∈ K.

Similarly parametrise Z ≅ 𝕋. The parametrisations of 𝔫,\overline{𝔫}, N and \overline{N} given above are

b ∈ ℝ ↦ ( \begin{matrix}0 & b/2 \\ 0 & 0 \end{matrix} ) ∈ 𝔫,
y ∈ ℝ ↦ ( \begin{matrix}0 & 0 \\ y/2 & 0 \end{matrix} ) ∈ \overline{𝔫},
b ∈ ℝ ↦ ( \begin{matrix}1 & b/2 \\ 0 & 1 \end{matrix} ) ∈ N,
y ∈ ℝ ↦ ( \begin{matrix}1 & 0 \\ y/2 & 1 \end{matrix} ) ∈ \overline{N}.

We shall be making use of both the Iwasawa decomposition of G and a corollary of the Bruhat lemma found in Wallach [15]. For any g ∈ G we have unique elements 𝔨_{I}(g) ∈ K, 𝔞_{I}(g) ∈ A^{+} and 𝔫_{I}(g) ∈ N such that g = 𝔨_{I}(g)𝔞_{I}(g)𝔫_{I}(g). Similarly

if J = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) and g ∈ G\ \{±J\} then there are unique elements \overline{𝔫}_{B}(g) ∈ \overline{N},

𝔪_{B}(g) ∈ M, 𝔞_{B}(g) ∈ A^{+} and 𝔫_{B}(g) ∈ N such that g = \overline{𝔫}_{B}(g)𝔪_{B}(g)𝔞_{B}(g)𝔫_{B}(g).

We mention in passing that M̃(2) is the twofold cover of M(2) = V ⋊ Z, the Euclidean motion group in two dimensions, the covering being given by the epimorphism (v, k) ↦ (v, kM).

3. Principal Series Representations

To certain coadjoint orbits of SL_{2}(ℝ) may associate a pair of unitary irreducible representations. We find realisations of these representations in L^{2}(ℝ) and L^{2}(𝕋), and explicit formulas for the operators which intertwine them.

The Killing form on G is

β (∑_{i=1}^{3} x_{i}e_{i}, ∑_{i=1}^{3} y_{i}e_{i}) = 2(x_{1}y_{1} + x_{2}y_{2} − x_{3}y_{3}).

It is preserved by the adjoint action, and thus intertwines the adjoint and coadjoint representations since if ξ = β(Y, ·) ∈ 𝔤^{∗} then

⟨Ad^{∗}(g)ξ, X⟩ = ⟨ξ, Ad(g^{−1})X⟩
= β (Y, Ad(g^{−1})X)
= β (Ad(g)Y, X) .

Therefore the coadjoint orbits are precisely the images of the adjoint orbits under the duality e_{i} ↦ ê_{i}. Fix ψ = Rê_{1} ∈ 𝔞^{+}^{∗} for some R > 0. The coadjoint orbit passing through ψ is a hyperboloid of revolution about the ê_{3} axis intersecting the ê_{1}ê_{2}–plane in a circle radius R :

O_{ψ} = {ξ = ∑_{i=1}^{3} ξ_{i}ê_{i} ∈ 𝔤^{∗} : ξ_{1}^{2} + ξ_{2}^{2} − ξ_{3}^{2} = R^{2}} .

Fix a unitary irreducible representation η of M. There are only two such representations, indexed by η′ ∈ {0, 1} such that η(−1) = −1^{η′}. Let μ ∈ 𝔞^{∗} be half the sum of the positive roots; namely ê_{1}/4. Define log : A^{+} → a by exp (log(a)) = a for all a in A^{+}. The pair (ψ, η) gives us a unitary irreducible representation of MAN :

η ⊗ e^{γ} ⊗ 1(g) = η (𝔪_{I}(g)) e^{⟨μ+iψ,log𝔞_{I}(g)⟩}

where we consider γ = μ + iψ an element of the complexification of 𝔞^{∗}. We take some liberties with notation, writing the second factor above for example as e^{γ} (𝔞_{I}(g)). Inducing up to the whole group yields a unitary irreducible representation of G :

σ_{γ,η} = Ind_{MAN}^{G} η ⊗ e^{γ} ⊗ 1.

More precisely, let

H = {f ∈ C^{∞}(G) : f(gg_{0}) = η ⊗ e^{γ} ⊗ 1(g_{0}^{−1})f(g) ∀g_{0} ∈ MAN, g ∈ G} .

Discard those f in H whose restrictions to K are not square integrable with respect to Haar measure on K. Complete with respect to the semi-norm

||f|| := ||f_{|_{K}}||_{L^{2}(K)}

to obtain a Hilbert space H_{γ,η}^{G}. Then the operation of left translation,

σ_{γ,η}(g)f(g_{0}) = f(g^{−1}g_{0}),

on H_{γ,η}^{G} provides a unitary irreducible representation. The representations thus formed comprise the principal series. For f ∈ H_{γ,η}^{G}, g = k ∈ K and g_{0} = an ∈ AN we have

f(kan) = f(gg_{0})
= η ⊗ e^{γ} ⊗ 1(n^{−1}a^{−1})f(k)
= η ⊗ e^{γ} ⊗ 1 (a^{−1}(an^{−1}a^{−1})) f(k)
= e^{γ}(a^{−1})f(k)

since an^{−1}a^{−1} is in N. Thus elements of H_{γ,η}^{G} are determined by their restrictions to K, and there is a bijection from H_{γ,η}^{G} to a subspace H^{K} of square integrable functions on f : K → ℂ satisfying

f(km) = η(m^{−1})f(k) ∀k ∈ K, m ∈ M.

