Acta Applicandae Mathematicae 81: 327–338, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.
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On Describing Invariant Subspaces of the Space of Smooth Functions on a Homogeneous Manifold S. S. PLATONOV Petrozavodsk State University, 185640 Petrozavodsk, Karelia, Russia. e-mail:
[email protected] Abstract. Let a Lie group G acts transitively on a manifold M, F a locally convex space consisting of functions on M, and π(g): f (x) → f (g −1 x) the quasi-regular representation of the group G on the topological vector space F . A vector subspace H ⊆ F is called an invariant subspace if it is closed and π-invariant. In the paper we give a survey of some results of the description of invariant subspaces of function spaces on homogeneous manifolods. Mathematics Subject Classifications (2000): Primary: 43A85; secondary: 22E30, 22E46. Key words: invariant subspaces, symmetric spaces, representations of Lie groups, quasi-regular representation.
1. Introduction: The General Problem of Describing Invariant Subspaces Let G be a Lie group that acts transitively on a smooth manifold M. For any g ∈ G and any function f (x) on M, we put (π(g)f )(x) := f (g −1 x).
(1.1)
A locally convex space F that consists of complex-valued functions (or distributions) on M will be called π -invariant if for any f (x) ∈ F and any g ∈ G we have π(g)f ∈ F and g → π(g)f is a continuous map from G to F . Then the operators π(g)|F (we shall denote them simply by π(g)) define the quasi-regular representation of the group G on the topological vector space F . A vector subspace H ⊆ F is called an invariant subspace if it is closed and π -invariant. One of the main problems of harmonic analysis on Lie groups is to describe all invariant subspaces for various Lie groups G, homogeneous manifolds M and various function spaces F distinguished by some restrictions of smoothness and growth. In the present paper we give a survey of some results concerning description of invariant subspaces of function spaces on homogeneous manifolds. In Section 1 we list some results of this kind. Let us note that we do not consider here the most thoroughly studied cases where G is compact or F is a Hilbert space and the quasi-regular representation is unitary. In Sections 2 and 3 we give a description of invariant subspaces in the case where homogeneous manifold is a Riemannian
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symmetric space of noncompact type of rank 1 or a tangent space to a Riemannian symmetric space of rank 1 and F = C ∞ (M) is the space of all smooth functions. In [23] Schwartz considered the case G = M = R, when R acts on R by translations and F = C(R) or F = C ∞ (R). In this case a closed vector subspace H ⊂ F is an invariant subspace if H is invariant with respect to the transformations f (x) → f (x + a) for any a ∈ R. For the sake of convinence we shall always assume H = F . For any µ ∈ C, r ∈ N let Vµ,r be the r-dimensional subspace spanned by the functions √ eiµx , xeiµx , . . . , x r−1 eiµx (i = −1) or, equivalently, Vµ,r is the set of all solutions to the equation (d/dx − µ)r f = 0. Let H be an invariant subspace of F , F = C(R) or F = C ∞ (R), µ ∈ C. We shall say that µ belongs to the spectrum of the invariant subspace H if Vµ,r ⊂ H for some r ∈ N. Denote rµ = max{r : Vµ,r ⊂ H }. We call the number rµ multiplicity of µ in spectrum. By σ (σ ⊂ C) denote the spectrum of the invariant subspace H and let each element µ ∈ σ contains in σ with multiplicity rµ . The next theorem follows from the results obtained by Schwartz in [23]. THEOREM 1. Let F be one of the function spaces C(R) or C ∞ (R) with their usual topologies, H be an invariant subspace of F , σ the spectrum of H . Then H is the closure in F of the linear span of the subspaces Vµ,r , where µ runs through σ and r = rµ is the multiplicity of µ in σ . We can also obtain a description of the spectra of invariant subspaces. Let σ be a subset of C and each element µ ∈ σ is contained in σ with finite multiplicity rµ . The set σ is a spectrum of some invariant subspace H if and only if σ satisfies the following condition: (S) There is a nonzero entire function (z) such that if µ is contained in σ with multiplicity rµ then µ is a zero of (z) of multiplicity rµ , and we have |(z)| A eB|Im z| (1 + |z|)C
