SEGAL-BARGMANN TRANSFORM AND PALEY-WIENER THEOREMS ON MOTION GROUPS 0 SUPARNASEN 1 0 2 Abstract. We study the Segal-Bargmann transform on a motion group Rn ⋉ n a K, where K is a compact subgroup of SO(n). A characterization of the Poisson J integrals associated to the Laplacian on Rn ⋉K is given. We also establish a 3 1 Paley-Wiener type theorem using the complexified representations. ] A MSC 2000 : Primary 22E30;Secondary 22E45. F . h Keywords : Segal-Bargmanntransform, Poissonintegrals, Paley-Wiener t a theorems. m [ 1 v 1. Introduction 9 1 1 The Segal-Bargmann transform, also called the coherent state transform, was 2 . 1 developed independently in the early 1960’s by Segal in the infinite-dimensional 0 0 context of scalar quantum field theories and by Bargmann in the finite-dimensional 1 v: context of quantum mechanics on Rn. We consider the following equivalent form of i X Bargmann’s original result. r a A function f L2(Rn) admits a factorization f(x) = g p (x) where g L2(Rn) t ∈ ∗ ∈ 1 2 x and pt(x) = (4πt)n2 e−|4t| (the heat kernel onRn) if andonly if f extends as anentire 1 2 function to Cn and we have f(z) 2e−|y2|t dxdy < (z = x+iy). In this (2πt)n/2 | | ∞ Cn Z case we also have 1 2 kgk22 = (2πt)n/2 |f(z)|2e−|y2|t dxdy. Cn Z The mapping g g p is called the Segal-Bargmann transform and the above t → ∗ says that the Segal-Bargmann transform is a unitary map from L2(Rn) onto (Cn) O The author was supported by Shyama Prasad Mukherjee Fellowship from Council of Scientific and Industrial Research, India. 1 2 SUPARNASEN 1 2 L2(Cn,µ), where dµ(z) = e−|y2|t dxdy and (Cn) denotes the space of (2πt)n/2 O \entire functions on Cn. In the paper [4], B. C. Hall introduced a generalization of the Segal-Bargmann transformonacompactLiegroup. IfK issuchagroup,thiscoherentstatetransform maps L2(K) isometrically onto the space of holomorphic functions in L2(G,µ ), t where G is the complexification of K and µ is an appropriate heat kernel measure t on G. The generalized coherent state transform is defined in terms of the heat kernel on the compact group K and its analytic continuation to the complex group G. Similar results have been proved by various authors. See [12], [6], [5], [8] and [7]. Next, consider the following result on R due to Paley and Wiener. A function f L2(R) admits a holomorphic extension to the strip x+iy : y < t such that ∈ { | | } sup f(x+iy) 2dx < s < t | | ∞ ∀ |y|≤sZR if and only if (1.1) e2sξ f(ξ) 2dξ < s < t | | | | ∞ ∀ R Z where f denotes the Fourier transeform of f. The condition (1.1) is the same as e ^1 e2s∆2f(ξ) 2dξ < s < t | | ∞ ∀ R Z where ∆ is the Laplacian on R. This point of view was explored by R. Goodman in Theorem 2.1 of [2]. The condition (1.1) also equals ei(x+iy)ξ 2 f(ξ) 2dξ < y < t. | | | | ∞ ∀ R Z Here ξ ei(x+iy)ξ may be seen as thee complexification of the parameters of the 7→ unitary irreducible representations ξ eixξ of R. This point of view also was fur- 7→ ther developed by R. Goodman (see Theorem 3.1 from [3]). Similar results were established for the Euclidean motion group M(2) of the plane R2 in [11]. Aim of this paper is to prove corresponding results in the context of general motion groups. SEGAL-BARGMANN TRANSFORM AND PALEY-WIENER THEOREMS ON MOTION GROUPS 3 Theplanofthispaperisasfollows: Inthefollowingsectionwerecalltherepresen- tation theory and Plancherel theorem of the motion group M. We also describe the Laplacian on M. In the next section we prove the unitarity of the Segal-Bargmann transform on M and we study generalized Segal-Bargmann transform which is an analogue of Theorem 8 and Theorem 10 in [4]. The fourth section is devoted to a study of Poisson integrals on M via a Gutzmer-type formula on M which is proved by using a Gutzmer formula for compact Lie groups established by Lassalle in 1978 (see [9]). This section is modelled after the work of Goodman [2]. In the final section we prove another characterization of functions extending holomorphically to the complexification of M which is an analogue of Theorem 3.1 of [3]. 