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Symmetric pairs, Gelfand pairs and pairs such that generating functions of zonal spherical functions define unitary operators—the case of the compact type
Journal of Functional Analysis 238 (2006) 427–446 www.elsevier.com/locate/jfa
Symmetric pairs, Gelfand pairs and pairs such that generating functions of zonal spherical functions define unitary operators—the case of the compact type Shigeru Watanabe The University of Aizu, Aizu-Wakamatsu City, Fukushima 965-8580, Japan Received 6 October 2005; accepted 15 February 2006 Available online 24 March 2006 Communicated by L. Gross
Abstract A new notion which is equivalent neither to that of Riemannian symmetric pair nor to that of Gelfand pair can be defined by generating functions. In the case of the compact type, this statement is shown. This notion also concerns an existence of a “good” parametrization for the spherical representations. These mean that generating functions can be dealt with systematically in the compact case. © 2006 Elsevier Inc. All rights reserved. Keywords: Riemannian symmetric pair; Gelfand pair; Zonal spherical function; Generating function
1. Introduction How possibility do generating functions have? The purpose of this paper is to give our answer to this question. Bargmann [1] constructed a unitary operator given by an integral operator whose kernel is a generating function of the Hermite polynomials. He also gave in [1] a similar construction for the Laguerre polynomials. Thus we naturally turn our interest to other orthogonal polynomials. In fact, he notices as follows [1, p. 203]: “It is worth noting that a similar interpretation may be given to other classical generating functions.” E-mail address: [email protected]. 0022-1236/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2006.02.008
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First of all, we turned our interest to the Gegenbauer polynomials which give essentially the zonal spherical functions on the pair (SO(n), SO(n − 1)). And in [9] we showed that a similar construction is possible for the Gegenbauer polynomials. Next, in [11] we gave a similar construction for the zonal spherical functions on the pair (U (n), U (n − 1)). On the other hand, Essadiq [2] gave a q-analogue of the Gegenbauer case. Let us return to the two cases that we constructed. As a matter of fact there is a similarity between them not only in the constructions but also in the forms of the generating functions. We shall explain in detail. Let N0 be the set of nonnegative integers, and let B be the unit open disk |z| < 1 in C. The set of all the zonal spherical functions on (SO(n), SO(n − 1)) is parametrized by N0 . Denote it by {ϕm | m ∈ N0 }. Further, for each m ∈ N0 we denote by dm the degree of the representation corresponding to ϕm . Then the generating function associated with the unitary operator is written as a series of the following form:
x ∈ SO(n), z ∈ B.
dm ϕm (x)zm ,
(1)
m∈N0
On the other hand, the set of all the zonal spherical functions on (U (n), U (n − 1)) is parametrized by N20 . As in the first case, define {ϕm | m ∈ N20 } and dm (m ∈ N20 ). Then the generating function in this case is written as a series of the following form:
x ∈ U (n), z ∈ B 2 ,
dm ϕm (x)zm ,
(2)
m∈N20
where zm = z1m1 z2m2 for z = (z1 , z2 ) and m = (m1 , m2 ). We are reasonably led to the question whether similar situations occur in the case of general Gelfand pair (G, K) of the compact type. That is, let G be a compact group, K a closed subgroup of G, and (G, K) a Gelfand pair. Further, suppose that the set of all the zonal spherical functions on (G, K) is parametrized by N0 , where is a positive integer and is not uniquely determined. And as in the case above, define {ϕm | m ∈ N0 } and dm (m ∈ N0 ). Then our question is as follows. Is it possible to find a generating function of the following form such that it is the kernel of an integral operator that is unitary?
dm ϕm (x)zm ,
x ∈ G, z ∈ B ,
(3)
m∈N0 m
where zm = z1m1 · · · z for z = (z1 , . . . , z ) and m = (m1 , . . . , m ). Remark that if we regard (3) as a function of x and denote it by Φz (x), then the following holds under a suitable condition: Φz ∗ Φw = Φzw ,
(4)
where zw = (z1 w1 , . . . , z w ) for z = (z1 , . . . , z ) and w = (w1 , . . . , w ). This follows from the relation (ϕm ∗ ϕm )(x) =
−1 ϕ (x), dm m 0,
m = m , m = m .
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Let us consider the question from a more general viewpoint. Let G be a compact group, and let K be a closed subgroup of G. We do not necessarily suppose that (G, K) is a Gelfand pair. Denote by dx the Haar measure on G, normalized by G dx = 1. Further, denote by L2 (G, K) the space of square integrable functions on G that are bi-invariant under K. Considering (4), we shall introduce the notion of a generating function associated with the pair (G, K), together with that of a Hilbert space of the Bargmann type. Definition 1. Let be a positive integer, and let (B , μB ) be a measure space. Denote by L2 (B , μB ) the Hilbert space of square integrable functions on B . If the space of analytic functions on B that belong to L2 (B , μB ) is a closed subspace of L2 (B , μB ), then it is called a Hilbert space of the Bargmann type of order . Definition 2. Suppose that there exists a positive integer and that a function Φ(z, x) defined on B × G satisfies the following conditions: (i) The function Φ(z, x) is continuous on B × G, for each x ∈ G it is analytic on B as a function of z, and for each z ∈ B it is bi-invariant under K as a function of x. (ii) For each z ∈ B define the function Φz (x) on G by Φz (x) = Φ(z, x). Then the following holds: (Φz ∗ Φw )(x) = Φzw (x) for all z, w ∈ B and all x ∈ G. (iii) For ϕ ∈ L2 (G, K) define (Φϕ)(z) = Φ z, x −1 ϕ(x) dx, z ∈ B . G
Then F(B ) = {Φϕ | ϕ ∈ L2 (G, K)} is a Hilbert space of the Bargmann type of order , and Φ is a unitary operator on L2 (G, K) to F(B ). We call Φ(z, x) a generating function associated with the pair (G, K). The first purpose of this paper is to give a sufficient condition for the pair (G, K) to have a generating function associated with itself. Main result 1. Let G be a compact connected Lie group, and let K be a closed subgroup of G. If the pair (G, K) is a Riemannian symmetric pair such that G/K is simply connected, then there exists a generating function of the form of (3) such that it is a generating function associated with (G, K). The second purpose of this paper is to give a necessary condition for the pair (G, K) to have a generating function associated with itself. Main result 2. Let G be a compact group, and let K be a closed subgroup of G. If the pair (G, K) has a generating function associated with itself, then it is a Gelfand pair. In this case the generating function Φ(z, x) associated with (G, K) is written as a series similar to the form of (3). Besides, if we impose a suitable condition on the measure μB related to F(B ), then Φ(z, x) is written as a series of the form of (3). The third purpose of this paper is to show that the notion of pair which has a generating function associated with itself is equivalent neither to that of Riemannian symmetric pair nor to that of Gelfand pair. This means that generating functions define a new notion. We also remark that this notion concerns an existence of a “good” parametrization for the spherical representations.
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The paper is organized as follows. In Section 3 we show Main result 1. In Section 4 we show Main result 2 and give the expansion of a generating function by the zonal spherical functions. In Section 5 we give some examples and mention the third purpose explained above. (The remark to the third purpose is explained at the end of Section 3.) 2. Notation and preliminaries We shall explain the notation and the terms which will be used throughout this paper. 2.1. General notation We shall use the notation N0 , R, C for the set of nonnegative integers, the field of real numbers and the field of complex numbers, respectively. For a fixed positive integer we denote by C the -dimensional complex space, and denote by N0 the set of all multi-indices m = (m1 , . . . , m ) with each mj ∈ N0 . For m = (m1 , . . . , m ) ∈ N0 and z = (z1 , . . . , z ) ∈ C m we write zm = z1m1 · · · z . For z = (z1 , . . . , z ) ∈ C and w = (w1 , . . . , w ) ∈ C we denote (z1 w1 , . . . , z w ) ∈ C by zw. We denote by B the unit open disk in C, and denote by B the open polydisk of z = (z1 , . . . , z ) ∈ C with each zj ∈ B. For a subset A ⊂ R or C we define A in the same way. We shall use the notation [a, b], [a, b) for the interval {x ∈ R | a x b}, the interval {x ∈ R | a x < b}, respectively. For ζ ∈ C let Re ζ be the real part of ζ , and ζ → ζ the usual conjugation in C. A function is assumed to be complex-valued. 2.2. Compact groups A compact group is a topological group whose underlying topology is compact Hausdorff. Let G be a compact group. Let dx denote the normalized Haar measure on G; that is, it has the property that G dx = 1. Let L2 (G) denote the Hilbert space of measurable functions ϕ on G with 2 ϕ G = ϕ(x) dx < ∞. G
The inner product is given by ϕ, ψ G =
ϕ(x)ψ(x) dx.
(5)
G
Let K be a closed subgroup of G. It is said that a function ϕ on G is bi-invariant under K if it satisfies Ly ϕ = ϕ, Ry ϕ = ϕ for any y, y ∈ K, where Ly , Ry (y, y ∈ G) are defined by (Ly ϕ)(x) = ϕ(y −1 x) (x ∈ G), (Ry ϕ)(x) = ϕ(xy ) (x ∈ G). Let L2 (G, K) denote the space of functions ϕ ∈ L2 (G) which are bi-invariant under K. Then L2 (G, K) is a closed subspace of L2 (G), that is, L2 (G, K) is also a Hilbert space with inner product (5). A representation of the compact group G means a continuous homomorphism of G to the group GL(V ) of invertible linear transformations on a finite-dimensional complex vector space V . Since G is compact, for an arbitrary representation of G there exists a G-invariant
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inner product on the representation space V , that is, V admits an inner product such that the representation is unitary. Given an irreducible representation ρ of G, let Vρ denote the representation space of ρ and let dρ denote the degree of ρ. Further, denote by VρK the subspace of Vρ consisting of elements w ∈ Vρ which satisfy ρ(x)w = w for any x ∈ K. If an irreducible representation ρ of G satisfies VρK = {0}, ρ is called a spherical representation of G with respect to the subgroup K. Let D(G, K) denote the set of all the equivalence classes of spherical representations of G with respect to K. For two continuous functions ϕ, ψ on G the convolution ϕ ∗ ψ is defined by (ϕ ∗ ψ)(x) = ϕ xy −1 ψ(y) dy, x ∈ G. G
The pair (G, K) is called a Gelfand pair if the convolution algebra of continuous functions on G which are bi-invariant under K is commutative. The pair (G, K) is a Gelfand pair if and only if dim VρK = 1 for all ρ ∈ D(G, K). 2.3. Riemannian symmetric pairs A Lie group is assumed to be differentiable of class C ∞ . Let G be a compact connected Lie group, and let K be a closed subgroup of G. The pair (G, K) is called a Riemannian symmetric pair if there exists an involutive C ∞ automorphism θ of G such that G0θ ⊂ K ⊂ Gθ , where Gθ is the set of fixed points of θ and G0θ is the identity component of Gθ . Let (G, K) be a Riemannian symmetric pair. Let g and k be the Lie algebras of G and K, respectively. The automorphism of g which is the differential of the automorphism θ of G will also be denoted by θ . Then we have
k = X ∈ g | θ (X) = X . We define the subspace m of g by
m = X ∈ g | θ (X) = −X . Then the Lie algebra g is decomposed into a direct sum of vector spaces as g = k + m. The subspace m is called the canonical complement of the pair (G, K). A maximal abelian subalgebra which is contained in m is called a Cartan subalgebra of the pair (G, K). Since G is compact, Cartan subalgebras have the same dimension. We call the dimension of a Cartan subalgebra the rank of the Riemannian symmetric pair (G, K). Let a be a Cartan subalgebra of the pair (G, K), and let t be a maximal abelian subalgebra of g containing a. We take a lexicographic order > on t, and denote by Σ + (G) the set of positive 1 roots with respect to the order >, and set δ = 2 α∈Σ + (G) α. Let ( , ) be a G-invariant inner product on g. For a representation ρ of G, an element λ ∈ t is called a weight of ρ if there exists a vector w = 0 in the representation space V such that √ ρ(H )w = 2π −1(λ, H )w, H ∈ t.
