Contractions of group representations via geometric quantisation

We propose a general framework to contract unitary dual of Lie groups via holomorphic quantization of their co-adjoint orbits. The sufficient condition for the contractability of a representation is expressed via cocycles on coadjoint orbits. This condition is checked explicitly for the contraction of ${\mathrm SU}_2$ into $\mathbb{H}$. The main tool is the geometric quantization. We construct two types of contractions that can be implemented on every matrix Lie group with diagonal contraction matrix.


Introduction
Physicists study certain degenerations of group laws which are called contractions. In this paper, we study Lie group contractions in the context of geometric quantization. The degeneration is understood in the sense that group law becomes "more abelian", i.e. more structure constants of Lie algebra becomes zero. The pioneering work of Inönü and Wigner [IW53] is a classical original source. Their motivation was simply to relate two different physics: Galilean and Lorentzian. Contractions of representation of Lie groups and Lie algebras and their unitary duals have attracted the attention of many authors [MN72,Ric86,RR86,Cah03,Doo83,Hig08,Hig11] and continue to be a field of active investigations, see for example [FP08,Gro04,BHS16,BHS17a].
The first general global definition of Lie group contraction has been given in [MN72]where the authors study the contraction of the principal continuous series of the de Sitter groups SO(n, 1) to positive mass representations of both the Euclidean group E(n) and the Poincaré group E 0 (n − 1, 1). In [Ric86], a contraction of SU (2) to the Heisenberg group H was studied in detail. The authors showed that the group action on the manifold depends smoothly on the contraction parameter ε > 0. They also established transference results linking Fourier analysis on these two groups. Applying [Ric86], Ricci and Rubin proved in [RR86] an analogue of de Leeuw's theorem relating L p -norms of Fourier multipliers on SU (2) and H. A short and elegant proof of the main result of [Ric86] was given in [Cah03] employing Berezin quantization. Furthermore, it was shown by [AJ85] that every semisimple Lie group can be contracted to its Cartan motion group.
Generally, one cannot expect that there always exists a contraction Φ ε : G 0 → G 1 for any Lie group G 0 and G 1 . Indeed, let g 1 and g 0 be the Lie algebras of G 1 and G 0 and U ε : g 1 → g 0 be differential of Φ ε . Then the existence of contraction U ε is equivalent to the fact that g 0 is the limit in the Zarissky topology on the affine variety of Lie algebras. It can be seen that not every two points in the affine variety of Lie algebras of dimension less than 6 are path-connected in the Zarissky topology [KN87,Theorem 1]. In other words, the existence of Lie algebra contraction U ε : g 0 → g 1 is equivalent to saying that g 0 is the limit of Lie algebras U ε g 1 in the Zarissky topology. Hence, we shall assume that there exists a Lie algebra contraction U ε of g 1 into g 0 .
The aim of the paper is to show that the existence of Lie group contraction combined with the geometric quantization yields a systematic approach to link harmonic analysis on two groups. The novelty of the paper is that we do not impose any restriction on the group nor on the contraction. In addition, we tried to answer the following question.
Question 1.1. Let G 0 and G 1 be two Lie groups and let Φ ε : G 1 → G 0 be a Lie group contraction. Does there exist a contraction Φ ε of G 1 into G 0 . In other words, does the following diagram commute (1.1) where GQ stands for the geometric quantization. We shall give meaning to Φ ε as an MNcontraction of G 1 into G 0 .
A similar problem has been treated in [BHS17b] in the context of the algebraic Harish-Chandra modules.
2. Geometric quantization 2.1. General theory. We first recall a few basic elements of the geometric quantization needed here. The procedure of quantization consists of assigning to a smooth function on a symplectic manifold (M, ω), an operator on a Hilbert space H with the following three properties. For each f : M → R, there should be an operatorf acting on H such that (1) the map f →f is linear; ( The main ingredient in the program of geometric quantization for a symplectic manifold (M, ω) of dimension 2n with symplectic (exact) 2-form ω is to construct a Hermitian line bundle π : B → M with Hermitian connection ∇ for which the curvature 2-form is equal to −1 ω. Locally, the connection ∇, or covariant derivative ∇ X for a vector field X on M , is given by where f is a function on M where the connection 1-form θ can be shown to be be locally related [Hal13,Proposition 23.5] to ω as dθ = ω. It is possible to construct such a line bundle on (M, ω) only is the symplectic manifold satisfies the integrality condition or is quantizable. Such a result is for example proved in [Hal13, Theorem 23.9, p.489] where the integrality condition is 1 2π where Σ arbitrary closed 2-surface in M . This condition is purely geometric and involves the computation of the holonomy of the line bundle for closed loop, which when contracted must be equal to 1. The condition (2.2) is obtained after using Stokes's theorem.