Clearly elements of H^{K} are in one to one correspondence with a collection H^{Z} of functions on 𝕋 ≅ Z square integrable with respect to Lebesgue measure. Indeed, transfering the action of G over to H^{Z} and completing yields a unitary representation of G in H_{γ,η}^{Z} = L^{2}(𝕋).

By a similar process replacing K with \overline{N}, or by directly relating ℝ and 𝕋, we obtain a representation of G in a Hilbert space H\overline{N} ; γ,η = L^{2}(ℝ). We now exhibit both realisations.

For f ∈ L^{2}(𝕋), F ∈ L^{2}(ℝ), y ∈ ℝ ≅ \overline{N}, θ ∈ 𝕋 ≅ Z, g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ G

and γ̃ = (1/4 + iR) we calculate

σ_{γ,η}^{Z}(g)f(θ) = [(d cos(θ/2) + b sin(θ/2))^{2} + (c cos(θ/2) + a sin(θ/2))^{2}]^{−2_{γ}}

×signum(d cos(θ/2) + b sin(θ/2))^{η′}

×f (2 tan^{−1} (\frac{c cos(θ/2) + a sin(θ/2)}{d cos(θ/2) + b sin(θ/2)}))

σ\overline{N} ; γ,η(g)F (y) = (d − by/2)^{−4_{γ}}signum(d − by/2)^{η′}F (2\frac{ay/2 − c}{d − by/2}) .

Next we calculate the representations of 𝔤 derived from the representations of G above. For X = ∑_{i=1}^{3} x_{i}e_{i} ∈ 𝔤 we have

dσ_{γ,η}^{Z}(X)f(θ) = 2γ̃ (x_{1} cos θ − x_{2} sin θ) f(θ)
+ (x_{2} cos θ + x_{1} sin θ − x_{3}) f′(θ)  (3.1)
dσ\overline{N} ; γ,η(X)F (y) = 2γ̃ (x_{1} + x_{2}y/2 + x_{3}y/2) F (y)
+ (x_{2}(y^{2}/4 − 1) + x_{3}(y^{2}/4 + 1) + yx_{1}) F′(y) (3.2)

where primes indicate differentiation with respect to θ and y respectively.

Proposition 3.1. The representations of G and 𝔤 on L^{2}(𝕋) and L^{2}(ℝ) are intertwined by I : D → C_{c}^{∞}(ℝ) where D is a dense subspace of C^{∞}(𝕋) and

I(f)(y) = (4 + y^{2})^{−2_{γ}} f (−2 tan^{−1}(y/2)) ,

for all f ∈ D.

Proof. We may take D = {f ∈ C^{∞}(𝕋) : π ∉ suppf}. Now I has inverse

I^{−1}(F)(θ) = (4 + tan^{2} θ/2)^{2_{γ}} F (−2 tan(θ/2))

for F ∈ C_{c}^{∞}(ℝ). Since I^{−1}(C_{c}^{∞}(ℝ)) ⊂ D, I(D) = C_{c}^{∞}(ℝ). Thus D and I(D) are dense subspaces of L^{2}(𝕋) and L^{2}(ℝ). As we are primarily interested in representations of 𝔤, we check directly that I respects the 𝔤 actions on these dense subspaces using the following convenient definitions. The remainder of the proof is left to the reader. Let

p(y) = −2 tan^{−1}(y/2)
κ_{0}(θ) = 2γ̃ (x_{1} cos θ − x_{2} sin θ)
κ_{1}(θ) = x_{2} cos θ + x_{1} sin θ − x_{3}
τ_{0}(y) = 2γ̃ (x_{1} + x_{2}y/2 + x_{3}y/2)
τ_{1}(y) = x_{2}(y^{2}/4 − 1) + x_{3}(y^{2}/4 + 1) + yx_{1}
φ(y) = (4 + y^{2})^{−2_{γ}}

and for f ∈ C_{c}^{∞}(ℝ), y ∈ ℝ put θ = p(y), F = I(f). Writing X · f for σ_{γ,η}^{Z}(X)f, X · F for σ\overline{N} ; γ,η(X)F and t = tan(θ/2) we have

(X · I(f)) (y) = τ_{0}(y)F (y) + τ_{1}(y)F′(y)
= τ_{0}(y)φ(y)f ∘ p(y) + τ_{1}(y) [φ′(y)f ∘ p(y) + φ(y)p′(y)f′ ∘ p(y)]
= φ(y)τ_{0} ∘ p^{−1}(θ) + φ(y)τ_{1} ∘ p^{−1}(θ)f(θ)

+τ_{1} ∘ p^{−1}(θ)f′(θ)p′ ∘ p^{−1}(θ)

= 2γ̃φ(y) \{[τ_{0}(−2t) + \frac{t}{1 + t^{2}} τ_{1}(−2t)] f(θ)

+ [\frac{−1}{1 + t^{2}} τ_{1}(−2t)] f′(θ)\}

= 2γ̃φ(y) \{[x_{1} − x_{2}t − x_{3}t + \frac{t}{1 + t^{2}} (x_{2}(t^{2} − 1) + x_{3}(t^{2} + 1)

−2tx_{1})] f(θ) − \frac{1}{1 + t^{2}} [x_{2}(t^{2} − 1) + x_{3}(t^{2} + 1) − 2tx_{1}] f′(θ)\}

