Journal of Lie Theory Volume 7 (1997) 29–60 ◯^{C}1997 Heldermann Verlag

The spherical transform for homogeneous vector bundles over Riemannian symmetric spaces

Roberto Camporesi

Communicated by G. Mauceri

Abstract. Let G/K be a Riemannian symmetric space of the noncompact type. For τ ∈ K̂, let E^{τ} be the homogeneous vector bundle over G/K associated with τ, and let C_{0}^{∞}(G, τ, τ) be the related convolution algebra of radial systems of sections of E^{τ}. Assuming C_{0}^{∞}(G, τ, τ) commutative, we use the theory of spherical functions of type τ on G to define a spherical transform for F ∈ C_{0}^{∞}(G, τ, τ). The corresponding inversion formula is obtained by using the Plancherel formula on G. The example of real hyperbolic spaces H^{N}(ℝ) is discussed. The Plancherel measure is written down explicitly in this case, and the vector bundles of Dirac spinors, symmetric traceless tensors, and p-forms on H^{N}(ℝ) are considered in detail.

1. Introduction

Let G be a connected noncompact semisimple Lie group with finite center, K a maximal compact subgroup, and G/K the corresponding Riemannian symmetric space of the noncompact type. Scalar harmonic analysis on G/K is by now well understood (see, e.g., Helgason’s books [12, 13]). In this paper we investigate the case of homogeneous vector bundles over G/K.

For a given irreducible unitary representation τ of K, we can let the induced bundle L^{2}(G, τ) sit in L^{2}(G) in a natural way. Indeed the left regular representation of G in L^{2}(G) is unitarily equivalent to the direct sum over K̂ of the induced representations π_{τ} = ind_{K}^{G}(τ), each π_{τ} occurring d_{τ} times (d_{τ} the dimension of τ). The direct sum ∑_{1}^{d_{τ}} L^{2}(G, τ) may be identified with the subspace L^{2}(G) ∗ d_{τ} χ̄_{τ} of L^{2}(G) (χ_{τ} is the character of τ), see [19] Lemma 5.1.

By the generalized Frobenius Reciprocity principle, if we have the Plancherel formula for G we also have the direct integral decomposition of any induced representation π_{τ} (K being compact) [18]. The Plancherel formula for f ∈ L^{2}(G) will produce a Plancherel formula for f ∈ L^{2}(G, τ).

Compared to the scalar case (when τ is the trivial representation of K), several new features arise in the case of bundles. For example the discrete series of G

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag (when they exist), and the generalized principal series representations (induced from nonminimal cuspidal parabolic subgroups of G) will enter the Plancherel formula for a generic vector bundle over G/K.

In this paper we use the theory of spherical functions of type τ on G, as developed by Godement, Harish Chandra and Warner [11, 28], to discuss the analysis of radial systems of sections of a homogeneous vector bundle E^{τ} over G/K.

Radial systems of sections of E^{τ} and the related convolution algebra C_{0}^{∞}(G, τ, τ) are defined in Section 2. This algebra may be identified with a certain subalgebra of C_{0}^{∞}(G), denoted I_{0,τ}(G). More precisely, I_{0,τ}(G) consists of those f ∈ C_{0}^{∞}(G) which are K -central and are invariant under convolution with d_{τ} χ̄_{τ}. The algebras C_{0}^{∞}(G, τ, τ) and I_{0,τ}(G) play the same role, for vector bundles, as the convolution algebra C_{0}^{#}(G) in the scalar case. (This is the algebra of compactly supported smooth functions on G which are biinvariant under K.)

In Section 3 we first define spherical functions of type τ on G, and discuss their functional and differential properties in the commutative case, namely when the convolution algebra C_{0}^{∞}(G, τ, τ) is commutative (for the given τ). The algebra D(G, τ) of invariant differential operators on E^{τ} is then also commutative [8]. The analysis of radial sections in this case is as close as possible to scalar harmonic analysis on Riemannian symmetric spaces G/K, or on the homogeneous spaces G/K with (G, K) a Gelfand pair.