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For an irreducible representation ρ of G, let λ(ρ) be the highest weight of ρ with respect to >. Then the degree dρ of ρ is given by Weyl’s dimension formula: dρ =
α∈Σ + (G)
(α, λ(ρ) + δ) . (α, δ)
Let g be the commutator subalgebra [g, g] of g, and let k = g ∩ k. We denote by G the connected Lie subgroup of G generated by g and set K = G ∩ K. Then the pair (G , K ) is a Riemannian symmetric pair and k is the Lie algebra of K . If furthermore, the quotient space G/K is simply connected, then G/K is diffeomorphic to G /K . 2.4. Completely monotonic multisequences Let be a positive integer. For each m = (m1 , . . . , m ) ∈ N0 and each j (1 j ) we set = (m1 , . . . , mj −1 , mj + 1, mj +1 , . . . , m ). Moreover, for a real multisequence {am }m∈N 0 define m(j )
Δj {am }m∈N = {am − am(j ) }m∈N , 0
0
1 j .
We shall say that the multisequence {am }m∈N is completely monotonic if all the terms of 0
Δn1 1 · · · Δn {am }m∈N
0
are nonnegative for each (n1 , . . . , n ) ∈ N0 . 3. Riemannian symmetric pair—a sufficient condition for a pair (G, K) to have a generating function associated with itself The purpose of this section is to prove Main result 1. Throughout this section let G be a compact connected Lie group, K a closed subgroup of G, and (G, K) a Riemannian symmetric pair of rank such that G/K is simply connected. First, we shall recall that D(G, K) is parametrized by N0 . We shall give explanations supplementary to the fact by necessity. Since the rank of (G, K) is and G/K is simply connected, the set of fundamental weights of (g , k ) consists of elements. We denote by {M1 , M2 , . . . , M } the set of fundamental weights of (g , k ). Then it is known (cf. [7]) that the mapping which sends ρ ∈ D(G, K) to the highest weight λ(ρ) ∈ a is a bijection of D(G, K) onto the semigroup in a
mj Mj mj ∈ N0 , 1 j .
j =1
For each m = (m1 , . . . , m ) ∈ N0 we take a spherical representation ρm ∈ D(G, K) such that λ(ρm ) =
j =1
mj Mj ,
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and fix it. It is clear that the mapping which sends m ∈ N0 to ρm ∈ D(G, K) is a bijection of N0 onto D(G, K). In addition, by Weyl’s dimension formula, we have (α, j =1 mj Mj + δ) j =1 mj (α, Mj ) + (α, δ) = . (6) dρm = (α, δ) (α, δ) + + α∈Σ (G)
α∈Σ (G)
In this formula we remark that (α, δ) > 0 and (α, Mj ) 0 (1 j ), and that for each Mj (1 j ) there exists an element α ∈ Σ + (G) such that (α, Mj ) > 0. This section is organized as follows. In Section 3.1 we construct a measure on [0, 1) as} sociated with the multisequence {dρ−1 . In Section 3.2 we construct a Hilbert space of the m m∈N0 Bargmann type related to the measure on [0, 1) given in the preceding subsection. In Section 3.3 we define a unitary operator given by a generating function of the form of (3), and prove Main result 1. } 3.1. Measure on [0, 1) associated with the multisequence {dρ−1 m m∈N
0
{dρ−1 } m m∈N0
In this subsection we show that the multisequence is completely monotonic and −1 give a measure on [0, 1) associated with {dρm }m∈N via the moment problem. 0 Let k be a positive integer, and let apq 0 (1 p k, 1 q ),
(1 r k).
br > 0
For (m1 , . . . , m ) ∈ N0 set A m1 , . . . , m ; (apq ); (br ); k = k
1
p=1 (ap1 m1
+ · · · + ap m + bp )
.
Then we have Lemma 1. The multisequence {A(m1 , . . . , m ; (apq ); (br ); k)} with respect to (m1 , . . . , m ) ∈ } N0 is completely monotonic. In particular, the multisequence {dρ−1 is completely monotonic. m m∈N 0
Proof. Let ap 0 (p = 1, 2), br > 0 (r = 1, 2), and set An =
1 , (a1 n + b1 )(a2 n + b2 )
n ∈ N0 .
Then we have 1 1 − (a1 n + b1 )(a2 n + b2 ) (a1 (n + 1) + b1 )(a2 n + b2 ) 1 1 − + (a1 (n + 1) + b1 )(a2 n + b2 ) (a1 (n + 1) + b1 )(a2 (n + 1) + b2 ) a1 = (a1 n + b1 )(a2 n + b2 )(a1 n + a1 + b1 ) a2 . + (a2 n + b2 )(a1 n + a1 + b1 )(a2 n + a2 + b2 )
An − An+1 =
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Applying this technique to A(m1 , . . . , m ; (apq ); (br ); k), it is easy to see that A m1 , . . . , m ; (apq ); (br ); k − A m1 + 1, m2 , . . . , m ; (apq ); (br ); k = a11 A m1 , . . . , m ; (∗∗); (∗); k + 1 + a21 A m1 , . . . , m ; (∗∗); (∗); k + 1 + · · · + ak1 A m1 , . . . , m ; (∗∗); (∗); k + 1 , where each (∗∗) is a matrix of (k + 1, )-type whose entries are nonnegative real numbers, and each (∗) is a (k + 1)-dimensional vector whose entries are positive real numbers. It follows from this formula that the multisequence {A(m1 , . . . , m ; (apq ); (br ); k)} with respect to (m1 , . . . , m ) ∈ N0 is completely monotonic. 2 Therefore, by [4, Theorem 1] we have Proposition 1. There exists a measure μ on [0, 1] such that
dρ−1 = m
t m dμ(t) for m ∈ N0 .
[0,1]
We shall show that μ([0, 1] \ [0, 1) ) = 0. For each j (1 j ) set
Fj = (t1 , . . . , t ) ∈ [0, 1] | tj = 1 . Since [0, 1] \ [0, 1) = j =1 Fj , it suffices to show that μ(Fj ) = 0 for each j . If m = (0, . . . , 0, mj , 0, . . . , 0) ∈ N0 , then by (6) and Proposition 1 we see that α∈Σ + (G)
(α, δ) = mj (α, Mj ) + (α, δ)
m
tj j dμ(t).
[0,1]
As we remarked in the formula (6), for each j there exists an element α ∈ Σ + (G) such that (α, Mj ) > 0. Hence, the left-hand side tends to zero as mj goes to infinity. On the other hand, m tj j tends to the characteristic function of the set Fj as mj → ∞. Therefore, by Lebesgue’s convergence theorem we can see that μ(Fj ) = 0. Lemma 2. Let μ be the measure given in Proposition 1. Then we have = dρ−1 m
t m dμ(t),
m ∈ N0 .
[0,1)
Let us consider the measure μ as a measure on [0, 1) . Let ([0, 1) , B) be the measurable space associated with the measure μ. We define the mapping T of [0, 1) onto itself by √ √ T (t1 , . . . , t ) = ( t1 , . . . , t ) and set
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B = E ⊂ [0, 1) | T −1 (E) ∈ B , ν(E) = μ T −1 (E) , E ∈ B . Then ([0, 1) , B , ν) is a measure space and satisfies h T −1 (r) dν(r) = [0,1)
h(t) dμ(t)
[0,1)
for any μ-integrable function h. In particular, we have 2m r dν(r) = t m dμ(t), [0,1)
m ∈ N0 ,
(7)
[0,1)
where 2m = (2m1 , . . . , 2m ) for m = (m1 , . . . , m ). 3.2. Measure on B and an associated Hilbert space of the Bargmann type In this subsection, first we define a measure on B related to the measure ν on [0, 1) given in the preceding subsection. Next we define a Hilbert space of the Bargmann type associated with the measure on B . In what follows, we shall denote by ν[0,1) the measure ν on [0, 1) . √
Set U = {e −1θ | θ ∈ R}, and let νU be the normalized Haar measure on the direct product Lie group U . Further, we denote by ν[0,1) ×U the product measure ν[0,1) × νU on the product set [0, 1) × U , and identify B with [0, 1) × U . Then a measure on B is induced from the identification. We shall denote it by νB . If f is a νB -integrable function, then f (r1 u1 , . . . , r u ) is ν[0,1) ×U -integrable as a function of (r, u) = (r1 , . . . , r , u1 , . . . , u ) ∈ [0, 1) × U and satisfies f (z) dνB (z) = f (r1 u1 , . . . , r u ) dν[0,1) ×U (r, u). (8) B
[0,1) ×U
Let L2 (B , νB ) be the Hilbert space of νB -measurable functions f on B with f B
2 =
f (z) dνB (z) < ∞. B
The inner product is given by f, g B =
f (z)g(z) dνB (z).
(9)
B
Let F(B , νB ) be the space of analytic functions on B which belong to L2 (B , νB ). Then F(B , νB ) is a closed subspace of the Hilbert space L2 (B , νB ), that is, F(B , νB ) is a Hilbert space of the Bargmann type of order with inner product (9). In the following we shall show it.
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Lemma 3. For f ∈ F(B , νB ) denote the power series expansion of f in B by
f (z) =
am z m .
m∈N0
Then we have
|am |2 dρ−1 f 2B . m
m∈N0
Proof. Let 0 < s < 1, and let Bs = {z ∈ B | |zj | < s, 1 j }. It is easy to see that Bs is νB -measurable. First, we notice that by (8) and the definition of νU , the following orthogonality relation holds: 2m dν r [0,1) (r), m = m , m m z z dνB (z) = [0,s) 0, m = m . Bs
Since the series expansion f (z) = m am zm converges uniformly on Bs , by considering the orthogonality relation above we have
m 2 2 2 f (z) 2 dν (z) = z |am | dνB (z) = |am | B
Bs
m∈N0
m∈N0
Bs
r 2m dν[0,1) (r).