(1.2)
for some A, B, C > 0. In particular, it follows from the condition (S) that the spectrum σ is finite or countable set. We note that an entire function (z) satisfies the condition (S) if and only if (z) is the Fourier transform of a compactly supported distribution. Theorem 1 and the condition (S) together give a complete description of invariant subspaces of C(R) or C ∞ (R). In [5] Gurevich constructed a counterexample that shows that this description of invariant subspaces does not hold in the case G = M = Rn , n 2, F = C(Rn ) or F = C ∞ (Rn ) since there are then invariant subspaces not containing functions of the form ei λ,x where λ = (λ1 , . . . , λn ) ∈ Cn , x = (x1 , . . . , xn ) ∈ Rn , λ, x =
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λ1 x1 + · · · + λn xn . Even in this case no complete description of the invariant subspaces is known, but there are a lot of papers containing a description of some special classes of invariant subspaces such as the subspaces of solutions to homogeneous differential equations with constant coefficients or to some convolution equations (see, for example, Malgrange [10], Ehrenpreis [3], Palamodov [12]). In [4] Ehrenpreis and Mautner obtained a description of invariant subspaces of the spaces C(M) and C ∞ (M) in the case M = SL(2, R) and G = SL(2, R) × SL(2, R), with respect to the action (g1 , g2 )x = g1−1 xg2 (that is, invariant subspaces are closed linear subspaces invariant with respect to left and right translations). These subspaces will be called biinvariant subspaces. The description of biinvariant subspaces was obtained using the Fourier transform on the group SL(2, R) and any biinvariant subspace was described in terms of zeros of the Fourier transforms of certain functions. For the case of other function spaces consisting of functions of exponential growth on the group SL(2, R), a description of biinvariant subspaces was obtained by Rashevskij in [22]. The main idea of Rashevskij is the reduction of the problem of describing biinvariant subspaces to the problem of describing closed vector subspaces of certain function spaces on R that are invariant with respect to the transformations 1 f (x) → (f (x + a) + f (x − a)) ∀a ∈ R. 2 Of considerable interest is the problem of describing invariant subspaces in the case where M is a Riemannian symmetric space of noncompact type, G is a group of isometries on M, and F = C ∞ (M) (see papers of Berenstein [1] and Wawrzynczyk [24]). This is closely related to problems of spectral synthesis on symmetric spaces and to the Pompeiu problem on symmetric spaces. For symmetric spaces of rank 2 there are no approaches yet to a description of all invariant subspaces. Berenstein and Gay in [2] extended to the counterexample of Gurevich [5] on the case of symmetric spaces of rank 2 and hence the problem of describing all invariant subspaces in this case is no less complicated than that of describing the invariant subspaces for M = G = Rn , n 2. For symmetric spaces of rank 1 a description of invariant subspaces of ‘general’ form was obtained by Wawrzynczyk in [24]. Let G be a Lie group that acts ransitively on a smooth noncompact manifold M (we always assume that the action is smooth), K is a maximal compact subgroup of G. We fix a point o ∈ M. We are called M a homogeneous manifold of rank 1 if there exists a one-parameter subgroup α(t), t ∈ R, of G such that any point x ∈ M can be represented in the form x = uα(t)o where u ∈ K, t ∈ R. The representations x = uα(t)o we shall call a polar decomposition of x. Examples of homogeneous manifolds of rank 1 are Riemannian symmetric spaces of rank 1 of noncompact type (see [7]) and semisimple pseudo-Riemannian symmetric spaces of rank 1 (see [11]). A special case of the last example is the case where M is a real semisimple Lie group of symmetric rank 1 (that is a rank of symmetric space
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M/K is 1, where K is a maximal compact subgroup of M), G = M × M and the group M × M acts on M by left and right translations, that is (g1 , g2 )x := g1 xg2−1 ,
x ∈ M, (g1 , g2 ) ∈ G = M × M.
