INVARIANT SUBSPACES IN CERTAIN FUNCTION

MaTeM C6OPHHK TOM 137(179)0988), Bbin. 4

Math. USSR Sbornik Vol. 65(1990), No. 2

INVARIANT SUBSPACES IN CERTAIN FUNCTION SPACES ON «-DIMENSIONAL LOBACHEVSKY SPACE UDC 517.954 S. S. PLATONOV ABSTRACT. Let Μ be «-dimensional Lobachevsky space, G the connected component of the identity in the group of isometries of M. In certain topological vector spaces of functions on Μ the author obtains a complete description of the closed subspaces invariant under the transformations f(x) —> f(gx), ζ € Μ, g e G. Bibliography: 14 titles.

§1. Introduction and formulation of the main results

In [1] Rashevskii solved the problem of describing the closed subspaces invariant with respect to right and left shifts in certain function spaces on the group SL(2,R), and he conjectured that a description of such subspaces of the analogous spaces could be obtained for all semisimple Lie groups of real rank 1. The first step in the solution of this problem was made in [2]. An analogous problem can be posed for symmetric spaces of rank 1. Let Μ be a Riemannian symmetric space of noncompact type, and Xo a particular point of M. Let \x\ denote the distance from x0 to a point x. Denote by Lpk {p > 1) the set of all measurable complex-valued functions f(x) on Μ such that

a

k

\f(x)\Pe~ ^dxj

\ l/P

o·^ is equipped with the topology of the inductive limit of the BS's Lpk. Denote by G the connected component of the identity in the group of isometries of Μ (the group of isometries is equipped with the compact-open topology). A closed subspace Η of Li (or any other topological vector space that is a function space on M) is said to be invariant if together with each function f{x) in Η all the functions f(gx) {g Ε G) are contained in H. PROBLEM. Describe all the invariant subspaces of Li if Μ is a symmetric space of noncompact type and of rank 1. 1980 Mathematics Subject Classification (1985 Revision). Primary 43A15; Secondary 46A12. ©1990 American Mathematical Society 0025-5734/90 $1.00 + $.25 per page 439

440

S. S. PLATONOV

This problem is one of the main problems in harmonic analysis on a group G (see [3], p. 134), because the invariant subspaces coincide precisely with the closed subspaces of LP that are invariant with respect to the quasiregular representation T(g):f(x)^f(g-lx).

(1.2)

We note also that we consider only proper subspaces (i.e., not coinciding with the whole space), and by an invariant subspace we always understand an invariant proper subspace. Our main result is a complete description of the invariant subspaces of L{ in the case when Μ is «-dimensional Lobachevsky space. We proceed to a more detailed description of the results. Denote by Ck the space of continuous functions f(x) on Μ such that \f{x)\e~k^ tends to zero as \x\ —> oo; Q. is a BS with the norm # * ( / ) = sup |/(x)|o Ck * s equipped with the topology of the inductive limit of the BS's Cfc. If in the definitions of the spaces Q , C , Lpk, and LP we consider functions f(x) with values in a finite-dimensional space Ε instead of complex-valued functions and we replace |/(x)| by ||/(x)|| (|| || is an arbitrary norm in E), then the resulting spaces will be denoted by Ck(E), Ct{E),Lpk{E), and LP(E), respectively. It is proved in §2 that the invariant subspaces of LP and C, are in a one-toone correspondence obtained by assigning to each subspace of C* its closure in Lp. Therefore, it suffices to describe the invariant subspaces of C*. In the space ft"+1 = {x = (x\... ,x" + 1 )} we introduce the bilinear form

[x,y] = -x'y'

x"yn+xn+ly"+l.

Let us take the upper skirt of the hyperboloid [x,x] = 1 (i.e., xn+\ > 0) as a realization of «-dimensional Lobachevsky space. Then the group G consists of the linear transformations A of the space R" +1 that preserve the form [x,y], have determinant 1, and are such that [Aen+\,en+\] > 0, where es is the vector with 5th coordinate 1 and the other coordinates 0 (i.e., G is the connected component of the identity of the group SO(«, 1)). We take the particular point Xo of Μ to be the point en+\. The distance between points x,y e Μ is found from the formula coshr = [x,y].