[0,s)
Therefore, we obtain f 2B
|am |
2
m∈N0
r 2m dν[0,1) (r).
[0,s)
By this inequality, (7) and Lemma 2 we have the assertion.
2
Lemma 4. Let f ∈ F(B , νB ). Then we have 2 f (z) f dρm zm , B
z ∈ B .
m∈N0
Proof. Applying the Schwarz inequality and Lemma 3 to the power series expansion of f in B , we have the assertion. (Note that m∈N dρm |zm |2 < ∞ for all z ∈ B , because dρm is a polyno0 mial of m1 , . . . , m .) 2 By Lemma 4 the convergence with respect to the norm of F(B , νB ) implies the uniform convergence on every compact subset of B . This shows that F(B , νB ) is a closed subspace of L2 (B , νB ), that is, F(B , νB ) is also a Hilbert space with inner product (9).
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Set vm (z) =
m ∈ N0 .
dρm zm ,
Then we have Lemma 5. The system {vm | m ∈ N0 } is a complete orthonormal system of F(B , νB ). Proof. In the same way as in the proof of Lemma 3 we see
zm zm
dνB (z) =
B
dρ−1 , m 0,
m = m , m = m ,
which implies that {vm | m ∈ N0 } is an orthonormal system of F(B , νB ). Next we show the completeness of the system. Let f be an element of F(B , νB ) with power series expan −1/2 sion f (z) = m am zm . Since m am zm = m am dρm vm (z) and m |am |2 dρ−1 < ∞ from m Lemma 3, the power series expansion of f converges with respect to the norm of F(B , νB ). This completes the proof. 2 The space F(B , νB ) has the reproducing kernel. This follows from Lemma 4. Lemma 6. For each w ∈ B there exists a unique gw ∈ F(B , νB ) such that f (w) = f, gw B
for f ∈ F B , νB .
Moreover, gw (z) has the following expansion: gw (z) =
vm (w)vm (z),
z ∈ B .
m∈N0
3.3. Unitary operator constructed by a generating function of the form of (3) In this subsection we construct a unitary operator given by an integral operator whose kernel is a generating function of the form of (3), and prove Main result 1. Denote by ϕm the zonal spherical function associated with ρm ∈ D(G, K). As is well known (cf. [3,7]), the system { dρm ϕm | m ∈ N0 } is a complete orthonormal system of L2 (G, K). Define Φ(z, x) =
dρm ϕm (x)zm ,
z ∈ B , x ∈ G.
m∈N0
Since for each m ∈ N0 we have |ϕm (x)| 1 (x ∈ G), the series in the right-hand side converges absolutely and uniformly on every compact subset of B × G. In particular, Φ(z, x) is continuous on B × G, and for each x ∈ G it is analytic on B as a function of z. Set ψm =
dρm ϕm ,
m ∈ N0 .
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Then we have the following expression:
Φ(z, x) =
(10)
ψm (x)vm (z).
m∈N0
We shall now define a linear operator on L2 (G, K) to F(B , νB ). For ϕ ∈ L2 (G, K) set (Φϕ)(z) =
Φ z, x −1 ϕ(x) dx,
z ∈ B .
(11)
G
Notice that for a fixed z ∈ B the right-hand side of (10) converges uniformly on G, and that for each m ∈ N0 we have ϕm (x −1 ) = ϕm (x) (x ∈ G). Then we see (Φϕ)(z) =
ϕ, ψm G vm (z).
(12)
m∈N0
Since m | ϕ, ψm G |2 < ∞ and m |vm (z)|2 < ∞ (z ∈ B ), the right-hand side of (12) converges absolutely on B and converges with respect to the norm of F(B , νB ). This implies Φϕ ∈ F(B , νB ). Hence, Φ is a linear operator on L2 (G, K) to F(B , νB ). Next we show that Φ is unitary. Applying the Parseval formula to (12), we obtain Φϕ 2B =
ϕ, ψm G 2 = ϕ 2 , G
m∈N0
which means Φ is an isometry. Further, let us take ϕ = ψm in (12). Then we obtain Φψm = vm ,
m ∈ N0 ,
which means Φ is surjective. Theorem 1. The operator Φ on L2 (G, K) to F(B , νB ) is unitary. For each z ∈ B define the function Φz (x) on G by Φz (x) = Φ(z, x). Then, Φz (x) is biinvariant under K and the following holds: (Φz ∗ Φw )(x) = Φzw (x) for all z, w ∈ B and all x ∈ G. The former is clear and the latter follows from the following relation: (ϕm ∗ ϕm )(x) =
ϕ (x), dρ−1 m m 0,
m = m , m = m .
Therefore, we can conclude that Φ(z, x) is a generating function associated with the pair (G, K). Theorem 2. Let G be a compact connected Lie group, and let K be a closed subgroup of G. If the pair (G, K) is a Riemannian symmetric pair such that G/K is simply connected, then it has a generating function associated with itself.
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Remark. Lemma 2 plays an important role in the discussion of this section. If a pair (G, K) has a measure μ on [0, 1) which satisfies Lemma 2, if not a Riemannian symmetric pair, Theorems 1 and 2 can be shown in the same way as above. In other words, the question whether a pair (G, K) has a generating function associated with itself concerns an existence of a “good” parametrization for D(G, K). 4. Gelfand pair—a necessary condition for a pair (G, K) to have a generating function associated with itself The purpose of this section is to prove Main result 2. Throughout this section let G be a compact group, let K be a closed subgroup of G, and suppose that the pair (G, K) has a generating function associated with itself. Further, denote by Φ(z, x) the generating function associated with (G, K) and suppose that it is defined on B × G, where is some positive integer. In Section 4.1 we show that (G, K) is a Gelfand pair. In Section 4.2 we give the expansion of Φ(z, x) by the zonal spherical functions on (G, K). That expansion is a series similar to the form of (3). 4.1. Proof of Main result 2 We shall make some preparations to show that (G, K) is a Gelfand pair. Let ρ ∈ D(G, K). Choose a G-invariant inner product , on the representation space Vρ , and take an orthonormal basis {e1 , . . . , eqρ } of VρK with respect to , , where qρ denotes the dimension of VρK . Further, define ρij (x) = ei , ρ(x)ej ,
x ∈ G, 1 i, j qρ .
As is well known (cf. [3,7]), the system { dρ ρij | ρ ∈ D(G, K), 1 i, j qρ } is a complete orthonormal system of L2 (G, K). Denoting by dk the normalized Haar measure on K, it is also known (cf. [3]) that
ρij (xkx ) dk =
qρ
ρip (x)ρpj (x ),
x, x ∈ G.
(13)
p=1
K
In the following, for ρ ∈ D(G, K) set ρˆij = dρ ρij (1 i, j qρ ). Since for fixed z, w ∈ B we have Φz ∗ Φw = Φzw from the condition (ii), it follows that Φz ∗ Φw , ρˆij G = Φzw , ρˆij G ,
ρ ∈ D(G, K), 1 i, j qρ .
(14)
Using ρij (x −1 ) = ρj i (x), it is easy to see that Φz , ρˆij G = (Φ ρˆj i )(z). Hence, the right-hand side of (14) is equal to (Φ ρˆj i )(zw). Let us compute the left-hand side of (14). Making use of (13), we have Φz ∗ Φw , ρˆij G = G
Φz (y) Φw y −1 x ρˆij (x) dx dy G
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= G
=
−1 −1 dx dy Φz (y) Φw x ρˆj i xy G
Φz (y) Φw x −1 ρˆj i xky −1 dk dx dy
G
=
G
K
qρ Φz (y) Φw x −1 dρ ρjp (x)ρpi y −1 dx dy
G
p=1
G
qρ 1 = (Φ ρˆpi )(z)(Φ ρˆjp )(w). dρ p=1
Therefore, we obtain the following lemma. Lemma 7. For ρ ∈ D(G, K) we have qρ 1 (Φ ρˆj i )(zw) = (Φ ρˆpi )(z)(Φ ρˆjp )(w), dρ p=1
z, w ∈ B , 1 i, j qρ .
Since each (Φ ρˆj i )(z) is analytic on B , it has the power series expansion in this polydisk. Let us denote it by (Φ ρˆj i )(z) =
a(ρˆj i , m)zm .
(15)
m∈N0
Rewrite the both sides of Lemma 7 by power series expansions, and for each m ∈ N0 compare the coefficients of zm and w m in both sides. Comparing the coefficients of zm , we have qρ 1 a(ρˆj i , m)w = a(ρˆpi , m)(Φ ρˆjp )(w), dρ p=1 m
w ∈ B .
Replacing w by z, we obtain qρ 1 a(ρˆj i , m)z = a(ρˆpi , m)(Φ ρˆjp )(z), dρ p=1 m
z ∈ B .
(16)
z ∈ B .
(17)
Comparing the coefficients of w m , we have qρ 1 a(ρˆj i , m)zm = (Φ ρˆpi )(z)a(ρˆjp , m), dρ p=1
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Lemma 8. For fixed ρ, i, j, m, let a(ρˆj i , m) = 0. Then we have a(ρˆpi , m) = 0,
1 p qρ .
Proof. By the assumption the left-hand side of (16) is equal to 0. On the other hand, since Φ is unitary, {Φ ρˆjp | 1 p qρ } is an orthonormal system. This proves the lemma. 2 We shall now show that (G, K) is a Gelfand pair, that is, qρ = 1 for all ρ ∈ D(G, K). For that purpose, suppose that there exists ρ ∈ D(G, K) such that qρ 2. First, we show that for each i, j , m the following holds: a(ρˆj i , m) = 0.
(18)
Notice that for each i, j, m the right-hand side of (16) is equal to that of (17): qρ
a(ρˆpi , m)(Φ ρˆjp )(z) =
p=1
qρ
a(ρˆjp , m)(Φ ρˆpi )(z).
(19)
p=1
Since qρ 2, in the left-hand side of (19) there exists a term with p = i. On the other hand, since Φ is unitary, {Φ ρˆj i | 1 i, j qρ } is an orthonormal system. Therefore, comparing the coefficients of (Φ ρˆjp )(z) with p = i, we have a(ρˆpi , m) = 0,
p = i.
By Lemma 8 we obtain (18). It follows from (15) and (18) that Φ ρˆj i = 0, which is a contradiction. Theorem 3. Let G be a compact group, and let K be a closed subgroup of G. If the pair (G, K) has a generating function associated with itself, then it is a Gelfand pair. Remark. Even if we replace the condition (iii) for Φ(z, x) by the following condition (iii) , Theorem 3 holds. The reason is that {Φ ρˆj i | 1 i, j qρ } is linearly independent under (iii) . (iii) For ϕ ∈ L2 (G, K) define (Φϕ)(z) =
Φ z, x −1 ϕ(x) dx,
z ∈ B .