This line bundle has a natural Hilbert space H consisting of square-integrable global sections with respect to the natural Liouville measure of M and a Hermitian structure on the line bundle B. It is an inner product in the fibres denoted by (·, ·) with the property to be smooth in the sense that for a smooth section v : M → B, (v, v) is a smooth function of M . We now have the Hilbert space and we need to find the map f →f satisfying the three conditions. It is given, for a section s : M → B, by where X f is the Hamiltonian vector field associated with the function f , i.e. ω(X f , Y ) = df (Y ) for any vector field Y , and f s is the multiplication operation. This operator obviously satisfies (1) and (2), the third condition is satisfied by using the properties of the covariant derivative ∇ X f , see [Woo80,Hal13] for more details.
We will now follow [Woo80] to expose a dynamical interpretation off , which will give us a useful formula for later. Let V f be the vector field on B given in a local trivialisation by where z = re iφ is the coordinate on the fibre of L and is the Lagrangian associated to the function f , of Legendre transform of the Hamiltonian f . This vector field is constructed such that it is invariant under gauge transformations and f → V f is a Lie algebra isomorphism from C ∞ (M ) to a subalgebra of vector fields on B. Let ξ t denote the flow of V f which projects to ρ t , the flow of X f . From these two flows, we can define the action of the flow ρ t on the sections s : M → B by mapping it toρ t such that ξ t ( ρ t s(m)) = s(ρ t m) .
(2.6) This formula shows thatρ t is the correct 'pull-back' of the sections such that the flow ξ t corresponds to only moving the base point with ρ t . In a local trivialisation, the sections s and ρ t s are represented by the complex functions ψ and ρ t ψ with the following relation It might happen that the quantum Hilbert space is further reducible. A notion of polarization imposes further restrictions.

following conditions
(1) If two complex vector fields X and Y lie in P (z) at each point z, then the commutator Definition 2.2 (The quantum Hilbert space). Let P be a polarization of M . A smooth section s of L → M is said to be polarized if ∇ X s = 0, ∀X ∈ P . (2.8) The quantum Hilbert space H P associated with P is the closure in the prequantum Hilbert space of the space of smooth, square integrable, polarized sections s of L → M .
A purely complex polarization P yields existence of a unique integrable almost complex structure J on M such that P (z) is the (1, 0)-tanget space of M for all z ∈ M .
Definition 2.4. Let P be a purely complex polarization of M and let J be the integrable almost complex structure associated with P such that J(z) = i Id on P (z) and J(z) = −i Id on P (z). We say that P is a Kähler polarization if the bilinear form For a thorough exposition of the geometric quantization we refer [Hal13, Kos70a, Kir90, Woo80].
2.2. Lie group representation. We now specify this construction to Lie group representations. Let G be a locally compact Lie group with Lie algebra g. Each ξ ∈ g generates a right-invariant vector field R ξ . Let µ ∈ g * and O µ be the coadjoint orbit through µ. A coadjoint orbit is a symplectic manifold with the symplectic form corresponding to the reduction of the canonical symplectic form on T * G using the momentum map J : T * G → g * . This form is exact and thus ω µ = dθ µ , where θ µ (ξ) = µ(ξ) .
(2.10) (O µ , dθ µ ) is a presymplectic manifold the symplectic manifold of right cosets G/G µ , where G µ is the stabilizer of µ, i.e. G µ := {g ∈ G : Ad * g µ = µ} , (2.11) and its Lie algebra is integral in the sense of (2.2). Let O µ be an integral coadjoint orbit, then there exists [Hal13, Theorem 23.9, p.489] a holomorphic line bundle B over O µ . From (2.10) the Lagrangian L µ associated with µ is given by L µ (ξ) = 1 2 µ(ξ). Let π ∈ G and let (U α , ϕ α ) be a local trivialization of O µ . Then it follows from (2.7) that the flow ρ t acts locally on the space of global holomorphic sections of B → O µ via the formula where m 1 = Ad * g −1 m. We used the fact that the Lagrangian L µ is invariant under the coadjoint flow and the flow on the base is given by the coadjoint action, see [MR10][Chapter 13].
The second element of the construction is a choice of polarization. A remarkable result [Ver70a,Ver70b] yields existence of polarizing subalgebras for solvable Lie algebras. These subalgebras can be constructed algorithmically [Ous15]. It is unknown to the authors whether there exist complex polarizations for general connected Lie groups. Therefore, we make the following assumption.
Assumption 2.5. Let G be a locally compact Lie group such that for every coadjoint orbit O there exists invariant Kähler polarization P ⊂ T O.

Contractions
3.1. Lie group contractions. Before using geometric quantization to contract unitary duals, let's recall classic results on Lie group contractions.