2. Preliminaries Let K be a compact, connected Lie group which acts as a linear group on a finite dimensional real vector space V. Let M be the semidirect product of V and K with the group law (x ,k ) (x ,k ) = (x +k x ,k k ) where x ,x V;k ,k K. 1 1 2 2 1 1 2 1 2 1 2 1 2 · ∈ ∈ M is called the motion group. Since K is compact, there exists a K-invariant inner product on V. Hence, we can assume that K is a connected subgroup of SO(n), where n = dimV. When K = 1 , M = V = Rn and if K = SO(n), M is ∼ { } the Euclidean motion group. Henceforth we shall identify V with Rn and K with a subgroup of SO(n). The group M may be identified with a matrix subgroup of GL(n+1,R) via the map k x (x,k) →   0 1 where x Rn and k K SO(n).   ∈ ∈ ⊆ n WenormalizetheHaarmeasuredmonM suchthatdm = dxdk,wheredx = (2π)−2 dx dx dx and dk is the normalized Haar measure on K. Let = L2(K) be the 1 2 n ··· H Hilbert space of all square integrable functions on K. Denote by , the Euclidean h· ·i 4 SUPARNASEN inner product on Rn. Let V be the dual space of V. Then we can identify V with Rn so that K acts on V naturally by k ξ,x = ξ,k 1 x where ξ V, x V, k K. − b h · i h · i ∈ ∈b ∈ For any ξ V let Uξ denote the induced representation of M by the unitary ∈ b b representation x ei<ξ,x> of V. Then for F and (x,k) M, b7→ ∈ H ∈ Uξ F(u) = ei x,uξ F(k 1u). h · i − (x,k) The representation Uξ is not irreducible. Any irreducible unitary representation of M is, however, contained in Uξ for some ξ V as an irreducible component. ∈ Let K be the isotropy subgroup of ξ V i.e. K = k K : k ξ = ξ . Consider ξ ∈ b ξ { ∈ · } σ K , the unitary dual of K . Denote by χ , d and σ the character, degree and ∈ ξ ξ b σ σ ij matrix coefficients of σ respectively. Let R be the right regular representation of K. c Define Pσ = d χ (w)R dw σ σ w ZKξ and Pσ = d σ (w)R dw γ σ γγ w ZKξ where dw is the normalized Haar measure on K . Then Pσ and Pσ are both or- ξ γ thogonal projections of . Let σ = Pσ and σ = Pσ . The subspaces σ are H H H Hγ γH Hγ invariant under Uξ for 1 γ d and the representations of M induced on σ ≤ ≤ σ Hγ under Uξ are equivalent for all 1 γ d . We fix one of them and denote it by Uξ,σ. σ ≤ ≤ Two representations Uξ,σ and Uξ′,σ′ are equivalent if and only if there exists an ele- ment k K such that ξ = k ξ and σ is equivalent to σk where σk(w) = σ(kwk 1) ′ ′ − ∈ · for w K . ξ ∈ The Mackey theory [10] shows that under certain conditions on K (for details refer to Section 6.6 of [1]), each Uξ,σ is irreducible and every infinite dimensional irreducible unitary representation is equivalent to one of Uξ,σ for some ξ Rn and ∈ σ K . Since = σ and σ = dσ σ, we have ∈ ξ H H H γ=1Hγ σM∈Kξ L c c Uξ = d Uξ,σ. ∼ σ σM∈Kξ c SEGAL-BARGMANN TRANSFORM AND PALEY-WIENER THEOREMS ON MOTION GROUPS 5 For any f L1(M) define the Fourier transform of f by ∈ f(ξ,σ) = f(m)Uξ,σdm. m ZM Then the Plancherel formulabgives f(m) 2dm = d f(ξ,σ) 2 dξ | | σ k kHS ZM ZRn σX∈Kξ c b where is the Hilbert-Schmidt norm of an operator. We will be working with HS k·k the generalized Fourier transform defined by f(ξ) = f(m)Uξ dm. m ZM Then we also have b f(m) 2dm = f(ξ) 2 dξ. | | k kHS ZM ZRn Let k and m be the Lie algebras of K and Mbrespectively. Then K X m = : X Rn, K k .   ∈ ∈  0 0     Let K ,K , ,K be a basis of k and X ,X , ,X be a Lie algebra basis of 1 2 N 1 2 n ··· ··· Rn. Define K 0 i M = for 1 i N i   ≤ ≤ 0 0   0 X i = for N +1 i N +n.   ≤ ≤ 0 0   Then it is easy to see that M ,M , ,M forms a basis for m. The Laplacian 1 2 N+n { ··· } ∆ = ∆ is defined by M ∆ = (M2 +M2 + +M2 ). − 1 2 ··· N+n A simple computation using the fact K SO(n) shows that ⊆ ∆ = ∆ ∆ Rn K − − 6 SUPARNASEN where ∆ and ∆ are the Laplacians on Rn and K respectively given by ∆ = Rn K Rn X2 +X2 + +X2 and ∆ = K2 +K2 + +K2. 1 2 ··· n K 1 2 ··· N 3. Segal-Bargmann transform and its generalisation Since ∆ and ∆ commute, it follows that the heat kernel ψ associated to ∆ is Rn K t given by the product of the heat kernels p on Rn and q on K. In other words t t 1 2 ψt(x,k) = pt(x)qt(k) = ne−|4xt| dπe−λ2πtχπ(k). (4πt)2 πXK ∈ b Here, for each unitary, irreducible representation π of K, d is the degree of π, λ π π is such that π(∆ ) = λ I and χ (k) = tr(π(k)) is the character of π. K π π − Denote by G the complexification of K. Let κ be the fundamental solution at the t identity of the following equation on G : du 1 = ∆ u, dt 4 G where ∆ is the Laplacian on G. It should be noted that κ is the real, positive heat G t kernel on G which is not the same as the analytic continuation of q on K. t Let (Cn G) be the Hilbert space of holomorphic functions on Cn G which H × × are square integrable with respect to µ ν(z,g) where dµ(z) = 1 Nne−|y2|t2dxdy on Cn (2πt)2 and dν(g) = κ (xg)dx on G. t ZK Then we have the following theorem : Theorem 3.1. If f L2(M), then f ψ extends holomorphically to Cn G. t ∈ ∗ × Moreover, the map C : f f ψ is a unitary map from L2(M) onto (Cn G). t t 7→ ∗ H × Proof. Let f L2(M). Expanding f in the K variable using the Peter-Weyl theo- ∈ − rem we obtain dπ f(x,k) = d fπ(x)φπ(k) π ij ij πXK iX,j=1 ∈ b SEGAL-BARGMANN TRANSFORM AND PALEY-WIENER THEOREMS ON MOTION GROUPS 7 where for each π K, d is the degree of π, φπ’s are the matrix coefficients of π and ∈ π ij fπ(x) = f(x,k)φπ(k)dk. Here, the convergence is understood in the L2-sense. ij bij ZK Moreover, by the universal property of the complexification of a compact Lie group (see Section 3 of [4]), all the representations of K, and hence all the matrix entries, extend to G holomorphically. Since ψ is K-invariant (as a function on Rn) a simple computation shows that t dπ f ∗ψt(x,k) = dπe−λ2πt fiπj ∗pt(x)φπij(k). πXK iX,j=1 ∈ b It is easily seen that fπ L2(Rn) for every π K and 1 i,j d . Hence fπ p ij ∈ ∈ ≤ ≤ π ij ∗ t extends toaholomorphicfunctiononCn andbytheunitarityoftheSegal-Bargmann b transform in Rn we have (3.1) fπ p (z) 2µ(y)dxdy = fπ(x) 2dx. | ij ∗ t | | ij | Cn Rn Z Z The analytic continuation of f ψ to Cn G is given by t ∗ × dπ f ∗ψt(z,g) = dπe−λ2πt fiπj ∗pt(z)φπij(g). πXK iX,j=1 ∈ b We claim that the above series converges uniformly on compact subsets of Cn G × so that f ψ extends to a holomorphic function on Cn G. We know from Section t ∗ × 4, Proposition 1 of [4] that the holomorphic extension of the heat kernel q on K is t given by qt(g) = dπe−λ2πtχπ(g). πXK ∈ b For each g G, define the function qg(k) = q (gk). Then qg is a smooth function on ∈ t t t K and is given by qtg(k) = dπe−λ2πtχπ(gk) πXK ∈ b dπ = dπe−λ2πt φπij(g)φπji(k). πXK iX,j=1 ∈ b 8 SUPARNASEN Since qg is a smooth function on K, we have for each g G, t ∈ dπ (3.2) qg(k) 2dk = d e λπt φπ(g) 2 < . | t | π − | ij | ∞ ZK