G
Then Φ is one-to-one on L2 (G, K). 4.2. Expansion of Φ(z, x) by the zonal spherical functions on (G, K) Let us consider the expansion of Φ(z, x) by the zonal spherical functions on (G, K).
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We shall preserve the notation used up to here. Since qρ = 1 for each ρ ∈ D(G, K), ρ11 is the zonal spherical function associated with ρ, and by (16) the following holds: 1 a(ρˆ11 , m)zm = a(ρˆ11 , m)(Φ ρˆ11 )(z), dρ
m ∈ N0 .
Moreover, since Φ ρˆ11 = 0, there exists m ∈ N0 such that a(ρˆ11 , m ) = 0. Hence, we have (Φ ρˆ11 )(z) =
dρ z m .
In particular, m ∈ N0 with a(ρˆ11 , m ) = 0 is uniquely determined by ρ. Denote it by mρ . Then the mapping which sends ρ ∈ D(G, K) to mρ ∈ N0 is an injection of D(G, K) into N0 , because Φ is an injection. Proposition 2. A generating function Φ(z, x) associated with the pair (G, K) has the following expansion:
Φ(z, x) =
dρ ρ11 (x)zmρ ,
ρ∈D (G,K)
where the series in the right-hand side converges absolutely on B × G. Proof. Since {ρˆ11 | ρ ∈ D(G, K)} is a complete orthonormal system of L2 (G, K), for each z ∈ B the function Φz ∈ L2 (G, K) can be written as
Φz =
Φz , ρˆ11 G ρˆ11 ,
(20)
ρ∈D (G,K)
where the series in the right-hand side converges with respect to the norm of L2 (G, K). On the other hand, we have
Φz , ρˆ11 G 2 < ∞.
ρ∈D (G,K)
Since Φz , ρˆ11 G =
dρ zmρ , (21) is rewritten as
2 dρ zmρ < ∞,
ρ∈D (G,K)
which holds for all z ∈ B . In particular, the following holds for all z ∈ B : ρ∈D (G,K)
dρ zmρ < ∞.
(21)
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This implies that
Φz , ρˆ11 G ρˆ11 (x)
ρ∈D (G,K)
converges absolutely for all z ∈ B and all x ∈ G. Therefore, by (20) we can conclude that Φz (x) =
Φz , ρˆ11 G ρˆ11 (x),
ρ∈D (G,K)
which converges absolutely on B × G, as desired.
2
Remark. Even if we replace the condition (iii) for Φ(z, x) by the condition (iii) stated in the remark following Theorem 3, Proposition 2 holds. Corollary 1. If a generating function Φ(z, x) associated with (G, K) satisfies the following condition (iv), then Φ(z, x) is written as a series of the form of (3). (iv) The set {zm | m ∈ N0 } is an orthogonal system of the Hilbert space F(B ). 5. Examples In this section, first we shall give some examples which are concerned with what we explained in introduction. In particular, the second one means that the converse statement of Main result 1 does not hold. The third one means that the converse statement of Main result 2 does not hold, and, moreover, gives an example of a Riemannian symmetric pair which does not have a generating function associated with itself. As a result, we can conclude that the notion of pair which has a generating function associated with itself is equivalent neither to that of Riemannian symmetric pair nor to that of Gelfand pair. For references on zonal spherical functions on (SO(n), SO(n−1)) or (U (n), U (n − 1)), see [5,8]. Example 1. Let SO(n) be the special orthogonal group of degree n. If n 3, the pair (SO(n), SO(n − 1)) is a Riemannian symmetric pair of rank 1 such that SO(n)/SO(n − 1) ∼ = S n−1 n−1 is simply connected, where S is the (n − 1)-dimensional unit sphere. In what follows, we assume that n 3. Set e1 = t (1, 0, . . . , 0) ∈ S n−1 . Then the identification SO(n)/SO(n − 1) ∼ = S n−1 is given by the mapping xSO(n − 1) → xe1 , x ∈ SO(n). On the other hand, the zonal (n−2)/2 , spherical functions on (SO(n), SO(n − 1)) are given by the Gegenbauer polynomials Cm m ∈ N0 . More precisely, we should explain as follows. For each m ∈ N0 define (n−2)/2
ϕm (x) =
Cm
((xe1 , e1 )) , (n−2)/2 Cm (1)
x ∈ SO(n),
where ( , ) denotes the canonical inner product on the real vector space Rn . Then the set of all the zonal spherical functions on (SO(n), SO(n − 1)) is equal to {ϕm | m ∈ N0 }. Let dm denote the
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degree of the representation corresponding to ϕm . Then, Theorem 1 in [9] can be translated into the following:
dm ϕm (x)zm =
m∈N0
1 − z2 , (1 − 2z(xe1 , e1 ) + z2 )n/2
x ∈ SO(n), z ∈ B,
is a generating function associated with the pair (SO(n), SO(n − 1)). Example 2. Let U (n) be the unitary group of degree n. If n 2, the pair (U (n), U (n − 1)) is a Gelfand pair, but it is not a Riemannian symmetric pair. In what follows, we assume that n 2. Denote by S(Cn ) the unit sphere in Cn and set e1 = t (1, 0, . . . , 0) ∈ S(Cn ). Then we have the identification U (n)/U (n − 1) ∼ = S(Cn ) by the mapping xU (n − 1) → xe1 , x ∈ U (n). On the other hand, the zonal spherical functions on (U (n), U (n − 1)) are given by the functions Gn−1 pq , p, q ∈ N0 , which have the following generating function (see [10, Theorem 1.1]): 1−n p q = Gn−1 1 − 2 Re(wξ ) + |w|2 pq (ξ )w w , p,q∈N0
w, ξ ∈ C, |w| < 1, |ξ | 1. That is to say, for each p, q ∈ N0 define ϕpq (x) =
Gn−1 pq ((xe1 , e1 )) Gn−1 pq (1)
,
x ∈ U (n),
where ( , ) denotes the canonical inner product on the complex vector space Cn . Then the set of all the zonal spherical functions on (U (n), U (n − 1)) is equal to {ϕpq | p, q ∈ N0 }. Let dpq denote the degree of the representation corresponding to ϕpq . Then, Theorem 1 in [11] can be translated into the following:
p q
dpq ϕpq (x)z1 z2 =
p,q∈N0
1 − z1 z2 , (1 − z1 (xe1 , e1 ) − z2 (e1 , xe1 ) + z1 z2 )n
x ∈ U (n), z1 , z2 ∈ B, is a generating function associated with the pair (U (n), U (n − 1)). √
Example 3. Let us consider the compact abelian group U = {e −1θ | θ ∈ R}, and let us take {1} as a closed subgroup of U . Then the Gelfand pair (U, {1}) does not have a generating function associated with itself. In what follows, we shall show it. For that purpose, we assume that (U, {1}) has a generating function associated with itself. Moreover, we denote it by Φ(z, x) and suppose that it is defined on B × U , where is some positive integer. First, since U is abelian, we have that dρ = 1 for all ρ ∈ D(U, {1}). Combining this with the unitarity of Φ, we obtain the following: m 2 z ρ dμ = 1, ρ ∈ D U, {1} , (22) B B
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where μB is a measure on B associated with Φ(z, x). Applying Lebesgue’s convergence theorem, it is easy to see that the left-hand side of (22) tends to zero as |mρ | goes to infinity, where |m| means m1 + · · · + m for m = (m1 , . . . , m ) ∈ N0 . This contradicts (22). Therefore, the Gelfand pair (U, {1}) does not have a generating function associated with itself. Since (U, {1}) is also a Riemannian symmetric pair, it is also an example of a Riemannian symmetric pair which does not have a generating function associated with itself. Finally, we give an example of application of Main result 1. Example 4. Let SU(n + 1) be the special unitary group of degree n + 1. Then S(U (1) × U (n)) is a closed subgroup of SU(n + 1). Set G = SU(n + 1), K = S U (1) × U (n) . Then the pair (G, K) is a Riemannian symmetric pair of rank 1 such that G/K is simply connected. (More precisely, G/K is diffeomorphic to the n-dimensional complex projective space including G-actions.) The zonal spherical functions on (G, K) are essentially given by the Jacobi (n−1,0) , m ∈ N0 (cf. [7]). Let dm denote the degree of the spherical representation polynomials Pm (n−1,0) corresponding to Pm . By Main result 1, we see that the series
(n−1,0)
dm
m∈N0
Pm
(t)
(n−1,0) Pm (1)
zm
(23)
gives a generating function associated with the pair (G, K). On the other hand, we have the (α,β) following generating function of the Jacobi polynomials Pm (t), m ∈ N0 (cf. [6]): (α + β + 1)m Pm(α,β) (t)zm (β + 1)m m∈N0 2tz + 2z α+β +1 α+β +2 −α−β−1 , , ; β + 1; = (1 + z) F 2 2 (z + 1)2
(24)
where (α)m = (α + m)/ (α) ( is the Gamma function) and F (a, b; c; x) is the hypergeometric function. Set Ψ (z, t) =
(n)m P (n−1,0) (t)zm . m! m
m∈N0
By formula (24), we have −n
Ψ (z, t) = (1 + z)
2tz + 2z n n+1 , ; 1; . F 2 2 (z + 1)2
Further, we see that the generating function (23) is equal to the following: 2z ∂Ψ (z, t) + Ψ (z, t). n ∂z
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References [1] V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Part I, Comm. Pure Appl. Math. 14 (1961) 187–214. [2] A. Essadiq, q-Analogue of the Watanabe unitary transform associated to the q-continuous Gegenbauer polynomials, Lett. Math. Phys. 53 (2000) 233–242. [3] S. Helgason, Groups and Geometric Analysis, Academic Press, New York, 1984. [4] T.H. Hildebrandt, I.J. Schoenberg, On linear functional operations and the moment problem for a finite interval in one or several dimensions, Ann. of Math. 34 (1933) 317–328. [5] K.D. Johnson, N.R. Wallach, Composition series and intertwining operators for the spherical principal series I, Trans. Amer. Math. Soc. 229 (1977) 137–173. [6] A.P. Prudnikov, Y.A. Brychkov, O.I. Marichev, Integrals and Series, Gordon & Breach, New York, 1986. [7] M. Takeuchi, Modern Spherical Functions, Transl. Math. Monogr., vol. 135, Amer. Math. Soc., Providence, RI, 1994. [8] N.J. Vilenkin, A.U. Klimyk, Representation of Lie Groups and Special Functions, vol. 2, Kluwer Acad. Publ., Dordrecht, 1993. [9] S. Watanabe, Hilbert spaces of analytic functions and the Gegenbauer polynomials, Tokyo J. Math. 13 (1990) 421– 427. [10] S. Watanabe, Generating functions and integral representations for the spherical functions on some classical Gelfand pairs, J. Math. Kyoto Univ. 33 (1993) 1125–1142. [11] S. Watanabe, Hilbert spaces of analytic functions associated with generating functions of spherical functions on U (n)/U (n − 1), Proc. Japan Acad. Ser. A 70 (1994) 323–325.