= 2γ̃φ(y) \{[x_{1} \frac{1 − t^{2}}{1 + t^{2}} − x_{2} \frac{−2t}{1 + t^{2}} ] f(θ)

− [x_{1} \frac{2t}{1 + t^{2}} + x_{2} \frac{1 − t^{2}}{1 + t^{2}} − x_{3}] f′(θ)\}

= 2γ̃φ(y) {[x_{1} cos θ − x_{2} sin θ] f(θ)

− [x_{1} sin θ + x_{2} cos θ − x_{3}] f′(θ)}

= φ(y) {κ_{0}(θ)f(θ) + κ_{1}(θ)f′(θ)}
= φ(y) (X · f) (p(y))
= I (X · f) (y),  (3.3)

showing that I respects the 𝔤 actions on (dense subspaces of) L^{2}(ℝ) and L^{2}(𝕋).

Next we give two examples of symbol calculi on the coadjoint orbits of G, both adapted to the orbits in the sense given in the introduction. The first is a special case of the construction given in Cahen [3] for real semisimple Lie groups with finite centre. We aim to compare this with a second calculus constructed using the Iwasawa decomposition and better suited to contractions.

4. The Ψ–Calculus on G–Orbits

The prescription for a symbol calculus on the adjoint orbits of SL_{2}(ℝ) given in [3] amounts to a transferral of the classical Weyl correspondence on ℝ^{2} ([16], [11], [14], [7]) to the orbit O_{ψ} by the parametrisation

Ψ(y, z) = (R + yz/2)ê_{1} + (−z/2 + Ry/2 + zy^{2}/8)ê_{2} + (−z/2 − Ry/2 − zy^{2}/8)ê_{3}

which is a diffeomorphism from ℝ^{2} to Õ_{ψ} = O_{ψ}\ {−Rê_{1}}. For this reason we shall refer to it as the Ψ–calculus. We now describe this calculus, essentially following Voros [14].

We say that f : Õ_{ψ} → ℂ is polynomial in z, or simply polynomial, if f ∘ Ψ(y, z) is a polynomial in z whose coefficients are smooth functions of y. For such an f we define a differential operator on C_{c}^{∞}(ℝ) by the formula

(A_{f}^{Ψ}ϕ) (y) = \frac{1}{2π} ∫_{ℝ×ℝ} e^{−itz}f ∘ Ψ (y + t/2, z) ϕ(y + t) dt dz,

for ϕ ∈ C_{c}^{∞}(ℝ). This is precisely Formula 5.1 of Cahen [3] in the special case G = SL_{2}(ℝ) with the identifications of Section 2. It is the correspondence introduced by Weyl in [16] applied to f ∘ Ψ. To see how the definition works, let f ∘ Ψ = u(y)z^{a} for some a ∈ ℕ, u ∈ ℂ^{∞}(ℝ). For ϕ ∈ C_{c}^{∞}(ℝ) define ϑ_{y}(t) = u(y + t/2)ϕ(y + t). Integrating by parts and applying the Fourier inversion theorem we have

(A_{f}^{Ψ}ϕ) (y) = \frac{1}{2π} ∫_{ℝ×ℝ} e^{−itz}u(y + t/2)z^{a}ϕ(y + t) dtdz
= \frac{1}{2π} ∫_{ℝ×ℝ} e^{−itz}z^{a}ϑ_{y}(t) dtdz
= \frac{1}{2π} ∫_{ℝ×ℝ} (−i\frac{d}{dt})^{a} ϑ_{y}(t) e^{−itz} dtdz
= [(−i\frac{d}{dt})^{a} ϑ_{y}]_{t=0}
= [(−i\frac{d}{dt})^{a} (u(y + t/2)ϕ(y + t))]_{t=0}
= ∑_{k=0}^{a} c_{k} \frac{d^{k}}{dy^{k}} ϕ(y)

where

c_{k} = (−i)^{a} ( \begin{matrix}a \\ k \end{matrix} ) 2^{(k−a)} \frac{d^{(a−k)}}{dy^{(a−k)}} u(y).

This permits an easy calculation of A_{f}^{Ψ} for polynomial f. As demanded by [3], the calculus is adapted to the representation (3.2). If X = ∑_{i=1}^{3} x_{i}e_{i} ∈ 𝔤 and X̃ defined by X̃(ξ) = ξ(X) for ξ ∈ 𝔤^{∗} we have

X̃ ∘ Ψ(y, z) = g(y) + h(y)z

where g(y) = 2R(x_{1} +x_{3}y/2+x_{2}y/2) and h(y) = x_{2}(y^{2}/4−1)+x_{3}(y^{2}/4+1)+x_{1}y. Denoting derivatives with respect to y by primes we therefore have

(iAΨ ; _{X}ϕ) (y) = i \{g(y)ϕ(y) − \frac{i}{2}h′(y)ϕ(y) − \frac{1}{i}h(y)ϕ′(y)\}

= 2 (1/4 + iR) (x_{1} + x_{2}y/2 + x_{3}y/2) ϕ(y)

+ (x_{1}y + x_{2}(y^{2}/4 − 1) + x_{3}(y^{2}/4 + 1)) ϕ′(y)

= τ_{0}(y)ϕ(y) + τ_{1}(y)ϕ′(y)
= (X · ϕ) (y).

for all ϕ ∈ C_{c}^{∞}(ℝ). That is,

AΨ ; _{X} = \frac{1}{i}dσ\overline{N} ; γ,η(X)  (4.4)

on C_{c}^{∞}(ℝ).