Another example of homogeneous manifold of rank 1 is a Euclidean n-dimensional space Rn if G = ISO(n) is the group of motions of Rn . The generalization of this example is the case where M = To X is a tangent space at a point o to a Riemannian symmetric space X of rank 1 and G is the Cartan group of motions (that is, G is semidirect product the vector group of the vector space To X and the compact group of isometries of X preserving the point o, see [8]). There exist and some other examples of homogeneous manifolds of rank 1. The homogeneous manifols of rank 1 is most promising for studying the problem of describing invariant subspaces of some function spaces. There is no solution of this problem for all homogeneous manifolds M of rank 1, but we have a solution of this problem for the cases where M is a Rimannian symmetric space of rank 1 or M is a tangent space to a Riemannian symmetric space of rank 1. These solutions werw obtained in [14, 16–21] and we will consider these results in Sections 2 and 3. Let us note that there exist a description of invariant subspaces for some other homogeneous manifolds of rank 1. So, a description of biinvariant subspaces of some function spaces on the group SL(2, C) and on the group of motions of Euclidean plane was obtained in [15] and in [13]. For simplicity we will assume that the function space F is the space E(M) = ∞ C (M) of all infinitely differential functions on M, but a description of invariant subspaces can be obtained similarly for many other function spaces (see [17]). 2. Invariant Subspaces in the Space of Smooth Functions on Riemannian Symmetric Spaces of Rank 1 Let M be a Riemannian symmetric space of noncompact type. We shall assume that M is realized as the quotient space G/K, where G is a real semisimple connected Lie group with finite centre, K is a compact subgroup of G, and the Riemannian metric on G/K is generated by the Killing form on the Lie algebra of the group G. Let E = C ∞ (M) be the space of all infinitely differentiable complex-valued functions on M. E is a complete locally convex space equipped with the standart topology. In this section we study the problem of describing invariant subspaces in the space E in the case where M = G/K is a Riemannian symmetric space of noncompact type of range 1. It is known from the classification of symmetric spaces (see, for example, [7]) that the Riemannian symmetric spaces of rank 1 and noncompact type are exhausted by the following spaces (n 2 everywhere): (1) SO0 (n, 1)/SO(n); (2) SU(n, 1)/U(n); (3) Sp(1, n)/Sp(1) × Sp(n);
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(4) the exeptional symmetric space (the Cayley space). Let E be a finite-dimensional vector space (all vector spaces are taken over C). By C ∞ (M, E) denote the set of all infiniely differentiable functions on M with values in E. The set C ∞ (M, E) is a complete locally convex vector space. Let be the set of equivalence classes of irreducible finite-dimensional representations of the group K. For λ ∈ let T λ : K → End(E λ ) be the corresponding irreducible representation of K, E λ the representation space, ·, · a K-invariant Hermitian form on E λ . By E (λ) we denote the set of functions F (x) ∈ C ∞ (M, E λ ) that satisfy F (ux) = T λ (u)F (x)
∀u ∈ K.
(2.1)
The set E (λ) is a closed vector-subspace of C ∞ (M, E λ ), and we equip it with the induced topology. Let 0 be the set of λ ∈ such that E (λ) = {0}. For any invariant subspace H ⊂ E (we always assume that H = E), we denote by H (λ) the set of all functions F (x) ∈ E (λ) such that the functions fξ (x) := F (x), ξ belong to H for each ξ ∈ E λ . The set H (λ) is a closed vector-subspace of E (λ) . The subspaces H (λ) are called cells of the invariant subspace H or simply cells. Nonzero cells can exist only for λ ∈ 0 . LEMMA 1. Any invariant subspace H ⊂ E can be uniquely recovered by the collection of its cells H (λ) , λ ∈ 0 . Namely, H is the closure in E of the linear span of the functions F (x), ξ , where F ∈ H (λ), ξ ∈ E λ , λ ∈ 0 . Proof. See [17], Lemma 3.2. 2 The general plan of describing invariant subspaces is as follows: first we shall describe the construction of all possible cells, and then determine conditions under which a collection of cells corresponds to a singe invariant subspace. We recall some standart notation of the theory of symmetric spaces (see [7, 8]). Let M = G/K be a connected semisimple Lie group with finite centre, and K a compact subgroup of G. Let g0 and k0 be the Lie algebras of G and K respectively, and let g0 = k0 + p0 be the Cartan decomposition of g0 . We choose a maximal Abelian subspace a0 of the space p0 and complete a0 to a maximal Abelian subalgebra h0 of g0 . Let g, k, p, a and h be the complexifications of the spaces g0 , k0 , p0 , a0 and h0 respectively. Then h is a Cartan subalgebra of g. We consider roots and restricted roots (that is, restrictions of the roots to a) with respect to this Cartan subalgebra. Let denote the set of restricted roots ( ⊂ a∗ , where a∗ is the space dual to a), and let + be the subset of positive restricted roots. Let Bg (X, Y ) be the Killing form of the Lie algebra g. For every functional α ∈ a∗ there is a unique element Hα ∈ a such that Bg (Hα , H ) = α(H ) for all H ∈ a. We define a bilinear form ·, · by putting α, β := Bg (Hα , Hβ ). Let ρ ∈ a∗ be the semisum of the positive restricted roots, and let be the Laplace–Beltrami operator on M (the Riemannian metric on M induced by the Killing form).