(1.4)

Let Κ = {g € G: gen+l = en+i} be the isotropy subgroup of the point en+i. The group Κ is isomorphic to SO(«). An arbitrary irreducible representation of SO(«) is determined by the highest weight, which can be identified with an integer tuple λ = (X\,...,Xk), k = [«/2] ([x] is the integer part of a number x) satisfying the conditions λ\ >λ2 > ••• > 4 > 0 (1.5) for η = 2k + 1 and Ai > A2 > •·· > A*-, > |ΑΛ|

(1.6)

for η = 2k (see [4]). Denote by Tl the irreducible representation of Κ with highest weight (/, 0,..., 0) (/ e Ζ for « = 2, and / e Z+ for « > 3, where Z+ is the set of nonnegative integers). Let El be the space of the representation Tl, and fix in E1 an invariant Hermitian form {ξ, η) (ξ, η e E').

INVARIANT SUBSPACES IN CERTAIN FUNCTION SPACES

441

Let Cll] be the set of all functions F(x) e C*(El) such that (1.7)

F{ux) = T'{u)F(x) VueK.

This space is equipped with the topology induced from C»(£'). For every invariant subspace Η c C» denote by /7 (/) the set of all functions F(x) e l) Cl such that the function
(1.9)

where Δ is the Laplace operator on M, and m = (n - l)/2. It will be shown later that dim V^ = r, and that V^ has a Jordan basis, i.e., a basis F\,...,FT such that AFi = -(A 2 '+ m2)F and AFk = -(λ2 + m2)Fk + Fk_{ for k > 2. THEOREM 1. For a particular I the cells of the invariant subspaces of C»'' are in a one-to-one correspondence with the admissible sets of complex numbers. Corresponding to a set {kj} is the cell // (/) that is the closure in dl) of the linear span of the subspaces V^, where λ runs through the set {Xj} and r is the multiplicity of λ in this set. (l)

Suppose that in each space Ci'' we fix a cell H of some invariant subspace, depending on / in general. The following theorem gives necessary and sufficient (/) conditions for all the cells // to be cells of a single invariant subspace H. 2. Suppose that in each space Ci'] (/ eZ+forn>3 and I e Ζ for η = 2) a cell H is fixed that is determined by the admissible set {λ^}. For the cells // (/) to be the cells of a single invariant subspace Η of C it is necessary and sufficient that the following conditions hold: (1) For I > 0 the sets {kf} and {λ{'+[)} can differ only by the multiplicity of the THEOREM (l)

number i(l + (n — l)/2), and the multiplicity of this number in {λ(/+1)} can be less by 1 than the multiplicity in {Xf}. (2) For I < 0 the sets {Xf} and {X{j~l)} can differ only by the multiplicity of the 1

number -il, and the multiplicity of this number in {Λ.^" '} can be less by 1 than its multiplicity in {X^}.

442

S. S. PLATONOV

Only condition (1) remains for η > 3. Combination of Theorems 1 and 2 gives a complete description of invariant subspaces. The main idea in the proof of Theorem 1 is that, because of condition (1.7), the functions in d!) are actually functions of a single independent variable, and the problem of describing the invariant cells can be reduced to Rashevskii's problem of describing the subspaces of a certain function space on R that are invariant with respect to the transformations φ(ί) —> \{). The construction of the operators X± and the determination of an explicit form of them required the creation of new algebraic methods (see §3), and, in particular, the determination of an explicit form reduced to the problem of determining certain Clebsch-Gordon coefficients of the tensor product decomposition Ελ El of the group SO(n), where Ελ is an arbitrary irreducible representation and El the representation with highest weight (1,0,...,0). §2. Reduction of the problem to the description of invariant subspaces in C,

Everywhere in this section Μ is an arbitrary symmetric space of noncompact type, and the spaces Lp* and C* were defined in §1. Recall that the distance from the particular point x 0 to a point χ e Μ is denoted by JJC|, and Κ is the isotropy subgroup of xoFor g e G we define the "norm" \g\ = \gxo\. Then it is obvious that for χ e Μ and g eG

|w| = 0 WueK;

\ux\ = \x\,

\uigu2\ = \g\ \gx\\x\-\g\,

Note that if f{x) e Lpk, then f{gx)

(2.2)

VuutaeK;

l

+ \x\,

(2.1) l

\g~ \ = \g\.