Then we define (in the commutative case) a spherical transform for radial systems of sections of E^{τ}, and derive an inversion formula from the (abstract) Plancherel formula on G. In this part of Section 3, we do not use the structure theory of semisimple Lie groups, and keep the notations as general as possible. In fact the results obtained here apply (in the commutative case) to any pair (G, K) of a locally compact unimodular Lie group G and a compact subgroup K, provided that (i) K is sufficiently large in G so that every irreducible unitary representation U of G is K -finite; (ii) there is a well defined theory of global characters on G, i.e., for every U ∈ Ĝ the operator U(f) = ∫_{G} f(x)U(x)dx is of trace class for all f ∈ C_{0}^{∞}(G), and the mapping Θ_{U} : f → Tr U(f) is a distribution on G. For example if G admits a uniformly large compact subgroup K (see the definition in [28] vol.I p. 305), then the conditions (i) and (ii) are satisfied. This includes all reductive pairs and all motion groups [28].

Another example is given by a pair (K, M) of a compact Lie group K and a closed subgroup M ⊂ K. It is clear that for each σ ∈ M̂ we can formulate a theory of spherical functions of type σ on K, and define a spherical transform for radial sections of the bundle E^{σ} over K/M, even if (K, M) is not a symmetric pair. This theory will be analogous (in the commutative case) to that developed in Section 3.

At the end of Section 3 we consider the semisimple case in more detail. The spherical transform and the inversion formula on C_{0}^{∞}(G, τ, τ) are written down explicitly in this case, using Harish Chandra’s Plancherel formula and Subquotient Theorem.

An important example of commutative algebras I_{0,τ}(G) is provided by the pairs (G, K) = (SO_{0}(N, 1), SO(N)) or (G, K) = (SU(N, 1), S(U(N) × U(1))), i.e., G/K is either a real or a complex hyperbolic space. In section 4 we discuss the example of real hyperbolic spaces H^{N}(ℝ). We make explicit the Plancherel measure and the inversion formula for the double cover Spin(N, 1) of SO_{0}(N, 1). This allows us to discuss also the spinor bundles over H^{N}(ℝ). The homogeneous vector bundles of Dirac spinors, symmetric traceless tensors, and p-forms over H^{N}(ℝ) are considered in detail. The spherical transform reduces to the Jacobi transform in this case. The approach using Jacobi functions analysis agrees with the representation theoretic approach presented here.

Acknowledgements

The author would like to thank E. Badertscher, A. Higuchi, E. Pedon, M. Reimann, and F. Ricci for interesting conversations on the subject of this paper.

2. Radial systems of sections

Let G be a connected noncompact semisimple Lie group with finite center, K ⊂ G a maximal compact subgroup, and G/K the corresponding Riemannian symmetric space of the noncompact type.

Let τ be an irreducible unitary representation of K on V_{τ}, and let E^{τ} be the homogeneous vector bundle over G/K defined by τ. It is well known (see, e.g.,

[26] sect. 5.3) that a cross section of E^{τ} may be identified with a vector-valued function f : G → V_{τ} which is right-K -covariant of type τ, i.e.,

f(gk) = τ(k^{−1})f(g), ∀g ∈ G, ∀k ∈ K.  (1)

We denote by C_{0}^{∞}(G, τ) the space of compactly supported smooth functions that are right-K -covariant of type τ, and by L^{2}(G, τ) the Hilbert space of square integrable such functions, with scalar product

⟨f_{1}, f_{2}⟩ = ∫_{G}⟨f_{1}(x), f_{2}(x)⟩dx.  (2)

By a radial system of sections we mean a map F : G → End(V_{τ}) such that

F (k_{1}gk_{2}) = τ(k_{2}^{−1})F (g)τ(k_{1}^{−1}), ∀g ∈ G, ∀k_{1}, k_{2} ∈ K.  (3)

For any v ∈ V_{τ}, the vector valued function f(g) = F (g)v satisfies (1) and defines a radial section of E^{τ}. [We follow here Badertscher and Reimann [1], who studied vector fields over the real hyperbolic spaces.]