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In what follows we assume that M is a symmetric space of rank 1. Then dimR a0 = 1. Given any functional µ ∈ a∗ and any positive integer r, we de(λ) the subspace of E (λ) consisting of all functions F (x) that satisfy the note by Vµ,r differential equation ( + µ, µ + ρ, ρ )r F = 0. (λ) (λ) = r for λ ∈ 0 and the space Vµ,r has a Jordan basis It can be shown that dim Vµ,r F1 , . . . , Fr (that is F1 = −( + µ, µ + ρ, ρ )F1 and Fk = −( + µ, µ + (λ) (λ) = V−µ,r . ρ, ρ )Fk + Fk−1 for k 2). Let us note that Vµ,r (λ) The subspaces Vµ,r are the simplest cells. The following theorem describes the structure of all cells in E (λ) .
THEOREM 2. Let H (λ) be an arbitrary cell in E (λ) , λ ∈ 0 . Then there is a unique finite or countable set σ ⊂ a∗ (each element µ ∈ σ may occur in σ with finite multiplicity) such that H (λ) is the closure in E (λ) of the linear span of the (λ) subspaces Vµ,r , where µ runs over σ and r is the multiplicity of µ in σ . The set σ may naturally be called the spectrum of the cell H (λ). There is a more detailed description of spectra of all possible cells. Since dim a∗ = 1, we can identify the set a∗ with C by choosing a basis in a∗ . It is known that the set + of positive restricted roots for a symmetric space of rank 1 consists of the roots β and 2β (that is + = {β, 2β}) whose multiplicities mβ , m2β depend on the space M (it may happen that m2β = 0). We choose the vector β for a basis of a∗ . Then we may assume that µ ∈ C. To choose one number from every pair of numbers µ and −µ, we require that µ ∈ C+ , where C+ is the set of complex numbers z such that Re z 0 and if Re z = 0, then Im z 0. We thus regard the spectrum σ as a subset of C+ . The possible spectra of cells are exactly those finite or countable sets σ ⊂ C+ (each element µ is contained in σ with finite multiplicity rµ ) that satisfy the condition (S) from Section 1. Suppose that we fix a cell H (λ) of some invariant subspace in every space E (λ) , λ ∈ 0 , and let σ (λ) be the spectrum of H (λ). Below, we shall obtain conditions on σ (λ) that are necessary and sufficient for H (λ) to be cells of a sigle invariant subspace. As a preliminary, we give explicit description of the set 0 for every type of symmetric spaces of rank 1. Every irreducible representation of a compact connected Lie group K is defined by its highest weight and we shall assume that is the set of the highest weights of irreducible representations of K. Let M be a symmetric space of type (I) (that is, M = SO0 (n, 1)/SO(n)), then K = SO(n). The highest weight λ of an irreducible representation of the group SO(n) can be identified with a tuple of integers λ = (λ1 , . . . , λm ) (m = [n/2] is the integer part of n/2) satisfying the conditions λ1 λ2 · · · λm−1 |λm | for n = 2m and λ1 λ2 · · · λm 0 for n = 2m + 1. Then 0 is the set of the highest weights of the form λ = (l, 0, . . . , 0) (that is λ2 = · · · = λm = 0), where l ∈ Z for n = 2 and l ∈ Z+ = {0, 1, 2, . . .} for n 3.