5. The Υ–Calculus on G–Orbits

Define Υ : 𝕋 × ℝ → O_{ψ} by

Υ(θ, z) = (R cos θ − \frac{z}{2} sin θ) ê_{1} − (R sin θ + \frac{z}{2} cos θ) ê_{2} − (\frac{z}{2}) ê_{3}.  (5.5)

This map is a slight modification of Ψ, where we take O_{ψ} = Ad(KAN)ψ in place of Õ_{ψ} = Ad(\overline{N}MAN)ψ. It is a diffeomorphism from 𝕋 × ℝ to O_{ψ}.

Say f : O_{ψ} → ℂ is polynomial in z if f ∘Υ(θ, z) is a polynomial in z whose coefficients are smooth functions of θ. For f supported in Õ_{ψ} this agrees with the definition of the previous section since

f ∘ Υ(θ, z) = f ∘ Ψ ∘ Ψ^{−1} ∘ Υ(θ, z)
= f ∘ Ψ (−2 tan(θ/2), R sin θ + \frac{z}{2} (cos θ + 1))

and

f ∘ Ψ(y, z) = f ∘ Υ ∘ Υ^{−1} ∘ Ψ(y, z)
= f ∘ Υ (−2 tan^{−1}(y/2), (1 + y^{2}/4) z + Ry)

so f ∘ Ψ is polynomial in z with smooth coefficients in y if and only if f ∘ Υ is polynomial in z with smooth coefficients in θ. For such an f define an operator A_{f}^{Υ} acting on ϕ ∈ C^{∞}(𝕋) by

(A_{f}^{Υ}ϕ) (θ) = \frac{1}{2π} ∫_{𝕋×ℝ} e^{itz}f ∘ Υ (θ + t/2, z) ϕ(θ + t) dt dz.  (5.6)

If f ∘ Υ(θ, z) = v(θ)z^{a} for some v ∈ C^{∞}(𝕋) and a ∈ ℕ then by similar manipulations to those of Section 4. we obtain

(A_{f}^{Υ}ϕ) (θ) = ∑_{k=0}^{a} c̃_{k} \frac{d^{k}}{dθ^{k}} ϕ(θ)

where

c̃_{k} = (i)^{a} ( \begin{matrix}a \\ k \end{matrix} ) 2^{(k−a)} \frac{d^{(a−k)}}{dθ^{(a−k)}} v(θ).

This calculus is also adapted to the orbit O_{ψ}. With X̃ as before

X̃ ∘ Υ(θ, z) = g̃(θ) + h̃(θ)z

with g̃(θ) = 2R(x_{1} cos θ − x_{2} sin θ) and h̃(θ) = −x_{1} sin θ − x_{2} cos θ + x_{3}. Thus, denoting derivatives with respect to θ by primes,

(iAΥ ; _{X}ϕ) (θ) = i \{g̃(y)ϕ(y) + \frac{i}{2}h̃′(θ)ϕ(θ) + ih̃(θ)ϕ′(θ)\}
= (i\frac{2R}{2γ′}κ_{0}(θ) + \frac{1}{2} \frac{1}{2γ′} κ_{0}(θ)) ϕ(θ) + κ_{1}(θ)ϕ′(θ)
= κ_{0}(θ)ϕ(θ) + κ_{1}(θ)ϕ′(θ)
= (X · ϕ) (θ).

That is,

AΥ ; _{X} = \frac{1}{i}dσ_{γ,η}^{Z}(X).  (5.7)

on C^{∞}(𝕋), showing that the Υ–calculus is adapted to the representation (3.1).

6. Non-equivalence of the Calculi

The operators of the Ψ-calculus act on C_{c}^{∞}(ℝ), whereas the operators of the Υ-calculus act on C^{∞}(𝕋). It is natural to ask whether, for every fixed symbol f on O_{ψ}, the operators A_{f}^{Ψ} and A_{f}^{Υ} are intertwined by the maps I and I^{−1} of Section 3. The two calculi would then be equivalent.

We conclude this section by showing that this is not the case. If A and B are operators on C_{c}^{∞}(ℝ) and C^{∞}(𝕋) respectively we say that A is equivalent to B and write A ∼ B if A = I ∘ B ∘ I^{−1}. Equations (4.4), (5.7) and (3.3) show that

AΨ ; _{X} ∼ AΥ ; _{X}

for all X ∈ 𝔤. No such equivalence holds for arbitrary symbols however. Let f = ê_{3}. As a special case of the above equation we have A_{f}^{Ψ} ∼ A_{f}^{Υ}, and it is easily seen that A_{f^{2}}^{Υ} = A_{f}^{Υ} ∘ A_{f}^{Υ}. Suppose A_{f^{2}}^{Ψ} ∼ A_{f^{2}}^{Υ} . This immediately gives

(A_{f^{2}}^{Ψ} − A_{f}^{Ψ} ∘ A_{f}^{Ψ}) ϕ(y) = (I ∘ A_{f^{2}}^{Υ} ∘ I^{−1} − I ∘ A_{f}^{Υ} ∘ I^{−1} ∘ I ∘ A_{f}^{Υ} ∘ I^{−1}) ϕ(y)
= I ∘ (A_{f^{2}}^{Υ} − A_{f}^{Υ} ∘ A_{f}^{Υ}) ∘ I^{−1}ϕ(y)
= 0.

which (we omit the calculation) is false for any ϕ in C_{c}^{∞}(ℝ) not identically zero at any point y ≠ 0. We conclude that A_{f^{2}}^{Ψ} is not equivalent to A_{f^{2}}^{Υ} .