The radial systems of sections generalize the notion of K -biinvariant functions on G. They are called radial because, due to the Cartan decomposition G = KAK, they are determined by their restriction to the vector subgroup A. We denote by C_{0}^{∞}(G, τ, τ) and by L^{2}(G, τ, τ) the obviously defined spaces of radial systems of sections, with scalar product

⟨F_{1}, F_{2}⟩ = ∫_{G} Tr [F_{1}(x)F_{2}(x)^{∗}]dx,

where ∗ denotes adjoint. For F_{1}, F_{2} ∈ C_{0}^{∞}(G, τ, τ), define the convolution by

(F_{1} ∗ F_{2})(x) = ∫_{G} F_{1}(y^{−1}x)F_{2}(y)dy.

This definition is arranged so that F_{1} ∗ F_{2} ∈ C_{0}^{∞}(G, τ, τ):

(F_{1} ∗ F_{2})(k_{1}xk_{2}) = ∫_{G} F_{1}(y^{−1}k_{1}xk_{2})F_{2}(y)dy
= τ(k_{2}^{−1}) ∫_{G} F_{1}(y^{−1}k_{1}x)F_{2}(y)dy
= τ(k_{2}^{−1}) ∫_{G} F_{1}(z^{−1}x)F_{2}(k_{1}z)dz
= τ(k_{2}^{−1}) ∫_{G} F_{1}(z^{−1}x)F_{2}(z)dz τ(k_{1}^{−1})
= τ(k_{2}^{−1})(F_{1} ∗ F_{2})(x)τ(k_{1}^{−1}).

In general on C_{0}^{∞}(G) (the space of compactly supported smooth functions on G), the convolution is defined by the usual rule

(f_{1} ∗ f_{2})(x) = ∫_{G} f_{1}(xy^{−1})f_{2}(y)dy = ∫_{G} f_{1}(z)f_{2}(z^{−1}x)dz.

The space C_{0}^{∞}(G, τ, τ) may be identified with a certain subalgebra of C_{0}^{∞}(G), which we now define. Let I_{0,τ}(G) denote the set of those f ∈ C_{0}^{∞}(G) which satisfy

f(kxk^{−1}) = f(x), ∀x ∈ G, ∀k ∈ K

(i.e., f is K -central), and

d_{τ} χ̄_{τ} ∗ f = f(= f ∗ d_{τ} χ̄_{τ}),  (4)

where d_{τ} and χ_{τ} are the dimension and the character of τ (a bar denotes complex conjugation, and the convolutions are over K).

Then I_{0,τ}(G) is a subalgebra of C_{0}^{∞}(G) and it is (anti)-isomorphic to C_{0}^{∞}(G, τ, τ). Indeed given F ∈ C_{0}^{∞}(G, τ, τ) define

f_{F} (x) ≡ d_{τ}Tr F (x).  (5)

It follows from (3) that f_{F} is K -central and moreover it satisfies (4). Indeed

(f_{F} ∗ d_{τ} χ̄_{τ})(x) = d_{τ} ∫_{K} f_{F} (xk)χ_{τ}(k)dk
= d_{τ}^{2} ∫_{K} Tr F (xk)χ_{τ}(k)dk
= d_{τ}^{2}Tr [∫_{K} τ(k^{−1})χ_{τ}(k)dk F (x)]
= d_{τ} Tr F (x) = f_{F} (x),

where we have used the Schur orthogonality relations for K with the normalization ∫_{K} dk = 1. Thus f_{F} ∈ I_{0,τ}(G). Viceversa, given f ∈ I_{0,τ}(G) put