(2.3)

e L"k for every g e G. Indeed,

\f(gx)\"e-k^

= J |/W*/ —• / , H° is dense in H; therefore, to different Η there correspond different subspaces H°. Let Ε be an arbitrary subspace of C , and let Η be the closure of Ε in Li. We show that H° - £\ If / e //°, then for some net we have that /„ —• / in Li and /, € £•. Then, by Lemma 2.1, ψη * fa —> ς»η * / (the limit is taken with respect to the directed set {„ * / —> / in C , it follows that / € £ . Let 0 be the Lie algebra of G. If X , , . . . , Xm is a basis for 0, then X 1 , . . . , Xm denotes the dual basis with respect to the Cartan form, i.e., (X',Xj) = S,j, where Sjj is the Kronecker symbol, and ( , ) is the Cartan form on (y) for PROPOSITION λ

446

S. S. PLATONOV

arbitrary φ e Hw and y e Ελ,λ € Λ. Then Η is a Harish-Chandra submodule ofV, and //« = nomA{E\ H) for all λ e Λ. See [2] (p. 198, Proposition 1.1) for a proof. The set Ρ(λ, μ) of operators can be described in a more explicit form. Let 0 = be a Cartan decomposition of the algebra 0. Since [&,φ] c φ, the complexification tyc is a ^-module. Denote this module by E, and let p\,... ,pn be a basis in E. LEMMA 3.1. An arbitrary element of Ρ (λ, μ) can be represented in the form L{p\,...,pn;a\,...,an) if λ Φ μ, and L(pu...,pn;au...,an) + cld if λ = μ, where c is a number and Id is the identity operator.

See [2] (p. 201, Corollary 1.2) for a proof. Denote the set of elements in Ρ(λ,μ) of the form L(pi,...,pn;a\,...,an) by Ρο(λ,μ) (note that Ρ0(λ,μ) Φ Ρ{λ,μ) only if λ = μ). Let ι ( Γ , Η ο ΐ η ( Ρ , ^ ) ) stand for the set of homomorphisms (intertwining operators) from the ^-module Ε* (Ε* is the module contragredient to E) to Ηοπι(Εμ,Ελ). PROPOSITION 3.2. As a vector space, Ρο(λ,μ) is isomorphic to the space s{E* ,ΥΙοτα{Εμ, Ελ)) of intertwining operators. To construct this isomorphism, choose a basis p\,...,pn in Ε and let p*,...,p* be the dual basis in E*. The isomorphism assigns to each operator L{p\,...,pn;au...,an) e Ρο(λ,μ) the intertwining operator φ:Ε* ^ Ηοτη(Ε"Ελ) with φ{ρ*) = at.

We apply Proposition 3.2 to the case when Ε is irreducible (this holds if (5 is a simple Lie algebra and the algebra Λ is semisimple; see [13], Theorem 8.8.3). 3.1. Suppose that the module Ε is irreducible. (1) Ρ0(λ, μ) Φ {0} if and only if the module Ελ ® Εμ has a submodule isomorphic to E*, where E* is the module contragredient to E, and Εμ is the module contragredient WE». (2) Suppose that Ελ ® Εμ contains a submodule isomorphic to E*. In this submodule there is a basisp*,...,p% dual to the basispu...,pn in E. Let e\,...,e^ be λ μ η μ a basis in Ε , ef,...,β%μ a basis in Ε , and ^,...,η μ the dual basis in Εμ. Then L(pi, ...,ρη;αι,...,αη) Ε Ρο{λ,μ) if and only if the matrices ofthe operators as satisfy the conditions COROLLARY

See [2] for a proof of Proposition 3.2 (p. 201, the proof of Proposition 1.2). We note only that the formulation of Proposition 1.2 in [2] is valid only for an irreducible Ε (it coincides with the formulation of Corollary 3.1), but the proof works in the general case. Corollary 3.1 can be obtained at once from Proposition 3.2 and the fact that the ^-module Ηοτη(Εμ,Ελ) is isomorphic to Ελ Εμ. Let E[, E2, and £3 be arbitrary ^-modules. Then the following sets of intertwining operators are isomorphic as vector spaces: s(E3,Hom(EuE2))

^ s(EuHom(Ei,E2)).

(3.4)

An isomorphism is obtained by assigning to each φ: E$ —» Hoxn(Ei,Ei) the intertwining operator ψ: Ε\ —* Hom(£3,£ 2 ) defined by VviGEu

V3GE3.

(3.5)

INVARIANT SUBSPACES IN CERTAIN FUNCTION SPACES

447

Let E\, Ei, and E} be irreducible ^-modules, and φ: Ετ, -> £Ί ® £ 2 a nonzero homomorphism. In each module £V we fix a basis e j ^ , ^ ' , — Let 3)

^(4 ) = Σ ^

(1)

2)

®^ ·

(3-6)

The coefficients c'kJ are called Clebsch-Gordon coefficients (CGC's). We use the notation C

k

e

- \i

e

>j

e

\k

I-

If E\ ®E2 contains only one submodule isomorphic to £3, then the homomorphism φ and the corresponding CGC's are determined to within multiplication by a common number. LEMMA 3.2. In the preceding notation let E^ be the module contragredient to E2, and let h' be the dual basis to the basis e\2) in E2. If' ψ'-Ε\ —> Ηοίη^,,ΕΊ) is a homomorphism of ^.-modules, then the matrix elements of the operator are

(3.7)

)) = ^\β^νΡ). PROOF.