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Let M be a symmetric space of type (II) (that is, M = SU(1, n)/U(n)), then K = U(n). The highest weight λ of an irreducible representation of the group U(n) is determined by a tuple of integers λ = (λ1 , . . . , λn ) satisfying the condition λ1 λ2 · · · λn . Then 0 is the set of the highest weights of the form λ = (k, d, . . . , d, l) (that is, λ1 = k, λ2 = · · · = λn−1 = d, λn = l), where k, d, l ∈ Z such that k d l and 3d = k + l. For symmetric spaces of types (III) and (IV) the groups K are semisimple and simply connected (in case (III), K = Sp(1)×Sp(n) and in case (IV), K = Spin(9)); therefore, the irreducible representations of the group K can be identified with the finite-dimensional irreducible representations of the corresponding semisimple Lie algebra k. Let k be an arbitrary semisimple Lie algebra over C, α1 , . . . , αn a system of simple roots of k, and ω1 , . . . , ωn the corresponding set of fundamental weights. The set of highest weights of finite-dimensional irreducible representations of k coincides with the set of all linear combinations λ = λ1 ω1 + · · · + λn ωn with coefficients λj ∈ Z+ . Therefore, we identify the highest weight λ with a tuple of nonnegative integers: λ = (λ1 , . . . , λn ), λj ∈ Z+ . In case (III), k = sp1(1) × sp(n), n 2. We choose simple roots α0 , α1 , . . . , αn of the Lie algebra k such that α0 is a root of the subalgebra sp(1), and α1 , . . . , αn are the simple roots of the subalgebra sp(n) = Cn , taken in the usual order determined, for example, by the Dynkin diagram o — o — · · · — o ⇐ o
α1
α2
αn−1
αn
In this case the set 0 consists of all the highest weights of k having the form λ = (k, k, l, 0, . . . , 0), where k, l ∈ Z+ (that is λ0 = λ1 = k, λ2 = l, and λ3 = · · · = λn = 0). In case (IV), k = so(9) = B4 . Let α1 , α2 , α3 , α4 be simple roots of the Lie algebra k, taken in order determined by the Dynkin diagram o — o — o ⇒ o
α1
α2
α3
α4
In this case the set 0 consists of the highest weights of the form λ = (k, 0, 0, l), where k, l ∈ Z+ . Suppose that in each space E (λ) , λ ∈ 0 , we fix a cell H (λ) of some invariant subspace that depends on λ in general, and let σ (λ) be the spectrum of H (λ). The following theorems give necessary and sufficient conditions for the cells H (λ) to be cells of a single invariant subspace. THEOREM 3. Suppose that M = SO0 (n, 1)/SO(n), λ = (l, 0, . . . , 0) ∈ 0 , and let δ = (1, 0, . . . , 0). The cells H (λ) correspond to a single invariant subspace if and only if the spectra σ (λ) satisfy the following conditions: (1) For l 0 the spectra σ (λ) and σ (λ + δ) must coincide, √ or differ only by the n−1 multiplicity of the number µ1 (λ) := i(l + 2 ), i = −1, and in the latter
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case the multiplicity of this number in the tuple σ (λ + δ) must be less by 1 than the multiplicity in σ (λ). (2) For l 0 the spectra σ (λ) and σ (λ − δ) must coincide, or differ only by the multiplicity of the number µ2 (λ) := i( 12 − l), and in the latter case the multiplicity of this number in the tuple σ (λ − δ) must be less by 1 than the multiplicity in σ (λ). For n 3 only the condition (1) remains. For the remaining types of symmetric spaces we first introduce integer vectors δ1 and δ2 and numbers µ1 (λ), µ2 (λ) ∈ C. In case (II), let δ1 = (2, 1, . . . , 1), δ2 = (−1, . . . , −1, −2) (δ1 , δ2 ∈ Zn ), λ = (k, d, . . . , d, l) ∈ 0 , µ1 (λ) = i(n + 2(k − d)), µ2 (λ) = i(n + 2(d − l)). In case (III), let δ1 = (1, 1, 0, . . . , 0), δ2 = (−1, −1, 1, 0, . . . , 0) (δ1 , δ2 ∈ Zn+1 ), λ = (k, k, l, 0, . . . , 0) ∈ 0 , µ1 (λ) = i(2k + 2l + 2n + 1), µ2 (λ) = i(2l + 2n − 1). In case (IV), let δ1 = (0, 0, 0, 1), δ2 = (1, 0, 0, −1), λ = (k, 0, 0, l) ∈ 0 , µ1 (λ) = i(2k + 2l + 11), µ2 (λ) = i(2k + 5). THEOREM 4. For symmetric spaces of types (II)–(IV) the cells H (λ) , λ ∈ 0 , correspond to a single invariant subspace if and only if the following conditions hold: (1) If λ+δ1 ∈ 0 , then the spectra σ (λ) and σ (λ+δ1 ) must coincide, or σ (λ+δ1 ) can be obtained from σ (λ) by decreasing the multiplicity of µ1 (λ) by 1. (2) If λ+δ2 ∈ 0 , then the spectra σ (λ) and σ (λ+δ2 ) must coincide, or σ (λ+δ2 ) can be obtained from σ (λ) by decreasing the multiplicity of µ2 (λ) by 1. In combination, Theorems 2–4 give a complete description of the invariant subspaces of E. The