7. Representations of the Motion Group

To certain coadjoint orbits of M̃(2) we associate a pair of unitary irreducible representations. This collection of representations is parametrised in the same way as the principal series representations of SL_{2}(ℝ). They are limits, in a precise sense, of these representations.

The coadjoint representation of M̃(2) is neatly expressed as

(v, k) · (p, f) = (k · p, k · f + v ∧ k · p)

for v ∈ V ,k ∈ K, p ∈ V^{∗} and f ∈ 𝔨^{∗}. Here k · f = Ad_{K}^{∗}(k)f, k · p = Ad_{G}(k)p and v ∧ p ∈ 𝔨^{∗} is defined by

⟨v ∧ p, A⟩ = ⟨p, A · v⟩

for A ∈ 𝔨 and A · v = [A, v]_{𝔤}. The wedge product for SL_{2}(ℝ) is

(v_{1}e_{1} + v_{2}e_{2}) ∧ (p_{1}ê_{1} + p_{2}ê_{2}) = (p_{1}v_{2} − p_{2}v_{1}) ê_{3}.

and the coadjoint orbits of M are cylinders centred along the ê_{3} axis. As in Section 3. we fix an orbit C_{ψ} passing through an element ψ = Rê_{1} ∈ 𝔞^{+}^{∗} and a unitary irreducible representation η of M. Just as was the case for the principal series representations of SL_{2}(ℝ), the pair (ψ, η) determines an irreducible unitary representation of M̃(2). We begin with the unitary irreducible representation e^{iψ} ⊗ η of V ⋊ M and induce up to a representation

ρ_{ψ,η} = Ind_{V ×M}^{V ⋊K} e^{iψ} ⊗ η

of the whole group M̃(2). For details see Dooley [5].

In a manner analogous to Section 3. we realise the representation in the Hilbert space H_{γ,η}^{Z} = L^{2}(𝕋). Again, details are found in [5]. The realisation on L^{2}(𝕋) is given by

(ρ_{ψ,η}^{Z}(v_{0}, θ_{0})ϕ) (θ) = e^{2iR(v_{1} cos θ+v_{2} sin θ)}ϕ (θ − θ_{0})

for ϕ ∈ L^{2}(𝕋), θ, θ_{0} ∈ 𝕋 and v_{0} = v_{1}e_{1} + v_{2}e_{2} ∈ V . Its derivative is

dρ_{ψ}^{Z}(ω, A)ϕ(θ) = 2iR (ω_{1} cos θ − ω_{2} sin θ) ϕ(θ) − Aϕ′(θ)  (7.8)

for ω = ω_{1}e_{1} + ω_{2}e_{2} ∈ V , A = Ae_{3} ∈ 𝔨, θ ∈ Z and ϕ ∈ L^{2}(𝕋).

8. The Γ–Calculus on V ⋊ K Orbits

Define the diffeomorphism Γ : 𝕋 × ℝ → C_{ψ} by

Γ(θ, z) = (R cos θ) ê_{1} − (R sin θ) ê_{2} − (z/2) ê_{3}.  (8.9)

Say f : C_{ψ} → ℂ is polynomial in z if f ∘ Γ is a polynomial in z whose coefficients are smooth functions of θ. For such an f define an operator A_{f}^{Γ} on C^{∞}(𝕋) by

A_{f}^{Γ}ϕ(θ) = \frac{1}{2π} ∫_{𝕋×ℝ} e^{itz}f ∘ Γ(θ + t/2, z)ϕ(θ + t) dz dt.  (8.10)

This is almost equivalent to Cahen’s procedure. The first difference is a factor of R in our parametrisation in the ê_{3} direction, which is compensated for by elimination of a factor of R in the exponent itz. The second difference is that we have replaced f ∘ Γ(θ, z) with f ∘ Γ(θ + t/2, z) in the integration.

We proceed in a similar fashion to Section 5. Suppose f(θ, z) = w(θ)z^{a} for some w ∈ ℂ^{∞}(𝕋), a ∈ ℕ. For ϕ ∈ C^{∞}(𝕋) define ϑ_{θ} : ℝ × ℝ → ℂ by ϑ_{θ}(t, z) = w(θ + t/2)ϕ(θ + t/2)χ_{(−π,π]}(t) where χ_{(−π,π]} is a characteristic function. We have

A_{f}^{Γ}ϕ(θ) = \frac{1}{2π} ∫_{ℝ×𝕋} e^{itz}z^{a}w(θ + t/2)ϕ(θ + t/2) dt dz
= \frac{1}{2π} ∫_{ℝ×ℝ} e^{itz}z^{a}ϑ_{θ}(t) dt dz
= \frac{1}{2π} ∫_{ℝ×ℝ} (i\frac{d}{dt})^{a} ϑ_{θ}(t) e^{itz} dt dz
= [(i\frac{d}{dt})^{a} ϑ_{θ}]_{t=0}
= [(i\frac{d}{dt})^{a} (w(θ + t/2)ϕ(θ + t))]_{t=0}
= ∑_{k=0}^{a} ˜ c̃_{k} \frac{d^{k}}{dθ^{k}} ϕ(θ)

where

˜c̃_{k} = (i)^{a} ( \begin{matrix}a \\ k \end{matrix} ) 2^{(k−a)} \frac{d^{(a−k)}}{dθ^{(a−k)}} w(θ).