Using (3.5), we get that

Then y,{ef) = Σ . , Λ ^ ' ® $\ and so Xk =

{e^^ef).

For the rest of the paper let Μ be «-dimensional Lobachevsky space, and let the groups G and Κ be as described in § 1. Every irreducible representation of the group Κ = SO(rt) is given by its highest weight λ = (λι,.,.,λ^) (k = [n/2]); see §1. Let ΕΛ be the space of the representation with weight λ. We introduce in Κ the sequence of subgroups Κ = Ko D K{ D K2 D • • • D Kn-i, where Km consists of the transformations leaving the vectors en,en-\,...,en-m+i fixed. It is clear that Km is isomorphic to SO(« — m). We can introduce an orthonormal Gel'fand-Tsetlin (GTs)-basis in the space El. (0) (1) ( 2) The elements of this basis are numbered by the schemes Λ = (A ,A ,...,/l """ ), (m) where λ™ = λ and A = (kf\...,λ[™}2]) is the highest weight of the subgroup Km, and these weights are connected by the relations

+λ)

-λ

ψ

(3.8)

(see [9], p. 56). Note that upon restriction of the representation from Κ to Km the corresponding basis vector Λ lies in the space of the representation of weight A(m). Let (5 and Ά be Lie algebras of the groups G and K; & is identified with the set of matrices of the form Ά χ< 0 where A is a skew-symmetric η χ η matrix, χ = (xu...,xn) is a row, x' is the transposed matrix, and 0 denotes the zero matrix. Then 8. and φ ( = Λ + φ is a Cartan decomposition) consist of the matrices

{(;?)}•

448

S. S. PLATONOV

The representation arising in the space
/,} = (

/(/

+" ~

2 )

^

β

.

(5.9)

Since T is a unitary representation, the operator / is skew-Hermitian; therefore, /o = {IVQ,V{) = -(υο,Ινι). Consequently, expansion in the basis vo,...,vm gives us that Ivx = -/(jwo-l .

INVARIANT SUBSPACES IN CERTAIN FUNCTION SPACES

453

We expand the right- and left-hand sides of (5.8) in the basis and take the coefficients of Vo. Then d?F°(a(t)R(

Tl{u)aiTl+l(u-1)

=^ ^ · .

j

(5.11)

j

LEMMA 5.2. For F e Q'>

(X{+]F)(x) = nds f

T!+l(u)a*nF(a(-s)u-ix)du

JK l

l+l

where a* e Hom(E ,E ) PROOF.

i=

(5.12) o

is the operator adjoint to an.

The operator X^ carries C#'' into C# / + 1 ) ; therefore, Λ± r (U

X) — 1

[U

)Λ+

t (X)

(j.ljj

for every u e K. It follows from (5.13) that

X(l]F(x)= f TM(u-1)X^)Tl(u)F(x)du.

(5.14)

JK x

The elementsp\,...,p n are conjugate with respect to the group K. Letp t = u,pnu~ . l l+{ l Then we get from (5.11) and the unitarity of T that a* = T (Ui) • a*T'(u~ ). According to (4.8),

(=1

i=0

Then (5.12) follows from (5.14), the fact that a(s) = exp{spn), and the invariance of the measure du. Denote by νιϋ,...,vlm (m depends on /) a GTs-basis in the space El. It will be assumed that the vectors νιϋ = v0 and υ[ = ν ι are chosen as above. The elements ρ ι,..., ρn are a basis in the space φ 0 . On the other hand, since 0),

(6.4)

(6.5)

Jo where Γ(α) is the gamma-function. If we make the change of variable u = cosh t-\, υ = cosh 5 - 1, then (6.5) becomes the Riemann-Liouville integral transformation (multiple integration; see [12], p. 85)

[\u-

Jo /ο

where h (cosh 5 - 1) = f{s). It follows at once from the properties of the RiemannLiouville transformation (see [12]) that Ψ/?(Ψα(/)) = Ψ ο + «(/), 1

sinlT tdtfaif) LEMMA

(6.6) a

=Ψα-ι(Α

> 1·

(6-7)