Let G 1 and G 0 be two locally compact connected Lie groups of the same dimension.
Definition 3.1. We say that the family {Φ ε } ε≥0 of differential maps The first two conditions are necessary for the limit of the third condition to be welldefined.
The Lie algebra can be contracted the differential This map is then used to contract the Lie algebra U ε : g 0 → g 1 . Now, we fix a contraction Φ ε : G 0 → G 1 of G 1 to G 0 as in Definition 3.1.

3.2.
Contraction of the unitary dual. Assume that there exists a contraction Φ ε : G 0 → G 1 . Then the differential U ε of Φ ε allows us to deform the coadjoint orbits of G 1 into the coadjoint orbits of G 0 . We express the adjoint action Ad 0 of G 0 on g 0 as follows By the duality, we obtain the coadjoint action The corresponding character of the contracted group G 0 is given by Following for example [ADO97], the coadjoint orbits of the contracted group G 0 can be obtained via the following procedure (1) Fix an element µ ∈ g * 0 and fix a coadjoint orbit O µ . (2) Transport the point µ to g * 1 by using the map (U * ε ) −1 . (3) The action of G 1 on µ ε = (U * ε ) −1 (µ) originates a G 1 -coadjoint orbitÕ ε ⊂ g * 1 . (4) The orbitÕ ε is translated to g * 0 by using the map U * ε , i.e.
The orbit method of Kirillov is a method in geometric representation theory which identifies the unitary representations of Lie groups with the canonical G-action on spaces of sections of specific line bundles over the coadjoint orbits. The general case of a Lie group is not yet fully understood. Vogan rigorously justified [Vog98a] the orbit method for reductive Lie groups. However, we shall restrict ourselves to solvable type I Lie groups to use results of Kostant [Kos70a] and Auslander [Aus67a]. Coadjoint orbits can be described by Ad G -invariant functions f : g * → C. However, it is not clear whether the holomorhic line bundles over the coadjoint orbits O ε are contracted properly. Therefore, we introduce  (1) for each v ∈ D there is an integer n(v) such that n ≥ n(v) implies v ∈ A n (H n ); (2) for each v ∈ D and g ∈ G 0 (3.11) lim n→0 A n π n (Φ εn (g))A −1 n v − σ(g)v H = 0. In Theorem 3.6 we find sufficient condition on elements π ∈ G to be contraction of a sequence {π ε } ⊂ G 1 . The condition is expressed in terms of the corresponding coadjoint orbits.
Theorem 3.6. Let G 1 be a compact Lie group and G 0 be a locally compact solvable type I Lie group. Let π ∈ G 0 be an infinite-dimensional irreducible unitary representation of G 0 . Then π is an M N -contraction of the sequence of unitary irreducible representations if O ε is an admissible contraction of {O π }.
Question 3.7. Is the converse true? In other words, let π be an M N -contraction of the sequence of unitary representations π ε ∈ G 1 . Then the coadjoint orbits O ε is an admissible contraction of {O π }.
Sketch of the argument for Question 3.7. Thinking locally, we should have M N -contraction on each U α , i.e. π ε (g)s α on each local section s α . Each local section give rise to cocycles c α,β . Adding a bit more mathematical rigour one should be able to conclude the statement of Question 3.7.
Prior to the proof of Theorem 3.6, we consider the classical contraction Φ ε : H 1 → SU 2 .
Example 3.8. Let G 1 = SU (2) and G 0 = H. A Lie group contraction Φ ε = exp(U ε ) : G 0 → G 1 is given by The coadjoint orbits {O s } of SU 2 are the 2-dimensional spheres with half-integer radius Using that O s is diffeomorphic to the complex projective line CP 1 we transport the coordinate atlas U 1 , U 2 ⊂ C on O s . There exists a holomorphic line bundle π : B → O s with the transition function c s (z) = z 2s (3.14) defines a bundle π : B → O s . The existence follows from the fact that O s is a quantizable symplectic manifold. The choice (3.14) picks up a specific bundle.
A global holomorphic section s of the bundle π : B → O s is given by two holomorphic functions s 1 : U 1 → C and s 2 : U 2 → C related by in respective coordinates and equating coefficients, we find thatã n = a n = 0 for m > 2s andã 0 = a 2s ,ã 1 = a 2s−1 , etc. . . Hence, the section s is given by a polynomial s(z) = 2s n=0 a n z n . (3.17) The coadjoint orbits O λ of the Heisenberg group H are given by This shows the condition (3.9) holds true.
The extension of Example 3.8 to a broader class of Lie groups (e.g. semisimple Lie groups) remains a challenging problem [TV17] and will be considered elsewhere.