In particular, if f = X̃ where X = (ω, A) = (ω_{1}e_{1} + ω_{2}e_{2}, Ae_{3}) ∈ V + 𝔨 then

X̃ ∘ Γ (θ, z) = 2R (ω_{1} cos θ − ω_{2} sin θ) + Az

and we calculate

iAΓ ; _{X}ϕ(θ) = 2iR (ω_{1} cos θ − ω_{2} sin θ) ϕ(θ) − Aϕ′(θ).

That is,

AΓ ; _{X} = \frac{1}{i}dρ_{ψ}^{Z}(X),  (8.11)

on C^{∞}(𝕋) showing that the calculus is adapted to the representation (7.8).

9. Contraction of G to V ⋊ K

In Section 1 we mentioned the contraction maps of Dooley defined by

π_{λ}(v, k) = exp_{G}(λv) · k
dπ_{λ}(ω, A) = λω + A

for v ∈ V ,k ∈ K, ω ∈ V , A ∈ 𝔨 and λ ∈ ℝ^{+}. Here we briefly illustrate how these may be used to contract the Lie algebra structure and representations of SL_{2}(ℝ) to those of its motion group M̃(2). These results are special cases of theorems in Dooley and Rice [5].

Proposition 9.1. For all X, Y ∈ m̃(2),

lim_{λ→0} dπ_{λ}^{−1} [dπ_{λ}(X), dπ_{λ}(Y)]_{𝔤} = [X, Y]_{m(2)}

Proposition 9.2. For all ϕ ∈ C^{∞}(𝕋),

sup_{θ∈𝕋} |dσ_{ψ/λ,η}^{Z} ∘ dπ_{λ}(ω, A)ϕ(θ) − dρ_{ψ}^{Z}(ω, A)ϕ(θ)| → 0

as λ → 0 for all (ω, A) ∈ m̃(2). Moreover the limit is obtained uniformly on compact subsets of m̃(2).

Proposition 9.3. For all ϕ ∈ C^{∞}(𝕋)

sup_{θ∈𝕋} |σ_{ψ/λ,η}^{Z} ∘ dπ_{λ}(v, θ′)ϕ(θ) − ρ_{ψ}^{Z}(v, θ′)ϕ(θ)| → 0

as λ → 0 uniformly on compact subsets of M̃(2).

Proof. These properties are routinely verified. We check only the second claim, which will be of greatest interest to us. Suppose ϕ ∈ C^{∞}(𝕋). If ω = ω_{1}e_{1} +ω_{2}e_{2} ∈ V and A = Ae_{3} ∈ 𝔨 then

dσ_{ψ/λ,η}^{Z} (dπ_{λ}(ω, A)) ϕ(θ) = dσ_{ψ/λ,η}^{Z} (λω, A) ϕ(θ)
= 2 (1/4 + iR/λ) (λω_{1} cos θ − λω_{2} sin θ) ϕ(θ)
+ (λω_{2} cos θ + λω_{1} sin θ − A) ϕ′(θ)
= 2iR (ω_{1} cos θ − ω_{2} sin θ) ϕ(θ) − Aϕ′(θ) + O(λ)
= dρ_{ψ}^{Z}(ω, A)ϕ(θ) + O(λ)

where

O(λ) = λ \{\frac{1}{2} (ω_{1} cos θ − ω_{2} sin θ) ϕ(θ) + (ω_{1} sin θ + ω_{2} cos θ) ϕ′(θ)\} .

Therefore

|dσ_{ψ/λ,η}^{Z} ∘ dπ_{λ}(ω, A)ϕ(θ) − ρ_{ψ}^{Z}(ω, A)ϕ(θ)| ≤ λ|ω| (|ϕ(θ)| + |ϕ′(θ)|)

for all ϕ ∈ C^{∞}(𝕋),θ ∈ 𝕋. Property 2 follows.

10. Contraction of the Υ–Calculus

Fix a V ⋊ K –orbit C_{ψ} passing through ψ ∈ 𝔞^{+}^{∗} and suppose f : C_{ψ} → ℂ is polynomial in z of degree d. That is,

f ∘ Γ(θ, z) = ∑_{a=0}^{d} u_{a}(θ)z^{a}

say, where each u_{a} ∈ C^{∞}(𝕋). For λ > 0 the G–orbit O_{ψ/λ} passing through ψ/λ is parametrised by

Υ_{λ}(θ, z) = (\frac{R}{λ} cos θ − \frac{z}{2} sin θ) ê_{1} − (\frac{z}{2} sin θ + \frac{R}{λ} cos θ) ê_{2} − (\frac{z}{2}) ê_{3}.

Suppose also that F = {f_{λ} : O_{ψ/λ} → ℂ}_{λ∈ℝ^{+}} is a family of polynomials in z of degree equal to the degree of f :

f_{λ} ∘ Υ_{λ}(θ, z) = ∑_{a=0}^{d} v_{a}^{λ}(θ)z^{a}

for v_{a}^{λ} ∈ C^{∞}(𝕋). If (v_{a}^{λ} − u_{a}) and all its derivatives of degree less than or equal to d tend uniformly to zero on 𝕋 as λ tends to zero then we say that the family F approximates f.