πXK iX,j=1 ∈ b Let L be a compact set in Cn G. For (z,g) L we have, × ∈ dπ (3.3) |f ∗ψt(z,g)| ≤ dπe−λ2πt |fiπj ∗pt(z)||φπij(g)|. πXK iX,j=1 ∈ b By the Fourier inversion fπ p (z) = fπ(ξ)e tξ2eiξ(x+iy)dξ ij ∗ t ij − | | · Rn Z f where z = x+iy Cn and fπ is the Fourier transform of fπ. Hence, if z varies in ∈ ij ij a compact subset of Cn, we have f fπ p (z) fπ e 2(tξ2+yξ)dξ | ij ∗ t | ≤ k ijk2 − | | · Rn Z C fπ . ≤ k ijk2 Using the above in (3.3) and applying Cauchy-Schwarz inequality we get dπ |f ∗ψt(z,g)| ≤ C dπ kfiπjk2e−λ2πt|φπij(g)| πXK iX,j=1 ∈ b 1 1 2 2 dπ dπ C d fπ(x) 2dx d e λπt φπ(g) 2 . ≤  π | ij |   π − | ij |  Rn πXK iX,j=1Z πXK iX,j=1 ∈ ∈  b   b  dπ Noting that f 2 = d fπ(x) 2dx and q is a smooth function on G we k k2 π | ij | t Rn πXK iX,j=1Z ∈ prove the claim using (b3.2). Applying Theorem 2 in [4] we get dπ f ψ (z,g) 2dν(g) = d fπ p (z) 2. | ∗ t | π | ij ∗ t | ZG πXK iX,j=1 ∈ b Integrating theaboveagainstµ(y)dxdy onCn andusing(3.1)weobtaintheisometry of C t f ψ (z,g) 2µ(y)dxdydν(g) = f 2. | ∗ t | k k2 ZCnZG SEGAL-BARGMANN TRANSFORM AND PALEY-WIENER THEOREMS ON MOTION GROUPS 9 To prove that the map C is surjective it suffices to prove that the range of C is t t dense in (Cn G). For this, consider functions of the form f(x,k) = h (x)h (k) 1 2 H × ∈ L2(M) where h L2(Rn), h L2(K). Then a simple computation shows that 1 2 ∈ ∈ f ψ (z,g) = h p (z)h q (g) for (z,g) Cn G. t 1 t 2 t ∗ ∗ ∗ ∈ × Suppose F (Cn G) be such that ∈ H × (3.4) F(z,g)h p (z)h q (g)µ(y)dxdydν(g) = 0 1 t 2 t ∗ ∗ ZCn G × h L2(Rn) and h L2(K). From (3.4) we have 1 2 ∀ ∈ ∀ ∈ F(z,g)h p (z)dµ(z) h q (g)dν(g) = 0, 1 t 2 t ∗ ∗ ZG(cid:18)ZCn (cid:19) which by Theorem 2 of [4] implies that F(z,g)h p (z)dµ(z) = 0. 1 t ∗ Cn Z Finally, an application of the surjectivity of Segal-Bargmann transform on Rn shows that F 0. Hence the proof. (cid:3) ≡ In [4] Brian C. Hall proved the following generalizations of the Segal-Bargmann transfoms for Rn and compact Lie groups : Theorem 3.2. (I) Let µ be any measurable function on Rn such that µ is strictly positive and locally bounded away from zero, • x Rn, σ(x) = e2xyµ(y)dy < . · • ∀ ∈ ∞ Rn Z Define, for z Cn ∈ eia(y) ψ(z) = e iyzdy, − · Rn σ(y) Z where aisa real valuedmeasurablefunctipon onRn.Thenthe mappingCψ : L2(Rn) → (Cn) defined by O C (z) = f(x)ψ(z x)dx ψ − Rn Z is an isometric isomorphism of L2(Rn) onto (Cn) L2(Cn,dxµ(y)dy). O T 10 SUPARNASEN (II) Let K be a compact Lie group and G be its complexification. Let ν be a measure on G such that ν is bi-K-invariant, • ν is given by a positive density which is locally bounded away from zero, • For each irreducible representation π of K, analytically continued to G, • 1 δ(π) = π(g 1) 2dν(g) < . − dimV k k ∞ π ZG d Define τ(g) = π Tr(π(g 1)U ) where g G and U ’s are arbitrary unitary − π π δ(π) ∈ πXK matrices. Then∈thbepmapping C f(g) = f(k)τ(k 1g)dk τ − ZK is an isometric isomorphism of L2(K) onto (G) L2(G,dν(w)). O \ A similar result holds for M. Let µ be any real-valued K-invariant function on Rn such that it satisfies the conditions of Theorem 3.2 (I). Define, for z Cn ∈ eia(y) ψ(z) = e iy.zdy, − Rn σ(y) Z p where a is a real valued measurable K-invariant function on Rn. Next, let ν, δ and τ be as in Theorem 3.2 (II). Also define φ(z,g) = ψ(z)τ(g) for z Cn, g G. It ∈ ∈ is easy to see that φ(z,w) is a holomorphic function on Cn G. Then it is easy to × prove the following analogue of Theorem 3.2 for M. Theorem 3.3. The mapping C f(z,g) = f(ξ,k)φ((ξ,k) 1(z,g))dξdk φ − ZM is an isometric isomorphism of L2(M) onto (Cn G) L2(Cn G,µ(y)dxdydν(g)). O × × \