6.2. Tfte invariant measure dx on the space Μ can be normalized in such

a way that /*

/*OO

/ 27(2a)!)-Alphas a nonzero radius of convergence in dl)d for every d e Z + ; here and in the following sums a varies from 0 to oo. Let d > I. The function h, as the image of F, is an analytic function, and it follows from the continuity of the mapping Φ ο Ο : C{J)d -» W? (see Lemma 6.1) that the series (6.22)

,

where Ah = d}h - m2h - y(cosh/ - l) r ~', has a nonzero radius of convergence in Our first goal is to show that the series 2

(6.23)

h

has nonzero radius of convergence in W® and converges to \{h{t + s) + h(t - s)). If (6.22) converges in W®, then it converges in some BS W£, and hence for some number C > 0 Ktk{Aah) < C2a(2a)\, (6.24) where η2,ι< is the norm in the BS Wg. It follows from (6.24) that for every λ e C the series

£>

ΓΑ

(6.25)

2

has nonzero radius of convergence. In particular, this is true for λ - m . Let Δι = A + m2. Since h(t) is an analytic function, the derivative d2rh is also an analytic function, and hence for some C > 0 C2Q-2'(2Q

for every a € Z+. From (6.26) and the fact that

- 2r)\

(6.26)

INVARIANT SUBSPACES IN CERTAIN FUNCTION SPACES

461

(for γ see Lemma 6.3) it is easy to get that for all β e Z + where P^(coshi) = a^ r _ 1 (cosh?) r ~ 1 + · · · + αββ, and the coefficients admit the following estimate: there exists a number C\ for which This implies that series (6.23) has nonzero radius of convergence ε in W®, and its sum S = \{h{t + s) + h(t - s)) differs from that of £(s 2c 7(2a)!)A?/j only by a linear combination of the functions cosl/ t, β = 0,..., r-1. Since Δ?Α e ή and cosh'31 e £, it follows that h\(t) = \{h{t + s) + h(t - s)) e f) for |s| < ε. The number ε here can depend on the function h e fj#, but it does not change if h(t) is replaced by h(t) = h{t + λ) + h(( - λ) VA G R, since the transformation h —• h is a continuous operator on W®. Then it follows from h\ e ft that \{hi{t + s) + hx{t-

s)) = \(h(t + 2s) + 2/2(0 + h(t - 2s)) e Sj.

Arguing similarly, we show by induction that for all t g Z + hk{t) = ±{h{t + ks) + h{t - ks)) e S). Finally, %(h{t + s) + h(t - s)) e Sj V5GR. So far we have considered the case when η = 2m + 1 is an odd number. If η = 2m + 1 is an even number, then m is a half-integer. Let l + m—l = k + j . As in the odd case, the transformation Φ is defined by Φ(/)(/) = (Γ(fc+ § ) ) " ' f

(coshi-cosh5) A : + 1 / 2 (sinh5)- / /(5)sinh5ii5

for t > 0, and we extend Φ(/) to negative values of t so that it is odd. Under this extension smooth functions go into smooth functions, as is easily seen from the fact that (cosh/ -cosh s)k+1/2

= (cosh i - c o s h 5) fc v / 2sinh^ -yjl - (sinhf/sinh^) 2 .

If //Ο c Cl'] is a cell of an invariant subspace and H«)d = H^ η C(J)d, then fl denotes the closed subspace of W} generated by the image Φ(ϋ(Η^ά)) and the functions sinh j,sinh jcosht,...,sinh^(cosht) k . If i / ^ and //j'' are cells of invariant subspaces of Ci'\ then the subspaces of Wj corresponding to them are denoted by Sj\ and 9)2, respectively. Arguing as in Proposition 6.2, we get that S)\ Φ f)2 if //(' φ H^'. Arguing as in Proposition 6.3, we get that the subspace Sj is invariant under the transformations (6.21). We proceed to the proof of Theorems 1 and 2 (see the Introduction). Assume that η = 2m + 1 is an odd integer. The case when η is even can be treated analogously. Recall (see §1) that V^ consists of all smooth functions F e Gi'' satisfying the condition (Δ + λ2 + m2)rF = 0. LEMMA

6.4. dim V^ — r, and the space V^ has a Jordan basis.

Let us verify that V^ c C{J\ To do this it suffices to verify that every function F € V^ is an analytic vector of the representation (4.1). But F is a solution PROOF.

462

S. S. PLATONOV

of the elliptic equation (Α + λ2 + m2)rF = 0, and a solution of an elliptic equation is an analytic vector (see [5], the remark on p. 418). Since the space Cil) is isomorphic to Si'\ it can be assumed that V^} c Sl'K The image