Let π be a fixed unitary irreducible representation of π associated with a coadjoint orbit O π ⊂ g 0 . Since O ε π is an admissible contraction of O π , there exists an admissible contraction {O ε π } ε>0 and let us denote by π ε be the representation corresponding to O ε π . Let (U α , ϕ α ) be a local trivialization of O π → B and let us fix an aribtrary element s ∈ H π . We denote by s α its localization to U α ⊂ O π . Abusing our notation we shall also write s α for the coordinate representation of s α , i.e. s α (z) = s α (ϕ −1 α (z)), z ∈ C r . Let us denote Ad * ε g : O π → O ε π , (3.24) given by Indeed, by (2.13) π ε (g)s α (m) = s α (Ad * ε g m)ψ µ (m, g) (3.26) where we denote Thus, we get where we used (3.27) and the fact that |ψ µ (m, g)| ≤ 1. This shows that The sections s and s ε are holomorphic. Hence, their localizations s α and s ε α admit Taylor expansions Since every coadjoint orbit O π ε is compact, the Hilbert space H ε of global holomorphic sections s ε is finite-dimensional in the view of the Riemann-Roch theorem. The dimension of the representation space is given [Kir04] by Hence, we get where N ε ≥ dim(π ε )). Let us choose We have (3.35) Since s ε α is a continuous function and using (3.5), we get that the second term in the last inequality of (3.35) goes to zero as ε → 0.
Composing (3.30) and (3.31) and (3.33), we get The last sum converges to zero uniformly by the holomorphicity of s α . This establishes (3.29). By the existence of contraction, we immediately obtain that lim ε→0 γ : m→Ad * g m L µ ds = γε : m→Ad * ε Φε(g) m L ε µ ds . This completes the proof.

Two classes of contractions
We exhibit here two type of contractions that can be implemented on any matrix Lie group and which encompass most of the contractions with a diagonal contraction matrix U ε . First note that in the limit ε → 0, the important terms in the matrix U ε will be the lowest powers in ε. Notice also that ε is arbitrary, so only the relative scaling between terms is important. From these considerations, we will study the following two form of the contraction matrices U ε .
where u is a matrix to be determined and does not depend on ε. The first contraction is the classical Inönü-Wigner contraction and the second will be called the anti-Inönü-Wigner contraction for reasons that will become obvious soon. First, we need to define a particular decomposition of the Lie algebra. Furthermore, these subspaces satisfy We now want to derive a geometrical condition on u that will ensure that we obtain a contraction. For this, we will need the following object.
Definition 4.2 (IW-tensor). We the following 2-tensor that we will call the IW-tensor.
In the case when T [u](ξ, η) = 0, it reduces to the well-known Nijenhuis tensor of complex geometry.
We will now write the conditions on u in term of this tensor and the subspaces defined above.
Theorem 4.3. The following conditions on u give valid Lie algebra contraction for a choice of splitting g 1 = V N ⊕ V R : ( Proof. We start by proving the IW contraction condition by rewriting where λ = ε 1−ε converges to 0 when ε → 0. We first rewrite the Lie bracket using (4.2) to get We then have to ensure that the inner term is in V R , the space where U is invertible in the limit ε → 0. First notice that from the definition of the contraction (4.2) we have the relation U −1 (λζ) = ζ − U −1 uζ for an arbitrary element ζ ∈ g. We can thus rewrite the previous equation as This computation supposes that u is diagonal, and will not apply for more general contractions. The, as U −1 is only defined on V R in the limit ε → 0, the condition for u to be a valid contraction, i.e. for the limit ε → 0 to exist translates to ta condition on the IW-tensor, The new Lie bracket is where λ = ε 1−ε . We then follow the same procedure as before. We first expand the bracket using (4.11) to get (4.12) We now have the relation U −1 (ελζ) = ζ − U −1 (εuζ), so we can rewrite We can go further in the computation to extract non-vanishing terms by using the relation U −1 (ελ(ελ −1 ζ)) = ελ −1 ζ − U −1 (εuελ −1 ζ) to get  In this case, the condition for existence of the contraction is given by (4.14) We can make a couple of interesting remarks at this point.
We can even go further and give an extra condition on the splitting as well the type of Lie algebra that results from the contraction.
Theorem 4.4. Given a finite dimensional complex Lie algebra g 1 , for any subspace u of a subalgebra of g 1 , letting V R = u yields an IWS contraction to a semi-direct algebra and letting V N = u yields an anti-IWS contraction to a nilpotent Lie algebra.
Proof. Let us first look at the anti-IWS contractions. Let V N ⊂ p, so that we have [V N , V N ] ⊂ V N and V N are nilpotent. One can easily check that T [u](ξ N , η N ) = 0, and that The last commutation trivially gives T [u](ξ N , η R ) = [ξ N , η R ] R − [ξ N , η R ] R = 0. We thus have that if V N ⊂ b then the contraction is anti-IWS.