Proposition 10.1. Let f be a polynomial symbol on a V ⋊ K –orbit C_{ψ} and suppose F = {f_{λ}}_{λ∈ℝ} is an approximating family of polynomial symbols on G– orbits O_{ψ/λ} as above. Let A_{f}^{Γ} (respectively A_{f_{λ}}^{Υ} ) denote the operator on C^{∞}(𝕋) corresponding to f (respectively f_{λ}) under the Γ–calculus on C_{ψ} (respectively Υ– calculus on O_{ψ/λ}). Then for all ϕ ∈ C^{∞}(𝕋)

sup_{θ∈𝕋} |A_{f_{λ}}^{Υ}ϕ(θ) − A_{f}^{Γ}ϕ(θ)| → 0

as λ → 0.

Proof. With the same notation as above and recalling the results of sections 5. and 7. we have

A_{f_{λ}}^{Υ}ϕ(θ) − A_{f}^{Γ}ϕ(θ) = ∑_{a=0}^{d} ∑_{k=0}^{d}(i)^{a} ( \begin{matrix}a \\ k \end{matrix} ) 2^{(k−a)} (\frac{d}{dθ} )^{(a−k)} (v_{a}^{λ} − u_{a})(θ) (\frac{d}{dθ} )^{k} (ϕ)(θ)

for ϕ ∈ C^{∞}(𝕋),θ ∈ 𝕋. As all derivatives of ϕ appearing above are uniformly bounded in 𝕋 and all derivatives of (v_{a}^{λ} − u_{a}) tend uniformly to zero in 𝕋, each term tends uniformly to zero on 𝕋.

Corollary 10.2. For all ϕ ∈ C^{∞}(𝕋) and (ω, A) ∈ m̃(2)

sup_{θ∈𝕋} |dσ_{ψ/λ,η}^{Z} ∘ dπ_{λ}(ω, A)ϕ(θ) − dρ_{ψ}^{Z}(ω, A)ϕ(θ)| → 0

as λ → 0.

Proof. For X = ω_{1}e_{1} + ω_{2}e_{2} + Ae_{3} ∈ m̃(2),λ ∈ ℝ^{+} let f = X̃ and define the family F = d {f_{λ} = π̃_{λ}(X)}_{λ∈ℝ^{+}} . Then

f_{λ} ∘ Υ_{λ}(θ, z) = dπ̃_{λ}(X) ∘ Υ_{λ}(θ, z)
= β(λω_{1}e_{1} + λω_{1}e_{2} + Ae_{3},

(\frac{R}{λ} cos θ − \frac{z}{2} sin θ) e_{1} − (\frac{R}{λ} sin θ + \frac{z}{2} cos θ) e_{2} − (\frac{z}{2}) e_{3})

= 2(Rω_{1} cos θ − Rω_{2} sin θ) + (A − λω_{1} sin θ − λω_{2} cos θ)z,

whereas

f ∘ Γ(θ, z) = X̃ ∘ Γ(θ, z)
= 2(Rω_{1} cos θ − Rω_{2} sin θ) + Az.

so F approximates f. By Proposition 1,

sup_{θ∈𝕋} |A_{f_{λ}}^{Υ}ϕ(θ) − A_{f}^{Γ}ϕ(θ)| → 0

as λ → 0 for all ϕ ∈ C^{∞}(𝕋). Since A_{f_{λ}}^{Υ} = AΥ ; _{π_{λ}(X)} = −i dσ_{γ,η}^{Z} ∘ dπ_{λ}(X) and

A_{f}^{Γ} = AΓ ; _{X} = −i dρ_{ψ}^{Z}(X) the result follows.

We have thus essentially recovered Proposition 9.2. Uniform convergence on compact subsets on m̃(2) can be demonstrated by a slight generalisation of Proposition 1, which we leave to the reader.

The form of the above proposition above was chosen purely to illustrate how the contraction results of Dooley and Rice [5] at the Lie algebra level follow from the contraction of adapted calculi. We have however, a more direct approach.

Proposition 10.3. Let f be a function on C_{ψ}, polynomial in z. Let f̃ = f ∘ Γ ∘ Υ^{−1}. Then f̃ is a function of O_{ψ}, polynomial in z and

A_{f}^{Γ} = AΥ ; _{f}

on C^{∞}(𝕋). Conversely, if f̃ is a function on O_{ψ}, polynomial in z and f = f̃∘ Υ ∘ Γ^{−1} then f is a function on C_{ψ} polynomial in z and

AΥ ; _{f} = A_{f}^{Γ}.

Proof. As f̃ ∘ Υ = f ∘ Γ, f̃ is polynomial in z if and only if f is polynomial in z. The equality of operators follows immediately from the defining equations (5.6) and (8.10).

11. Non-polynomial Symbols

In Section 5 we said that f : O_{ψ} → ℂ was polynomial in z if f ∘Υ was polynomial in z. In the same manner [3] we define L^{p} functions, rapidly decreasing functions and distributions on O_{ψ}. Of interest to us here will be the Hilbert spaces L^{2}(O_{ψ}) consisting of those functions f on O_{ψ} for which f ∘ Υ ∈ L^{2}(𝕋 × ℝ). The aim of this section is to extend the Υ calculus to these functions. We first recall a few facts from [13].

The family L_{2}(H) of (compact) operators A on a Hilbert space H satisfying TrA^{∗}A < ∞ is a Hilbert space with respect to the inner product

(A, B) = Tr(A^{∗}B).

Such operators are called Hilbert-Schmidt. In the case H = L^{2}(M, dμ) for ⟨M, μ⟩ a measure space the Hilbert-Schmidt operators are precisely those with kernels K_{A} in L^{2}(M × M, dμ ⊗ dμ), the map A ↦ K_{A} being a Hilbert space isometry from L_{2}(L^{2}(M, dμ)) to L^{2}(M × M, dμ ⊗ dμ) [13].