Symmetric pairs, Gelfand pairs and pairs such that generating functions of zonal spherical functions define unitary operators—the case of the compact type Shigeru Watanabe The University of Aizu, Aizu-Wakamatsu City, Fukushima 965-8580, Japan Received 6 October 2005; accepted 15 February 2006 Available online 24 March 2006 Communicated by L. Gross
Abstract A new notion which is equivalent neither to that of Riemannian symmetric pair nor to that of Gelfand pair can be defined by generating functions. In the case of the compact type, this statement is shown. This notion also concerns an existence of a “good” parametrization for the spherical representations. These mean that generating functions can be dealt with systematically in the compact case. © 2006 Elsevier Inc. All rights reserved. Keywords: Riemannian symmetric pair; Gelfand pair; Zonal spherical function; Generating function
1. Introduction How possibility do generating functions have? The purpose of this paper is to give our answer to this question. Bargmann [1] constructed a unitary operator given by an integral operator whose kernel is a generating function of the Hermite polynomials. He also gave in [1] a similar construction for the Laguerre polynomials. Thus we naturally turn our interest to other orthogonal polynomials. In fact, he notices as follows [1, p. 203]: “It is worth noting that a similar interpretation may be given to other classical generating functions.” E-mail address: [email protected]. 0022-1236/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2006.02.008
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First of all, we turned our interest to the Gegenbauer polynomials which give essentially the zonal spherical functions on the pair (SO(n), SO(n − 1)). And in [9] we showed that a similar construction is possible for the Gegenbauer polynomials. Next, in [11] we gave a similar construction for the zonal spherical functions on the pair (U (n), U (n − 1)). On the other hand, Essadiq [2] gave a q-analogue of the Gegenbauer case. Let us return to the two cases that we constructed. As a matter of fact there is a similarity between them not only in the constructions but also in the forms of the generating functions. We shall explain in detail. Let N0 be the set of nonnegative integers, and let B be the unit open disk |z| < 1 in C. The set of all the zonal spherical functions on (SO(n), SO(n − 1)) is parametrized by N0 . Denote it by {ϕm | m ∈ N0 }. Further, for each m ∈ N0 we denote by dm the degree of the representation corresponding to ϕm . Then the generating function associated with the unitary operator is written as a series of the following form:
x ∈ SO(n), z ∈ B.
dm ϕm (x)zm ,
(1)
m∈N0
On the other hand, the set of all the zonal spherical functions on (U (n), U (n − 1)) is parametrized by N20 . As in the first case, define {ϕm | m ∈ N20 } and dm (m ∈ N20 ). Then the generating function in this case is written as a series of the following form:
x ∈ U (n), z ∈ B 2 ,
dm ϕm (x)zm ,
(2)
m∈N20
where zm = z1m1 z2m2 for z = (z1 , z2 ) and m = (m1 , m2 ). We are reasonably led to the question whether similar situations occur in the case of general Gelfand pair (G, K) of the compact type. That is, let G be a compact group, K a closed subgroup of G, and (G, K) a Gelfand pair. Further, suppose that the set of all the zonal spherical functions on (G, K) is parametrized by N0 , where is a positive integer and is not uniquely determined. And as in the case above, define {ϕm | m ∈ N0 } and dm (m ∈ N0 ). Then our question is as follows. Is it possible to find a generating function of the following form such that it is the kernel of an integral operator that is unitary?
dm ϕm (x)zm ,
x ∈ G, z ∈ B ,
(3)
m∈N0 m
where zm = z1m1 · · · z for z = (z1 , . . . , z ) and m = (m1 , . . . , m ). Remark that if we regard (3) as a function of x and denote it by Φz (x), then the following holds under a suitable condition: Φz ∗ Φw = Φzw ,
(4)
where zw = (z1 w1 , . . . , z w ) for z = (z1 , . . . , z ) and w = (w1 , . . . , w ). This follows from the relation (ϕm ∗ ϕm )(x) =
−1 ϕ (x), dm m 0,
m = m , m = m .
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Let us consider the question from a more general viewpoint. Let G be a compact group, and let K be a closed subgroup of G. We do not necessarily suppose that (G, K) is a Gelfand pair. Denote by dx the Haar measure on G, normalized by G dx = 1. Further, denote by L2 (G, K) the space of square integrable functions on G that are bi-invariant under K. Considering (4), we shall introduce the notion of a generating function associated with the pair (G, K), together with that of a Hilbert space of the Bargmann type. Definition 1. Let be a positive integer, and let (B , μB ) be a measure space. Denote by L2 (B , μB ) the Hilbert space of square integrable functions on B . If the space of analytic functions on B that belong to L2 (B , μB ) is a closed subspace of L2 (B , μB ), then it is called a Hilbert space of the Bargmann type of order . Definition 2. Suppose that there exists a positive integer and that a function Φ(z, x) defined on B × G satisfies the following conditions: (i) The function Φ(z, x) is continuous on B × G, for each x ∈ G it is analytic on B as a function of z, and for each z ∈ B it is bi-invariant under K as a function of x. (ii) For each z ∈ B define the function Φz (x) on G by Φz (x) = Φ(z, x). Then the following holds: (Φz ∗ Φw )(x) = Φzw (x) for all z, w ∈ B and all x ∈ G. (iii) For ϕ ∈ L2 (G, K) define (Φϕ)(z) = Φ z, x −1 ϕ(x) dx, z ∈ B . G
Then F(B ) = {Φϕ | ϕ ∈ L2 (G, K)} is a Hilbert space of the Bargmann type of order , and Φ is a unitary operator on L2 (G, K) to F(B ). We call Φ(z, x) a generating function associated with the pair (G, K). The first purpose of this paper is to give a sufficient condition for the pair (G, K) to have a generating function associated with itself. Main result 1. Let G be a compact connected Lie group, and let K be a closed subgroup of G. If the pair (G, K) is a Riemannian symmetric pair such that G/K is simply connected, then there exists a generating function of the form of (3) such that it is a generating function associated with (G, K). The second purpose of this paper is to give a necessary condition for the pair (G, K) to have a generating function associated with itself. Main result 2. Let G be a compact group, and let K be a closed subgroup of G. If the pair (G, K) has a generating function associated with itself, then it is a Gelfand pair. In this case the generating function Φ(z, x) associated with (G, K) is written as a series similar to the form of (3). Besides, if we impose a suitable condition on the measure μB related to F(B ), then Φ(z, x) is written as a series of the form of (3). The third purpose of this paper is to show that the notion of pair which has a generating function associated with itself is equivalent neither to that of Riemannian symmetric pair nor to that of Gelfand pair. This means that generating functions define a new notion. We also remark that this notion concerns an existence of a “good” parametrization for the spherical representations.
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The paper is organized as follows. In Section 3 we show Main result 1. In Section 4 we show Main result 2 and give the expansion of a generating function by the zonal spherical functions. In Section 5 we give some examples and mention the third purpose explained above. (The remark to the third purpose is explained at the end of Section 3.) 2. Notation and preliminaries We shall explain the notation and the terms which will be used throughout this paper. 2.1. General notation We shall use the notation N0 , R, C for the set of nonnegative integers, the field of real numbers and the field of complex numbers, respectively. For a fixed positive integer we denote by C the -dimensional complex space, and denote by N0 the set of all multi-indices m = (m1 , . . . , m ) with each mj ∈ N0 . For m = (m1 , . . . , m ) ∈ N0 and z = (z1 , . . . , z ) ∈ C m we write zm = z1m1 · · · z . For z = (z1 , . . . , z ) ∈ C and w = (w1 , . . . , w ) ∈ C we denote (z1 w1 , . . . , z w ) ∈ C by zw. We denote by B the unit open disk in C, and denote by B the open polydisk of z = (z1 , . . . , z ) ∈ C with each zj ∈ B. For a subset A ⊂ R or C we define A in the same way. We shall use the notation [a, b], [a, b) for the interval {x ∈ R | a x b}, the interval {x ∈ R | a x < b}, respectively. For ζ ∈ C let Re ζ be the real part of ζ , and ζ → ζ the usual conjugation in C. A function is assumed to be complex-valued. 2.2. Compact groups A compact group is a topological group whose underlying topology is compact Hausdorff. Let G be a compact group. Let dx denote the normalized Haar measure on G; that is, it has the property that G dx = 1. Let L2 (G) denote the Hilbert space of measurable functions ϕ on G with 2 ϕ G = ϕ(x) dx < ∞. G
The inner product is given by ϕ, ψ G =
ϕ(x)ψ(x) dx.
(5)
G
Let K be a closed subgroup of G. It is said that a function ϕ on G is bi-invariant under K if it satisfies Ly ϕ = ϕ, Ry ϕ = ϕ for any y, y ∈ K, where Ly , Ry (y, y ∈ G) are defined by (Ly ϕ)(x) = ϕ(y −1 x) (x ∈ G), (Ry ϕ)(x) = ϕ(xy ) (x ∈ G). Let L2 (G, K) denote the space of functions ϕ ∈ L2 (G) which are bi-invariant under K. Then L2 (G, K) is a closed subspace of L2 (G), that is, L2 (G, K) is also a Hilbert space with inner product (5). A representation of the compact group G means a continuous homomorphism of G to the group GL(V ) of invertible linear transformations on a finite-dimensional complex vector space V . Since G is compact, for an arbitrary representation of G there exists a G-invariant
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inner product on the representation space V , that is, V admits an inner product such that the representation is unitary. Given an irreducible representation ρ of G, let Vρ denote the representation space of ρ and let dρ denote the degree of ρ. Further, denote by VρK the subspace of Vρ consisting of elements w ∈ Vρ which satisfy ρ(x)w = w for any x ∈ K. If an irreducible representation ρ of G satisfies VρK = {0}, ρ is called a spherical representation of G with respect to the subgroup K. Let D(G, K) denote the set of all the equivalence classes of spherical representations of G with respect to K. For two continuous functions ϕ, ψ on G the convolution ϕ ∗ ψ is defined by (ϕ ∗ ψ)(x) = ϕ xy −1 ψ(y) dy, x ∈ G. G
The pair (G, K) is called a Gelfand pair if the convolution algebra of continuous functions on G which are bi-invariant under K is commutative. The pair (G, K) is a Gelfand pair if and only if dim VρK = 1 for all ρ ∈ D(G, K). 2.3. Riemannian symmetric pairs A Lie group is assumed to be differentiable of class C ∞ . Let G be a compact connected Lie group, and let K be a closed subgroup of G. The pair (G, K) is called a Riemannian symmetric pair if there exists an involutive C ∞ automorphism θ of G such that G0θ ⊂ K ⊂ Gθ , where Gθ is the set of fixed points of θ and G0θ is the identity component of Gθ . Let (G, K) be a Riemannian symmetric pair. Let g and k be the Lie algebras of G and K, respectively. The automorphism of g which is the differential of the automorphism θ of G will also be denoted by θ . Then we have
k = X ∈ g | θ (X) = X . We define the subspace m of g by
m = X ∈ g | θ (X) = −X . Then the Lie algebra g is decomposed into a direct sum of vector spaces as g = k + m. The subspace m is called the canonical complement of the pair (G, K). A maximal abelian subalgebra which is contained in m is called a Cartan subalgebra of the pair (G, K). Since G is compact, Cartan subalgebras have the same dimension. We call the dimension of a Cartan subalgebra the rank of the Riemannian symmetric pair (G, K). Let a be a Cartan subalgebra of the pair (G, K), and let t be a maximal abelian subalgebra of g containing a. We take a lexicographic order > on t, and denote by Σ + (G) the set of positive 1 roots with respect to the order >, and set δ = 2 α∈Σ + (G) α. Let ( , ) be a G-invariant inner product on g. For a representation ρ of G, an element λ ∈ t is called a weight of ρ if there exists a vector w = 0 in the representation space V such that √ ρ(H )w = 2π −1(λ, H )w, H ∈ t.
