The Paley-wiener Theorem and the Local Huygens’ Principle for Compact Symmetric Spaces

We prove a Paley-Wiener Theorem for a class of symmetric spaces of the compact type, in which all root multiplicities are even. This theorem characterizes functions of small support in terms of holomorphic extendability and exponential type of their (discrete) Fourier transforms. We also provide three independent new proofs of the strong Huygens’ principle for a suitable constant shift of the wave equation on odd-dimensional spaces from our class. Introduction In the context of spherical harmonic analysis, the compactness of a symmetric space U/K is reflected by the discreteness of its dual space, which is the set of irreducible K-spherical unitary representations of U . The same set parameterizes the set of (elementary) spherical functions. Thus the spherical Fourier transforms of K-invariant functions on U/K are functions on a discrete set. Likewise, the formula for spherical inversion, which recovers a sufficiently regular function on the symmetric space in terms spherical functions, is given by a series. This structural discreteness can be overcome for functions with “small support”, by relating them to functions on the tangent space q to U/K at the base point x0 = {K}. This procedure can be easily illustrated in the Euclidean setting: consider a smooth function f : S → C, where S denotes the unit circle. View f as a periodic functions on R by t 7→ f(e) and assume that f has small support, say in [−R,R]+2πZ, where 0 < R < π. We can then regard f as a smooth function on the real line with support in [−R,R] by setting it equal to 0 outside of the fundamental period [−π, π). By the classical Paley-Wiener theorem on R, the Fourier transform of f is an entire function of exponential type R. It therefore provides a holomorphic extension of the Fourier transform of f as a function on S. Likewise, if F is a holomorphic function on C of exponential type R, 0 < R < π, then the inversion formula for the continuous Fourier transform gives a function f1 with support in [−R,R], and we can define a function f on S by f(e) = f1(t). The possibility of characterizing central smooth functions with “small support” on compact Lie groups by means of the entire extension and exponential growth of their Fourier transform was first proven by Gonzalez in [7]. In this paper we extend the local Paley-Wiener theorem to all compact symmetric spaces U/K with even multiplicities: the K-invariant smooth functions on U/K with “small support” will be characterized in terms of holomorphic extendibility and exponential growth of their spherical Fourier transform. Moreover, the exponential growth of the transformed function will be linked to the size of the support of the function on the symmetric space. The given characterization relies on the fact that the spherical functions on a compact symmetric space extend holomorphically to the complexified symmetric space. Their restrictions to the noncompact dual symmetric spaces G/K are in turn spherical functions on G/K. This allows us to use known information on the spherical functions on G/K and classical Fourier Date: November 13, 2004. 2000 Mathematics Subject Classification. Primary 53C35; Secondary 35L05. GO was supported by NSF grants DMS-0070607, DMS-0139783, and DMS-0402068, and by the DFG-Schwerpunkt “Global Methods in Complex Geometry.” TB was partially supported by the Erwin Schrödinger Institute. 1 2 THOMAS BRANSON, GESTUR ÓLAFSSON, AND ANGELA PASQUALE analysis on the Lie algebra of a maximal abelian subspace of q. In particular, the classical Paley-Wiener theorem is used to obtain the required holomorphic extension of compact spherical Fourier transforms of a K-invariant function on U/K with a small support. Properties of holomorphic extendibility for spherical functions on symmetric spaces have been the objects of intensive recent study, with different approaches and perspectives. See e.g. [14], [20], and [17]. The situation which we consider in this paper corresponds to symmetric spaces with even multiplicities. It is rather special because of the existence of shift operators providing explicit formulas for the spherical functions by relating them to exponential functions [16]. These shift operators are suitable multiples of Opdam’s shift operators. The multiplying factor has been chosen so, as to cancel the singularities of the coefficients of Opdam’s shift operators along the walls of the Weyl chambers. The resulting operators are differential operators with holomorphic coefficients. Hence, we can read off the properties of holomorphic extendibility of the spherical functions directly from these formulas. Furthermore, the shift operators allow us, as mentioned above, to reduce several problems in harmonic analysis on symmetric spaces of even multiplicities to the corresponding problems in Euclidean harmonic