Lemma 11.1. Equation (5.6) of Section 5 defines a map from L^{2}(O_{ψ}) to L_{2}(L^{2}(𝕋)). Further,

∥A_{f}^{Υ}∥_{L_{2}(L^{2}(𝕋))} ≤ ∥f∥_{L^{2}(O_{ψ})}

holds for all f ∈ L^{2}(O_{ψ}).

Proof. (Similar to [3], [14] and [11]). Define the partial Fourier transform F_{2} : L^{2}(𝕋 × ℝ) → L^{2}(𝕋 × ℝ) by

F_{2}(f)(θ, z) = \frac{1}{2π} ∫_{ℝ} f(θ, z)e^{−ixz}dz.

Then F_{2} is an isometry. Define also U : L^{2}(𝕋 × ℝ) → L^{2}(𝕋 × 𝕋) by

U(f)(θ, θ′) = f ((θ + θ′)/2, θ − θ′) .

As is easily checked, ∥U(f)∥_{L^{2}(𝕋×𝕋)} ≤ ∥f∥_{L^{2}(𝕋×ℝ)}. Finally let K = U ∘ F_{2}. Then for f ∈ L^{2}(O_{ψ}) we have

∥K(f ∘ Υ)∥_{L^{2}(𝕋×𝕋)} ≤ ∥F_{2}(f ∘ Υ)∥_{L^{2}(𝕋×ℝ)} = ∥f ∘ Υ∥_{L^{2}(𝕋×ℝ)} =: ∥f∥_{L^{2}(O_{ψ})}.

So K(f ∘ Υ) is in L^{2}(𝕋 × 𝕋) and is thus the kernel of a Hilbert-Schmidt operator on L^{2}(𝕋). Denoting this operator by Ã_{f} we then have

∥Ã_{f}∥_{L_{2}(L^{2}(𝕋))} = ∥K(f ∘ Υ)∥_{L^{2}(𝕋×𝕋)} ≤ ∥f∥_{L^{2}(O_{ψ})},

so if we can show that Ã_{f} = A_{f}^{Υ} we are done. Using Fubini’s theorem and the substitution t = θ′ − θ we have

Ã_{f}ϕ(θ) = ∫_{𝕋} K(f ∘ Υ)(θ, θ′)ϕ(θ′) dθ′
= ∫_{𝕋} \frac{1}{2π} ∫_{ℝ} f ∘ Υ ((θ + θ′)/2, z) e^{−iz(θ−θ′)} dz ϕ(θ′) dθ′
= \frac{1}{2π} ∫_{𝕋×ℝ} f ∘ Υ(θ + t/2)ϕ(θ + t)e^{izt} dtdz
= A_{f}^{Υ}ϕ(θ)

for all ϕ ∈ C^{∞}(𝕋), θ ∈ 𝕋.

We may similarly extend the Ψ–calculus (see [3], [11] or [14]) and the Γ–calculus to functions in L^{2}(O_{ψ}) and L^{2}(C_{ψ}) respectively. Taking this as done, we continue in the same vein as Section 10.

Proposition 11.2. Suppose f ∈ L^{2}(C_{ψ}) and F = {f_{λ} : O_{ψ/λ} → ℂ}_{λ∈ℂ} such that f_{λ} ∈ L^{2}(O_{ψ/λ}). We say that F approximates f in L^{2} sense if

∥f_{λ} ∘ Υ_{λ} − f ∘ Γ∥_{L^{2}(𝕋×ℝ)} → 0

as λ → 0. In this case A_{f_{λ}}^{Υ} tends to A_{f}^{Γ} in Hilbert-Schmidt norm.

Proof. By Lemma 11.1 the operators A_{f_{λ}}^{Υ} tend in Hilbert-Schmidt norm to the operator A_{f∘Γ∘Υ^{−1}}^{Υ} . By inspection of the defining equations (5.6) and (8.10), this operator is precisely A_{f}^{Γ}.

Proposition 11.3. If f ∈ L^{2}(C_{ψ}) then f̃:= f ∘ Γ ∘ Υ^{−1} is in L^{2}(O_{ψ}) and

A_{f}^{Γ} = AΥ ; _{f}.

Conversely if f̃ ∈ L^{2}(O_{ψ}) then f = f̃∘ Υ ∘ Γ^{−1} ∈ L^{2}(C_{ψ}) and

AΥ ; _{f} = A_{f}^{Γ}.

Proof. As f̃∘Υ = f ∘Γ, f̃ is an element of L^{2}(O_{ψ}) if and only if f is an element of L^{2}(C_{ψ}). Equality of operators follows directly from the defining equations (5.6) and (8.10).

12. Concluding Remarks

The authors expect the results of this paper to apply to all pairs (G, K) discussed in the introduction. For G a general semisimple Lie group with finite centre the subgroup M is of course no longer discrete (we recall that M was the centraliser of A in K, where K,A and N provide an Iwasawa decomposition). Whilst this introduces a fair amount of work in construction of the adapted calculi, it should not complicate the contraction process at all, since M may be treated in an identical fashion in both cases (as in Cahen [3]).

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School of Mathematics University of New South Wales NSW 2052 Australia A.Dooley@unsw.edu.au Department of Mathematics Stanford University Stanford CA 94305 U.S.A. cotton@gauss.Stanford.edu

Received September 6, 1996 and in final form October 23, 1996