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For an irreducible representation ρ of G, let λ(ρ) be the highest weight of ρ with respect to >. Then the degree dρ of ρ is given by Weyl’s dimension formula: dρ =
α∈Σ + (G)
(α, λ(ρ) + δ) . (α, δ)
Let g be the commutator subalgebra [g, g] of g, and let k = g ∩ k. We denote by G the connected Lie subgroup of G generated by g and set K = G ∩ K. Then the pair (G , K ) is a Riemannian symmetric pair and k is the Lie algebra of K . If furthermore, the quotient space G/K is simply connected, then G/K is diffeomorphic to G /K . 2.4. Completely monotonic multisequences Let be a positive integer. For each m = (m1 , . . . , m ) ∈ N0 and each j (1 j ) we set = (m1 , . . . , mj −1 , mj + 1, mj +1 , . . . , m ). Moreover, for a real multisequence {am }m∈N 0 define m(j )
Δj {am }m∈N = {am − am(j ) }m∈N , 0
0
1 j .
We shall say that the multisequence {am }m∈N is completely monotonic if all the terms of 0
Δn1 1 · · · Δn {am }m∈N
0
are nonnegative for each (n1 , . . . , n ) ∈ N0 . 3. Riemannian symmetric pair—a sufficient condition for a pair (G, K) to have a generating function associated with itself The purpose of this section is to prove Main result 1. Throughout this section let G be a compact connected Lie group, K a closed subgroup of G, and (G, K) a Riemannian symmetric pair of rank such that G/K is simply connected. First, we shall recall that D(G, K) is parametrized by N0 . We shall give explanations supplementary to the fact by necessity. Since the rank of (G, K) is and G/K is simply connected, the set of fundamental weights of (g , k ) consists of elements. We denote by {M1 , M2 , . . . , M } the set of fundamental weights of (g , k ). Then it is known (cf. [7]) that the mapping which sends ρ ∈ D(G, K) to the highest weight λ(ρ) ∈ a is a bijection of D(G, K) onto the semigroup in a
mj Mj mj ∈ N0 , 1 j .
j =1
For each m = (m1 , . . . , m ) ∈ N0 we take a spherical representation ρm ∈ D(G, K) such that λ(ρm ) =
j =1
mj Mj ,
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and fix it. It is clear that the mapping which sends m ∈ N0 to ρm ∈ D(G, K) is a bijection of N0 onto D(G, K). In addition, by Weyl’s dimension formula, we have (α, j =1 mj Mj + δ) j =1 mj (α, Mj ) + (α, δ) = . (6) dρm = (α, δ) (α, δ) + + α∈Σ (G)
α∈Σ (G)
In this formula we remark that (α, δ) > 0 and (α, Mj ) 0 (1 j ), and that for each Mj (1 j ) there exists an element α ∈ Σ + (G) such that (α, Mj ) > 0. This section is organized as follows. In Section 3.1 we construct a measure on [0, 1) as} sociated with the multisequence {dρ−1 . In Section 3.2 we construct a Hilbert space of the m m∈N0 Bargmann type related to the measure on [0, 1) given in the preceding subsection. In Section 3.3 we define a unitary operator given by a generating function of the form of (3), and prove Main result 1. } 3.1. Measure on [0, 1) associated with the multisequence {dρ−1 m m∈N
0
{dρ−1 } m m∈N0
In this subsection we show that the multisequence is completely monotonic and −1 give a measure on [0, 1) associated with {dρm }m∈N via the moment problem. 0 Let k be a positive integer, and let apq 0 (1 p k, 1 q ),
(1 r k).
br > 0
For (m1 , . . . , m ) ∈ N0 set A m1 , . . . , m ; (apq ); (br ); k = k
1
p=1 (ap1 m1
+ · · · + ap m + bp )
.
Then we have Lemma 1. The multisequence {A(m1 , . . . , m ; (apq ); (br ); k)} with respect to (m1 , . . . , m ) ∈ } N0 is completely monotonic. In particular, the multisequence {dρ−1 is completely monotonic. m m∈N 0
Proof. Let ap 0 (p = 1, 2), br > 0 (r = 1, 2), and set An =
1 , (a1 n + b1 )(a2 n + b2 )
n ∈ N0 .
Then we have 1 1 − (a1 n + b1 )(a2 n + b2 ) (a1 (n + 1) + b1 )(a2 n + b2 ) 1 1 − + (a1 (n + 1) + b1 )(a2 n + b2 ) (a1 (n + 1) + b1 )(a2 (n + 1) + b2 ) a1 = (a1 n + b1 )(a2 n + b2 )(a1 n + a1 + b1 ) a2 . + (a2 n + b2 )(a1 n + a1 + b1 )(a2 n + a2 + b2 )
An − An+1 =
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Applying this technique to A(m1 , . . . , m ; (apq ); (br ); k), it is easy to see that A m1 , . . . , m ; (apq ); (br ); k − A m1 + 1, m2 , . . . , m ; (apq ); (br ); k = a11 A m1 , . . . , m ; (∗∗); (∗); k + 1 + a21 A m1 , . . . , m ; (∗∗); (∗); k + 1 + · · · + ak1 A m1 , . . . , m ; (∗∗); (∗); k + 1 , where each (∗∗) is a matrix of (k + 1, )-type whose entries are nonnegative real numbers, and each (∗) is a (k + 1)-dimensional vector whose entries are positive real numbers. It follows from this formula that the multisequence {A(m1 , . . . , m ; (apq ); (br ); k)} with respect to (m1 , . . . , m ) ∈ N0 is completely monotonic. 2 Therefore, by [4, Theorem 1] we have Proposition 1. There exists a measure μ on [0, 1] such that
dρ−1 = m
t m dμ(t) for m ∈ N0 .
[0,1]
We shall show that μ([0, 1] \ [0, 1) ) = 0. For each j (1 j ) set
Fj = (t1 , . . . , t ) ∈ [0, 1] | tj = 1 . Since [0, 1] \ [0, 1) = j =1 Fj , it suffices to show that μ(Fj ) = 0 for each j . If m = (0, . . . , 0, mj , 0, . . . , 0) ∈ N0 , then by (6) and Proposition 1 we see that α∈Σ + (G)
(α, δ) = mj (α, Mj ) + (α, δ)
m
tj j dμ(t).
[0,1]
As we remarked in the formula (6), for each j there exists an element α ∈ Σ + (G) such that (α, Mj ) > 0. Hence, the left-hand side tends to zero as mj goes to infinity. On the other hand, m tj j tends to the characteristic function of the set Fj as mj → ∞. Therefore, by Lebesgue’s convergence theorem we can see that μ(Fj ) = 0. Lemma 2. Let μ be the measure given in Proposition 1. Then we have = dρ−1 m
t m dμ(t),
m ∈ N0 .
[0,1)
Let us consider the measure μ as a measure on [0, 1) . Let ([0, 1) , B) be the measurable space associated with the measure μ. We define the mapping T of [0, 1) onto itself by √ √ T (t1 , . . . , t ) = ( t1 , . . . , t ) and set
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B = E ⊂ [0, 1) | T −1 (E) ∈ B , ν(E) = μ T −1 (E) , E ∈ B . Then ([0, 1) , B , ν) is a measure space and satisfies h T −1 (r) dν(r) = [0,1)
h(t) dμ(t)
[0,1)
for any μ-integrable function h. In particular, we have 2m r dν(r) = t m dμ(t), [0,1)
m ∈ N0 ,
(7)
[0,1)
where 2m = (2m1 , . . . , 2m ) for m = (m1 , . . . , m ). 3.2. Measure on B and an associated Hilbert space of the Bargmann type In this subsection, first we define a measure on B related to the measure ν on [0, 1) given in the preceding subsection. Next we define a Hilbert space of the Bargmann type associated with the measure on B . In what follows, we shall denote by ν[0,1) the measure ν on [0, 1) . √
Set U = {e −1θ | θ ∈ R}, and let νU be the normalized Haar measure on the direct product Lie group U . Further, we denote by ν[0,1) ×U the product measure ν[0,1) × νU on the product set [0, 1) × U , and identify B with [0, 1) × U . Then a measure on B is induced from the identification. We shall denote it by νB . If f is a νB -integrable function, then f (r1 u1 , . . . , r u ) is ν[0,1) ×U -integrable as a function of (r, u) = (r1 , . . . , r , u1 , . . . , u ) ∈ [0, 1) × U and satisfies f (z) dνB (z) = f (r1 u1 , . . . , r u ) dν[0,1) ×U (r, u). (8) B
[0,1) ×U
Let L2 (B , νB ) be the Hilbert space of νB -measurable functions f on B with f B
2 =
f (z) dνB (z) < ∞. B
The inner product is given by f, g B =
f (z)g(z) dνB (z).
(9)
B
Let F(B , νB ) be the space of analytic functions on B which belong to L2 (B , νB ). Then F(B , νB ) is a closed subspace of the Hilbert space L2 (B , νB ), that is, F(B , νB ) is a Hilbert space of the Bargmann type of order with inner product (9). In the following we shall show it.
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Lemma 3. For f ∈ F(B , νB ) denote the power series expansion of f in B by
f (z) =
am z m .
m∈N0
Then we have
|am |2 dρ−1 f 2B . m
m∈N0
Proof. Let 0 < s < 1, and let Bs = {z ∈ B | |zj | < s, 1 j }. It is easy to see that Bs is νB -measurable. First, we notice that by (8) and the definition of νU , the following orthogonality relation holds: 2m dν r [0,1) (r), m = m , m m z z dνB (z) = [0,s) 0, m = m . Bs
Since the series expansion f (z) = m am zm converges uniformly on Bs , by considering the orthogonality relation above we have
m 2 2 2 f (z) 2 dν (z) = z |am | dνB (z) = |am | B
Bs
m∈N0
m∈N0
Bs
r 2m dν[0,1) (r).