proofs of Theorems 2–4 see in [14, 16] for symmetric spaces of type (I), in [17] for symmetric spaces of type (II), in [19] for symmetric spaces of type (III), and in [20] for a symmetric space of type (IV). From Theorems 2–4 it is possible to get a description of some special invariant subspaces of E. We shall give a description of irreducible and indecomposable invariant subspaces of E. An invariant subspace H is called irreducible if H contains no invariant subspace H1 ⊂ H other than H and zero subspace. Each irreducible invariant subspace is determined by a complex number µ ∈ C+ . Let 0 = (0, . . . , 0) be the highest weight of the trivial one-dimensional representation of the group K. The spectrum σ (0) must consist of the number µ with multiplicity 1, and the remaining spectra σ (λ), λ ∈ 0 , are defined in such a way that we get the smallest invariant subspace whose spectra satisfy conditions (1) and (2) of Theorems 3 and 4. If µ = µ1 (λ) and µ = µ2 (λ) for any λ ∈ 0 then all the spectra σ (λ) must consist of one number µ with multiplicity 1. The corresponding irreducible invariant subspace consists of all functions f ∈ E satisfying the differential equation ( + µ, µ + ρ, ρ )F = 0.
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Hence this invariant subspace is an eigenspace of the Laplace–Beltrami operator. (Here we use the identification of a∗ and C+ introduced above.) We note that the irreducibility of these eigenspaces for the Laplace–Beltrami operator was obtained by Helgason [6]. An invariant subspace H is called indecomposable if H = H1 + H2 for any nonzero invariant subspaces H1 and H2 such that H1 ∩ H2 = {0}. Here H1 + H2 is the closure of the algebraic sum of the subspaces. An invariant subspace H is indecomposable if and only if all its spectra σ (λ), λ ∈ 0 , consist of single number µ ∈ C+ not depending on λ with some multiplicity. This multiplicity does not depend on λ, except for the critical cases when µ has the form µ = µ1 (λ) or µ = µ2 (λ) for some λ ∈ 0 . In the critical cases the multiplicity of µ can vary according to λ in such a way that the conditions of Theorems 3 and 4 hold. If all the spectra σ (λ) consist of the number µ with constant multiplicity r, then the corresponding indecomposable invariant subspace consists of all functions f ∈ E satisfying the differential equation ( + µ, µ + ρ, ρ )r F = 0.
3. Invariant Subspaces in the Space of Smooth Functions on a Tangent Space Let M be a Riemannian symmetric space of noncompact type. Let X = To M be a tangent space to the manifold M at some point o ∈ M. We shall assume that M is realised as the quotient space G/K, G being a real semisimple connected Lie group with finite centre, K is a compact subgroup of G. Let o = eK ∈ G/K = M where e is the unit element of G. The group G acts transitively on G/K by left translations and K is the isotropy subgroup of the point o. The vector space X = To M is an Abelian Lie group. Let k∗ ξ := (dk)o ξ be the induced action of K on X where dk is the differential of k and ξ ∈ X. We can form semidirect product G0 = X K with respect to this action of K on X. The group G0 is called the Cartan motion group of X (see [7]). The group G0 acts transitively on X by (ξ, k)x := ξ + k∗ x where (ξ, k) ∈ G0 , x ∈ X. In the case M = SO0 (n, 1)/SO(n) we can identify the tangent space X = To M with n-dimensional Euclidean space Rn and the Cartan motion group is the group ISO(n) of motions of Rn . In what follows we consider the case where X = To M is the tangent space to a Riemannian symmetric space M of rank 1 at the point o, the Cartan motion group G0 acts on X, a function space F = E(X) is the space of all C ∞ -functions on X. In this section we study the problem of describing invariant subspaces in E(X). The general plan of describing of the invariant subspaces is similar to that of Section 2. As in Section 2 let be the set of equivalence classes of irreducible finite-dimensional representations of the group K or, equivalently, the set of highest weights of irreducible representations of K. For any λ ∈ let T λ (u) be the
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corresponding irreducible representation of K, E λ the representation space, ·, · a K-invariant Hermitian form on E λ . For any λ ∈ we denote by E (λ) the set of functions F (x) ∈ C ∞ (X, E λ ) that satisfy F (ux) = T λ (u)F (x)
∀u ∈ K.