analysis. Our proof depends heavily on the assumption that all root multiplicities are even, and it is not possible to generalize it to obtain local Paley-Wiener type theorems for general compact symmetric spaces. On the other hand, the same proof can be employed for several other even-valued multiplicity functions which are not geometric. We will not explore this avenue further in the present article. The relation between spherical transforms on compact and noncompact symmetric spaces investigated in this paper also yields a representation of smooth functions with “small support” on the compact space as integrals of spherical functions of the noncompact dual. These integral formulas are the key ingredient for studying the solutions of the wave equation on Riemannian symmetric spaces of compact type. From exponential estimates for the solutions, we deduce in Section 4 that the strong Huygens’ principle is valid on these spaces. The (strong ) Huygens’ principle states that, in odd dimensions, the light at time t0 at a location x influences at later times t1 only those locations which have distance exactly t1 − t0 from x. Hence, if a wave is supported in the sphere {x | ‖x‖ ≤ R} at the initial time 0, then it will be supported in the annulus {x | t − R ≤ ‖x‖ ≤ t+ R} at time t. In particular, at times t > R, the wave will vanish inside the sphere {x | ‖x‖ < t−R}. Several different authors have proven the validity of Huygens’ principle on odd dimensional Riemannian symmetric spaces with even multiplicities of either noncompact or compact type. Here “light” is to be interpreted as a solution of a suitable wave equation, obtained by a certain constant shift of the d’Alembertian. Their proofs use a variety of different methods. The first results in this direction were given by Helgason [8, 11], see also [13], who proved Huygens’ principle for symmetric spaces G/K for which either G is complex or G = SO0(n, 1), and for compact groups. In the general case of odd dimensional Riemannian symmetric spaces of the noncompact type with even multiplicities, the validity of Huygens’ principle was stated without proof by Solomantina [21]. A proof by Radon transform methods was provided by Ólafsson and Schlichtkrull [18]. An independent proof was obtained by Helgason [12] by means of his Fourier transform. In [4] the authors proved an exponential decay property for solutions of the wave equation with compactly supported initial data. This method implied another independent proof of the Huygens’ principle for odd dimensional symmetric spaces with even multiplicities; see [4]. Finally, a completely different approach based on Heckman-Opdam’s shift operators and explicit formulas for the fundamental solutions was provided by Chalykh and Veselov in [5]. The formulas of Chalykh and Veselov give the fundamental solution of the wave equation in polar coordinates. By replacing hyperbolic functions with their trigonometric counterparts, one can also deduce formulas for the fundamental solutions of the wave equation on compact symmetric spaces. These formulas will be valid for small values of time. Using this argument, Chalykh and Veselov state that Huygens’ principle holds also on Riemannian symmetric spaces of the compact type. PALEY-WIENER THEOREM 3 In the context of Riemannian symmetric spaces, Huygens’ principle has been much less studied for compact type than for noncompact type. In [19], Ørsted used conformal properties of wave operators and of Lorentzian spaces covered by R × S to establish Huygens’ principle for the wave, Dirac, and Maxwell equations on S. His proof makes it clear that analogues will be valid for other linear differential operators with suitable hyperbolicity and conformal properties. A different proof for the wave equation on the odd sphere S were given by Lax and Phillips [15]. Branson [1] extended the LaxPhillips proof to an infinite class of hyperbolic equations on the odd sphere. Helgason proved Huygens’ principle for the compact group case, see [11]. Finally, Branson and Ólafsson [3] proved that the local Huygens’ principle for a compact symmetric space U/K is valid if and only if Huygens’ principle holds for the non-compact dual space G/K. In this article we provide three independent new proofs of a local version of the strong Huygens’ principle for compact symmetric spaces U/K. One of these methods comes from exponential estimates for the smooth solutions of K-invariant Cauchy problems for the modified wave equations on U/K. These estimates are obtained by methods similar to those introduced for the noncompact setting in [4]. It is nevertheless important to mention that the use of the shift operators indeed reduces the proof of the of Huygens’ principle on Riemannian symmetric spaces of either type (compact or noncompact) to the validity of the same principle in the Euclidean setting. The proof presented in this