[0,s)
Therefore, we obtain f 2B
|am |
2
m∈N0
r 2m dν[0,1) (r).
[0,s)
By this inequality, (7) and Lemma 2 we have the assertion.
2
Lemma 4. Let f ∈ F(B , νB ). Then we have 2 f (z) f dρm zm , B
z ∈ B .
m∈N0
Proof. Applying the Schwarz inequality and Lemma 3 to the power series expansion of f in B , we have the assertion. (Note that m∈N dρm |zm |2 < ∞ for all z ∈ B , because dρm is a polyno0 mial of m1 , . . . , m .) 2 By Lemma 4 the convergence with respect to the norm of F(B , νB ) implies the uniform convergence on every compact subset of B . This shows that F(B , νB ) is a closed subspace of L2 (B , νB ), that is, F(B , νB ) is also a Hilbert space with inner product (9).
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Set vm (z) =
m ∈ N0 .
dρm zm ,
Then we have Lemma 5. The system {vm | m ∈ N0 } is a complete orthonormal system of F(B , νB ). Proof. In the same way as in the proof of Lemma 3 we see
zm zm
dνB (z) =
B
dρ−1 , m 0,
m = m , m = m ,
which implies that {vm | m ∈ N0 } is an orthonormal system of F(B , νB ). Next we show the completeness of the system. Let f be an element of F(B , νB ) with power series expan −1/2 sion f (z) = m am zm . Since m am zm = m am dρm vm (z) and m |am |2 dρ−1 < ∞ from m Lemma 3, the power series expansion of f converges with respect to the norm of F(B , νB ). This completes the proof. 2 The space F(B , νB ) has the reproducing kernel. This follows from Lemma 4. Lemma 6. For each w ∈ B there exists a unique gw ∈ F(B , νB ) such that f (w) = f, gw B
for f ∈ F B , νB .
Moreover, gw (z) has the following expansion: gw (z) =
vm (w)vm (z),
z ∈ B .
m∈N0
3.3. Unitary operator constructed by a generating function of the form of (3) In this subsection we construct a unitary operator given by an integral operator whose kernel is a generating function of the form of (3), and prove Main result 1. Denote by ϕm the zonal spherical function associated with ρm ∈ D(G, K). As is well known (cf. [3,7]), the system { dρm ϕm | m ∈ N0 } is a complete orthonormal system of L2 (G, K). Define Φ(z, x) =
dρm ϕm (x)zm ,
z ∈ B , x ∈ G.
m∈N0
Since for each m ∈ N0 we have |ϕm (x)| 1 (x ∈ G), the series in the right-hand side converges absolutely and uniformly on every compact subset of B × G. In particular, Φ(z, x) is continuous on B × G, and for each x ∈ G it is analytic on B as a function of z. Set ψm =
dρm ϕm ,
m ∈ N0 .
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Then we have the following expression:
Φ(z, x) =
(10)
ψm (x)vm (z).
m∈N0
We shall now define a linear operator on L2 (G, K) to F(B , νB ). For ϕ ∈ L2 (G, K) set (Φϕ)(z) =
Φ z, x −1 ϕ(x) dx,
z ∈ B .
(11)
G
Notice that for a fixed z ∈ B the right-hand side of (10) converges uniformly on G, and that for each m ∈ N0 we have ϕm (x −1 ) = ϕm (x) (x ∈ G). Then we see (Φϕ)(z) =
ϕ, ψm G vm (z).
(12)
m∈N0
Since m | ϕ, ψm G |2 < ∞ and m |vm (z)|2 < ∞ (z ∈ B ), the right-hand side of (12) converges absolutely on B and converges with respect to the norm of F(B , νB ). This implies Φϕ ∈ F(B , νB ). Hence, Φ is a linear operator on L2 (G, K) to F(B , νB ). Next we show that Φ is unitary. Applying the Parseval formula to (12), we obtain Φϕ 2B =
ϕ, ψm G 2 = ϕ 2 , G
m∈N0
which means Φ is an isometry. Further, let us take ϕ = ψm in (12). Then we obtain Φψm = vm ,
m ∈ N0 ,
which means Φ is surjective. Theorem 1. The operator Φ on L2 (G, K) to F(B , νB ) is unitary. For each z ∈ B define the function Φz (x) on G by Φz (x) = Φ(z, x). Then, Φz (x) is biinvariant under K and the following holds: (Φz ∗ Φw )(x) = Φzw (x) for all z, w ∈ B and all x ∈ G. The former is clear and the latter follows from the following relation: (ϕm ∗ ϕm )(x) =
ϕ (x), dρ−1 m m 0,
m = m , m = m .
Therefore, we can conclude that Φ(z, x) is a generating function associated with the pair (G, K). Theorem 2. Let G be a compact connected Lie group, and let K be a closed subgroup of G. If the pair (G, K) is a Riemannian symmetric pair such that G/K is simply connected, then it has a generating function associated with itself.
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Remark. Lemma 2 plays an important role in the discussion of this section. If a pair (G, K) has a measure μ on [0, 1) which satisfies Lemma 2, if not a Riemannian symmetric pair, Theorems 1 and 2 can be shown in the same way as above. In other words, the question whether a pair (G, K) has a generating function associated with itself concerns an existence of a “good” parametrization for D(G, K). 4. Gelfand pair—a necessary condition for a pair (G, K) to have a generating function associated with itself The purpose of this section is to prove Main result 2. Throughout this section let G be a compact group, let K be a closed subgroup of G, and suppose that the pair (G, K) has a generating function associated with itself. Further, denote by Φ(z, x) the generating function associated with (G, K) and suppose that it is defined on B × G, where is some positive integer. In Section 4.1 we show that (G, K) is a Gelfand pair. In Section 4.2 we give the expansion of Φ(z, x) by the zonal spherical functions on (G, K). That expansion is a series similar to the form of (3). 4.1. Proof of Main result 2 We shall make some preparations to show that (G, K) is a Gelfand pair. Let ρ ∈ D(G, K). Choose a G-invariant inner product , on the representation space Vρ , and take an orthonormal basis {e1 , . . . , eqρ } of VρK with respect to , , where qρ denotes the dimension of VρK . Further, define ρij (x) = ei , ρ(x)ej ,
x ∈ G, 1 i, j qρ .
As is well known (cf. [3,7]), the system { dρ ρij | ρ ∈ D(G, K), 1 i, j qρ } is a complete orthonormal system of L2 (G, K). Denoting by dk the normalized Haar measure on K, it is also known (cf. [3]) that
ρij (xkx ) dk =
qρ
ρip (x)ρpj (x ),
x, x ∈ G.
(13)
p=1
K
In the following, for ρ ∈ D(G, K) set ρˆij = dρ ρij (1 i, j qρ ). Since for fixed z, w ∈ B we have Φz ∗ Φw = Φzw from the condition (ii), it follows that Φz ∗ Φw , ρˆij G = Φzw , ρˆij G ,
ρ ∈ D(G, K), 1 i, j qρ .
(14)
Using ρij (x −1 ) = ρj i (x), it is easy to see that Φz , ρˆij G = (Φ ρˆj i )(z). Hence, the right-hand side of (14) is equal to (Φ ρˆj i )(zw). Let us compute the left-hand side of (14). Making use of (13), we have Φz ∗ Φw , ρˆij G = G
Φz (y) Φw y −1 x ρˆij (x) dx dy G
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= G
=
−1 −1 dx dy Φz (y) Φw x ρˆj i xy G
Φz (y) Φw x −1 ρˆj i xky −1 dk dx dy
G
=
G
K
qρ Φz (y) Φw x −1 dρ ρjp (x)ρpi y −1 dx dy
G
p=1
G
qρ 1 = (Φ ρˆpi )(z)(Φ ρˆjp )(w). dρ p=1
Therefore, we obtain the following lemma. Lemma 7. For ρ ∈ D(G, K) we have qρ 1 (Φ ρˆj i )(zw) = (Φ ρˆpi )(z)(Φ ρˆjp )(w), dρ p=1
z, w ∈ B , 1 i, j qρ .
Since each (Φ ρˆj i )(z) is analytic on B , it has the power series expansion in this polydisk. Let us denote it by (Φ ρˆj i )(z) =
a(ρˆj i , m)zm .
(15)
m∈N0
Rewrite the both sides of Lemma 7 by power series expansions, and for each m ∈ N0 compare the coefficients of zm and w m in both sides. Comparing the coefficients of zm , we have qρ 1 a(ρˆj i , m)w = a(ρˆpi , m)(Φ ρˆjp )(w), dρ p=1 m
w ∈ B .
Replacing w by z, we obtain qρ 1 a(ρˆj i , m)z = a(ρˆpi , m)(Φ ρˆjp )(z), dρ p=1 m
z ∈ B .
(16)
z ∈ B .
(17)
Comparing the coefficients of w m , we have qρ 1 a(ρˆj i , m)zm = (Φ ρˆpi )(z)a(ρˆjp , m), dρ p=1
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Lemma 8. For fixed ρ, i, j, m, let a(ρˆj i , m) = 0. Then we have a(ρˆpi , m) = 0,
1 p qρ .
Proof. By the assumption the left-hand side of (16) is equal to 0. On the other hand, since Φ is unitary, {Φ ρˆjp | 1 p qρ } is an orthonormal system. This proves the lemma. 2 We shall now show that (G, K) is a Gelfand pair, that is, qρ = 1 for all ρ ∈ D(G, K). For that purpose, suppose that there exists ρ ∈ D(G, K) such that qρ 2. First, we show that for each i, j , m the following holds: a(ρˆj i , m) = 0.
(18)
Notice that for each i, j, m the right-hand side of (16) is equal to that of (17): qρ
a(ρˆpi , m)(Φ ρˆjp )(z) =
p=1
qρ
a(ρˆjp , m)(Φ ρˆpi )(z).
(19)
p=1
Since qρ 2, in the left-hand side of (19) there exists a term with p = i. On the other hand, since Φ is unitary, {Φ ρˆj i | 1 i, j qρ } is an orthonormal system. Therefore, comparing the coefficients of (Φ ρˆjp )(z) with p = i, we have a(ρˆpi , m) = 0,
p = i.
By Lemma 8 we obtain (18). It follows from (15) and (18) that Φ ρˆj i = 0, which is a contradiction. Theorem 3. Let G be a compact group, and let K be a closed subgroup of G. If the pair (G, K) has a generating function associated with itself, then it is a Gelfand pair. Remark. Even if we replace the condition (iii) for Φ(z, x) by the following condition (iii) , Theorem 3 holds. The reason is that {Φ ρˆj i | 1 i, j qρ } is linearly independent under (iii) . (iii) For ϕ ∈ L2 (G, K) define (Φϕ)(z) =
Φ z, x −1 ϕ(x) dx,
z ∈ B .
G
Then Φ is one-to-one on L2 (G, K). 4.2. Expansion of Φ(z, x) by the zonal spherical functions on (G, K) Let us consider the expansion of Φ(z, x) by the zonal spherical functions on (G, K).