Let 0 be the set of λ ∈ such that E (λ) = {0}. For any invariant subspace H ⊂ E(X) (we always assume that H = E(X)) and λ ∈ we define the cell H (λ) ⊂ E (λ) as in Section 2. Nonzero cells can exist only for λ ∈ 0 and any invariant subspace H ⊂ E(X) can be uniquely recovered by the collection of its cells H (λ) (see Lemma 1). X is an Euclidean space with scalar product induced by Riemannian metric on M. Let be differential Laplace operator on the Euclidean space X. Given any (λ) the subspace of E (λ) consisting µ ∈ C and any positive integer r, we denote by Vµ,r of all functions F (x) that satisfy the differential equation ( + µ2 )r F = 0. It can (λ) (λ) be proved that dim Vµ,r = r for λ ∈ 0 and the space Vµ,r has a Jordan basis 2 2 F1 , . . . , Fr (that is, F1 = −µ F1 and Fk = −µ Fk + Fk−1 for k 2). We note (λ) (λ) = V−µ,r and, hence, we may assume that µ ∈ C+ . that Vµ,r (λ) is the simplest invariant cell in E (λ) . The structure of general The subspace Vµ,r invariant cell in E (λ) is described by the folloving theorem. THEOREM 5. For any invariant cell H (λ) ⊂ E (λ) there is a unique finite or countable collection σ ⊂ C+ (each element µ ∈ σ may occur in σ with finite multiplicity) such that H (λ) is the closure in E (λ) of the linear span of the subspaces (λ) , where µ runs over σ , and r is the multiplicity of µ in σ . Vµ,r The colection σ may naturally be called the spectrum of the cell H (λ). The possible spectra are exactly those finite or countable sets σ ⊂ C+ that satisfy the condition (S) from Section 1. Suppose that in each space E (λ) , λ ∈ 0 , we fix a cell H (λ) of some invariant subspace that depend on λ in general, and let σ (λ) be the spectrum of H (λ). We shall give necessary and sufficient conditions for the spectra σ (λ) such that the cells H (λ) are cells of a single invariant subspace. Let r0(λ) denote the multiplicity of the number 0 in σ (λ) (r0(λ) ∈ Z+ ), and let α+ := max(α, 0), α ∈ R. THEOREM 6. Suppose that X = To M, M = SO0 (1, n)/SO(n), λ = (l, 0, . . . , 0) ∈ 0 , and δ = (1, 0, . . . , 0). The cells H (λ) correspond to a single invariant subspace if and only if the following conditions hold: (1) The spectra σ (λ) can be different only by the multiplicities r0(λ) for various λ ∈ 0 . (2) For any l 0 the multiplicity r0(λ+δ) must be equal to r0(λ) or (r0(λ) − 1)+ .
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(3) For any l 0 the multiplicity r0(λ−δ) must be equal to r0(λ) or (r0(λ) − 1)+ . For n 3 only the conditions (1) and (2) remain. For the remaining types of symmetric spaces of rank 1 we shall use the notation δ1 and δ2 as in Section 2. THEOREM 7. Let X = To M, and M is a symmetric space of type (II), (III) or (IV). The cells H (λ), λ ∈ 0 , correspond to a single invariant subspace if and only if the following conditions hold: (1) The spectra σ (λ) can be differrent only by the multiplicity r0(λ) for various λ ∈ 0 . (2) If λ + δ1 ∈ 0 , then the multiplicity r0(λ+δ1 ) must be equal to r0(λ) or (r0(λ) − 1)+ . (3) If λ + δ2 ∈ 0 , then the multiplicity r0(λ+δ2 ) must be equal to r0(λ) or (r0(λ) − 1)+ . In combination, Theorems 5–7 give a complete description of the invariant subspaces of E(X). The proofs of these theorems see in [18] for the spaces of type (I) and in [21] for the other types of spaces. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
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