paper is therefore easier than that in [4]. Another proof of the local strong Huygens’ principle is in the spirit of the paper of Chalykh and Veselov [5]. The formulas for the spherical functions proven in Theorem 2.9 permit us to derive an explicit formula for the solution of the wave equation corresponding to a given smooth initial condition. The tools for writing down these formulas appear in the proof of the local Paley-Wiener theorem. An essential property in our argument is that our shift operators, which link spherical functions to exponential functions, have regular (indeed analytic) coefficients. This fact was not proven in [5]. Our paper is organized as follows. In Section 1 we recall some structure theory of Riemannian symmetric spaces of the compact type. The spherical functions and spherical representations are introduced in Section 2. Theorem 2.9 proves the existence of differential shift operators. These provide explicit formulas for the spherical functions on compact symmetric spaces. The main theorem in this paper is the local Paley-Wiener theorem, which is stated and proven in Section 3. The integral formula for functions with “small support” is given by Corollary 3.16. Finally, Section 4 contains the proofs of the local strong Huygens’ principle on Riemannian symmetric spaces of the compact type. 1. Symmetric spaces 1.1. Compact symmetric spaces. In this section we recall some facts about compact symmetric spaces. We use [9], Chapter VII, and [22], Chapter II, as standard references. Let U be a connected compact Lie group with center Z and Lie algebra u. Denote by z the center of u. Then u = z⊕ u, where u := [u, u] is semisimple. Let exp : u → U be the exponential map. If z 6= {0}, then we set Γ0 := {X ∈ z | expX = e}, where e denotes the identity of U . Then Γ0 is a full rank lattice in z and T := z/Γ0 is isomorphic to the identity connected component of Z. We will from now on write T = Z0. Denote by U ′ the analytic subgroup of U with Lie algebra u. Then U ′ is semisimple with finite center and U = TU ′ ≃ T ×F U ′ where F = T ∩U ′ is a finite central subgroup of U . We will for simplicity assume that F is trivial. Thus U ≃ T × U . Let τ : U → U be a non-trivial analytic involution. Set U τ := {u ∈ U | τ(u) = u}, and define K be the identity connected component of U τ . Then U/K is a connected compact symmetric space (also called Riemannian symmetric space of the compact type). The derived involution of τ on u will be denoted by the same letter τ . Thus τ(exp(X)) = exp(τ(X)) for all X ∈ u. 4 THOMAS BRANSON, GESTUR ÓLAFSSON, AND ANGELA PASQUALE Let k denote the Lie algebra ofK. We shall assume that U acts effectively on U/K, i.e. that k∩z = {0}. Then k = u := {X ∈ u | τ(X) = X} ⊂ u . Set q := {X ∈ u | τ(X) = −X} . Then u = k ⊕ q and z ⊆ q. For a real vector space V we denote by V ∗ its dual and by VC := V ⊗R C its complexification. If V is a Euclidean vector space with inner product 〈·, ·〉 and W ⊆ V is a subspace, then W denotes the orthogonal complement of W in V . We identify W ∗ with the space {f ∈ V ∗ | f |W⊥ = 0}. The complex linear extension to VC of a linear map φ : V → V will be denoted by the same symbol φ. For λ ∈ V ∗ define hλ ∈ V by λ(H) = 〈H,hλ〉. For λ 6= 0 we set Hλ := 2〈hλ, hλ〉hλ. Then λ(Hλ) = 2. Finally we define an inner product on V ∗ by 〈λ, μ〉 := 〈hλ, hμ〉 = λ(hμ) = μ(hλ) . Recall that the Killing form κ on u is negative definite on u. Fix an inner product 〈·, ·〉 on z and define a U -invariant inner product on u by 〈Z1 +X1, Z2 +X2〉 := 〈Z1, Z2〉 − κ(X1, X2) , Z1, Z2 ∈ z , X1, X2 ∈ u ′ . Let b ⊆ q be a maximal abelian subspace and set b1 := b ∩ u. Then b = z ⊕ b1 . Set a := ib ⊆ uC and a1 = ib1. Then, by restriction, 〈·, ·〉 defines an inner product on a, and hence we can apply the above notational conventions to (a, 〈·, ·〉). In particular Hλ ∈ a is well defined for all nonzero λ ∈ a. For α ∈ b C = a C let uC := {X ∈ uC | ∀H ∈ b : [H,X ] = α(H)X} and set mα := dimC u α C . If u C 6= {0}, then α is called a root and mα is its multiplicity. We denote by ∆ the set of roots and by W = W (∆) the corresponding Weyl group. Recall that W is generated by the reflections sα with α ∈ ∆. Here sα(H) := H − α(H)Hα. If α ∈ ∆, then uC ⊆ u ′ C , α|zC = 0, and α ∈ ib1 = a ∗ 1. Hence α is real valued on a and α|z = 0. Choose X ∈ a so that α(X) 6= 0 for all roots α. Then ∆ := {α ∈ ∆ | α(X) > 0} is a set of positive roots. We denote by Σ the corresponding set of simple roots. 1.2. Integration on U/K. We now fix our normalization of measures. If L is a locally compact Hausdorff topological group, then dl denotes a left invariant (Haar) measure on L. When L is a compact group we normalize dl so that the volume of L is 1. In this case, if M is a closed (and hence compact) subgroup of L, then we normalize the invariant measure d(lM) on L/M so that L/M has volume 1. We then have for all f ∈ L(L/M) and g ∈ L(L): ∫