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We shall preserve the notation used up to here. Since qρ = 1 for each ρ ∈ D(G, K), ρ11 is the zonal spherical function associated with ρ, and by (16) the following holds: 1 a(ρˆ11 , m)zm = a(ρˆ11 , m)(Φ ρˆ11 )(z), dρ
m ∈ N0 .
Moreover, since Φ ρˆ11 = 0, there exists m ∈ N0 such that a(ρˆ11 , m ) = 0. Hence, we have (Φ ρˆ11 )(z) =
dρ z m .
In particular, m ∈ N0 with a(ρˆ11 , m ) = 0 is uniquely determined by ρ. Denote it by mρ . Then the mapping which sends ρ ∈ D(G, K) to mρ ∈ N0 is an injection of D(G, K) into N0 , because Φ is an injection. Proposition 2. A generating function Φ(z, x) associated with the pair (G, K) has the following expansion:
Φ(z, x) =
dρ ρ11 (x)zmρ ,
ρ∈D (G,K)
where the series in the right-hand side converges absolutely on B × G. Proof. Since {ρˆ11 | ρ ∈ D(G, K)} is a complete orthonormal system of L2 (G, K), for each z ∈ B the function Φz ∈ L2 (G, K) can be written as
Φz =
Φz , ρˆ11 G ρˆ11 ,
(20)
ρ∈D (G,K)
where the series in the right-hand side converges with respect to the norm of L2 (G, K). On the other hand, we have
Φz , ρˆ11 G 2 < ∞.
ρ∈D (G,K)
Since Φz , ρˆ11 G =
dρ zmρ , (21) is rewritten as
2 dρ zmρ < ∞,
ρ∈D (G,K)
which holds for all z ∈ B . In particular, the following holds for all z ∈ B : ρ∈D (G,K)
dρ zmρ < ∞.
(21)
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This implies that
Φz , ρˆ11 G ρˆ11 (x)
ρ∈D (G,K)
converges absolutely for all z ∈ B and all x ∈ G. Therefore, by (20) we can conclude that Φz (x) =
Φz , ρˆ11 G ρˆ11 (x),
ρ∈D (G,K)
which converges absolutely on B × G, as desired.
2
Remark. Even if we replace the condition (iii) for Φ(z, x) by the condition (iii) stated in the remark following Theorem 3, Proposition 2 holds. Corollary 1. If a generating function Φ(z, x) associated with (G, K) satisfies the following condition (iv), then Φ(z, x) is written as a series of the form of (3). (iv) The set {zm | m ∈ N0 } is an orthogonal system of the Hilbert space F(B ). 5. Examples In this section, first we shall give some examples which are concerned with what we explained in introduction. In particular, the second one means that the converse statement of Main result 1 does not hold. The third one means that the converse statement of Main result 2 does not hold, and, moreover, gives an example of a Riemannian symmetric pair which does not have a generating function associated with itself. As a result, we can conclude that the notion of pair which has a generating function associated with itself is equivalent neither to that of Riemannian symmetric pair nor to that of Gelfand pair. For references on zonal spherical functions on (SO(n), SO(n−1)) or (U (n), U (n − 1)), see [5,8]. Example 1. Let SO(n) be the special orthogonal group of degree n. If n 3, the pair (SO(n), SO(n − 1)) is a Riemannian symmetric pair of rank 1 such that SO(n)/SO(n − 1) ∼ = S n−1 n−1 is simply connected, where S is the (n − 1)-dimensional unit sphere. In what follows, we assume that n 3. Set e1 = t (1, 0, . . . , 0) ∈ S n−1 . Then the identification SO(n)/SO(n − 1) ∼ = S n−1 is given by the mapping xSO(n − 1) → xe1 , x ∈ SO(n). On the other hand, the zonal (n−2)/2 , spherical functions on (SO(n), SO(n − 1)) are given by the Gegenbauer polynomials Cm m ∈ N0 . More precisely, we should explain as follows. For each m ∈ N0 define (n−2)/2
ϕm (x) =
Cm
((xe1 , e1 )) , (n−2)/2 Cm (1)
x ∈ SO(n),
where ( , ) denotes the canonical inner product on the real vector space Rn . Then the set of all the zonal spherical functions on (SO(n), SO(n − 1)) is equal to {ϕm | m ∈ N0 }. Let dm denote the
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degree of the representation corresponding to ϕm . Then, Theorem 1 in [9] can be translated into the following:
dm ϕm (x)zm =
m∈N0
1 − z2 , (1 − 2z(xe1 , e1 ) + z2 )n/2
x ∈ SO(n), z ∈ B,
is a generating function associated with the pair (SO(n), SO(n − 1)). Example 2. Let U (n) be the unitary group of degree n. If n 2, the pair (U (n), U (n − 1)) is a Gelfand pair, but it is not a Riemannian symmetric pair. In what follows, we assume that n 2. Denote by S(Cn ) the unit sphere in Cn and set e1 = t (1, 0, . . . , 0) ∈ S(Cn ). Then we have the identification U (n)/U (n − 1) ∼ = S(Cn ) by the mapping xU (n − 1) → xe1 , x ∈ U (n). On the other hand, the zonal spherical functions on (U (n), U (n − 1)) are given by the functions Gn−1 pq , p, q ∈ N0 , which have the following generating function (see [10, Theorem 1.1]): 1−n p q = Gn−1 1 − 2 Re(wξ ) + |w|2 pq (ξ )w w , p,q∈N0
w, ξ ∈ C, |w| < 1, |ξ | 1. That is to say, for each p, q ∈ N0 define ϕpq (x) =
Gn−1 pq ((xe1 , e1 )) Gn−1 pq (1)
,
x ∈ U (n),
where ( , ) denotes the canonical inner product on the complex vector space Cn . Then the set of all the zonal spherical functions on (U (n), U (n − 1)) is equal to {ϕpq | p, q ∈ N0 }. Let dpq denote the degree of the representation corresponding to ϕpq . Then, Theorem 1 in [11] can be translated into the following:
p q
dpq ϕpq (x)z1 z2 =
p,q∈N0
1 − z1 z2 , (1 − z1 (xe1 , e1 ) − z2 (e1 , xe1 ) + z1 z2 )n
x ∈ U (n), z1 , z2 ∈ B, is a generating function associated with the pair (U (n), U (n − 1)). √
Example 3. Let us consider the compact abelian group U = {e −1θ | θ ∈ R}, and let us take {1} as a closed subgroup of U . Then the Gelfand pair (U, {1}) does not have a generating function associated with itself. In what follows, we shall show it. For that purpose, we assume that (U, {1}) has a generating function associated with itself. Moreover, we denote it by Φ(z, x) and suppose that it is defined on B × U , where is some positive integer. First, since U is abelian, we have that dρ = 1 for all ρ ∈ D(U, {1}). Combining this with the unitarity of Φ, we obtain the following: m 2 z ρ dμ = 1, ρ ∈ D U, {1} , (22) B B
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where μB is a measure on B associated with Φ(z, x). Applying Lebesgue’s convergence theorem, it is easy to see that the left-hand side of (22) tends to zero as |mρ | goes to infinity, where |m| means m1 + · · · + m for m = (m1 , . . . , m ) ∈ N0 . This contradicts (22). Therefore, the Gelfand pair (U, {1}) does not have a generating function associated with itself. Since (U, {1}) is also a Riemannian symmetric pair, it is also an example of a Riemannian symmetric pair which does not have a generating function associated with itself. Finally, we give an example of application of Main result 1. Example 4. Let SU(n + 1) be the special unitary group of degree n + 1. Then S(U (1) × U (n)) is a closed subgroup of SU(n + 1). Set G = SU(n + 1), K = S U (1) × U (n) . Then the pair (G, K) is a Riemannian symmetric pair of rank 1 such that G/K is simply connected. (More precisely, G/K is diffeomorphic to the n-dimensional complex projective space including G-actions.) The zonal spherical functions on (G, K) are essentially given by the Jacobi (n−1,0) , m ∈ N0 (cf. [7]). Let dm denote the degree of the spherical representation polynomials Pm (n−1,0) corresponding to Pm . By Main result 1, we see that the series
(n−1,0)
dm
m∈N0
Pm
(t)
(n−1,0) Pm (1)
zm
(23)
gives a generating function associated with the pair (G, K). On the other hand, we have the (α,β) following generating function of the Jacobi polynomials Pm (t), m ∈ N0 (cf. [6]): (α + β + 1)m Pm(α,β) (t)zm (β + 1)m m∈N0 2tz + 2z α+β +1 α+β +2 −α−β−1 , , ; β + 1; = (1 + z) F 2 2 (z + 1)2
(24)
where (α)m = (α + m)/ (α) ( is the Gamma function) and F (a, b; c; x) is the hypergeometric function. Set Ψ (z, t) =
(n)m P (n−1,0) (t)zm . m! m
m∈N0
By formula (24), we have −n
Ψ (z, t) = (1 + z)
2tz + 2z n n+1 , ; 1; . F 2 2 (z + 1)2
Further, we see that the generating function (23) is equal to the following: 2z ∂Ψ (z, t) + Ψ (z, t). n ∂z
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References [1] V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Part I, Comm. Pure Appl. Math. 14 (1961) 187–214. [2] A. Essadiq, q-Analogue of the Watanabe unitary transform associated to the q-continuous Gegenbauer polynomials, Lett. Math. Phys. 53 (2000) 233–242. [3] S. Helgason, Groups and Geometric Analysis, Academic Press, New York, 1984. [4] T.H. Hildebrandt, I.J. Schoenberg, On linear functional operations and the moment problem for a finite interval in one or several dimensions, Ann. of Math. 34 (1933) 317–328. [5] K.D. Johnson, N.R. Wallach, Composition series and intertwining operators for the spherical principal series I, Trans. Amer. Math. Soc. 229 (1977) 137–173. [6] A.P. Prudnikov, Y.A. Brychkov, O.I. Marichev, Integrals and Series, Gordon & Breach, New York, 1986. [7] M. Takeuchi, Modern Spherical Functions, Transl. Math. Monogr., vol. 135, Amer. Math. Soc., Providence, RI, 1994. [8] N.J. Vilenkin, A.U. Klimyk, Representation of Lie Groups and Special Functions, vol. 2, Kluwer Acad. Publ., Dordrecht, 1993. [9] S. Watanabe, Hilbert spaces of analytic functions and the Gegenbauer polynomials, Tokyo J. Math. 13 (1990) 421– 427. [10] S. Watanabe, Generating functions and integral representations for the spherical functions on some classical Gelfand pairs, J. Math. Kyoto Univ. 33 (1993) 1125–1142. [11] S. Watanabe, Hilbert spaces of analytic functions associated with generating functions of spherical functions on U (n)/U (n − 1), Proc. Japan Acad. Ser. A 70 (1994) 323–325.