F_{f}(x) ≡ ∫_{K} τ(k)f(kx)dk.  (6)

Then

F_{f}(k_{1}xk_{2}) = ∫_{K} τ(k)f(kk_{1}xk_{2})dk = ∫_{K} τ(k)f(kxk_{2})dk τ(k_{1}^{−1})
= ∫_{K} τ(k)f(k_{2}kx)dk τ(k_{1}^{−1}) = τ(k_{2}^{−1}) ∫_{K} τ(k)f(kx)dk τ(k_{1}^{−1})
= τ(k_{2}^{−1})F_{f}(x)τ(k_{1}^{−1}),

i.e., F_{f} is in C_{0}^{∞}(G, τ, τ). We have the following result (see [28] vol.II, p.3, Example 1):

Proposition 2.1. The map f → F_{f} is a linear bijection of I_{0,τ}(G) onto C_{0}^{∞}(G, τ, τ). Its inverse is the map F → f_{F}. These maps satisfy

F_{f_{1}∗f_{2}} = F_{f_{2}} ∗ F_{f_{1}},  (7)
f_{F_{1}∗F_{2}} = f_{F_{2}} ∗ f_{F_{1}}.  (8)

As a corollary of this proposition we see that the convolution algebra C_{0}^{∞}(G, τ, τ) is commutative if and only if the convolution algebra I_{0,τ}(G) is commutative.

Now let Ĝ (K̂) denote the set of equivalence classes of irreducible unitary representations of G (K). In what follows, we shall identify a class [U] ∈ Ĝ ([τ] ∈ K̂) with a representative U (τ) in that class, and we shall write (somewhat incorrectly) U ∈ Ĝ (τ ∈ K̂). Let m(τ, U) denote the multiplicity of τ in U|_{K}. The following result, which characterizes the commutative case, is well known (for the proof see [11] p.522 the Corollary to Th.8, or [28] vol.II p.9 Prop. 6.1.1.6):

Proposition 2.2. The following conditions are equivalent:

1) I_{0,τ}(G) is commutative;

2) m(τ, U) ≤ 1 ∀U ∈ Ĝ.

For a symmetric pair (G, K), it is known that m(τ, U) ≤ d_{τ}, ∀U, ∀τ [11]. Therefore the two conditions above are satisfied when τ is the trivial representation of K (i.e., in the scalar case, see [12]). It is natural to begin the investigation of vector bundles from the commutative case. Of course for a given G, I_{0,τ}(G) may or may not be commutative depending on τ ∈ K̂. An example of commutative algebras I_{0,τ}(G) may be given as follows. We say that the compact subgroup K is multiplicity free in G if each irreducible unitary representation of K is contained in each irreducible unitary representation of G at most once, i.e., m(τ, U) ≤ 1, ∀τ ∈ K̂, ∀U ∈ Ĝ. In this case, the conditions of Prop. 2.2 are satisfied for each τ ∈ K̂. The following result is classical.

Theorem 2.3. Let the symmetric pair (G, K) be either (SO_{0}(n, 1), SO(n)) or (SU(n, 1), S(U(n) × U(1))). Then K is multiplicity free in G.

A simple proof of this result may be found in [16], where it is proved that (G × K, diag(K)) is a Gelfand pair. The result then follows by noting that the multiplicity of τ in U is the same as the multiplicity of the trivial representation of diag(K) in U ⊗ τ̌ (τ̌ the contragredient representation of τ).

3. The spherical transform in the commutative case

We keep the notations of Section 2. Thus we fix a Riemannian symmetric pair (G, K), and an irreducible unitary representation (τ, V_{τ}) of K. In this section we assume that the conditions of Proposition 2.2 are satisfied for the given τ, so that the convolution algebra C_{0}^{∞}(G, τ, τ) is commutative.

We denote by Ĝ(τ) the set of those U in Ĝ which contain τ upon restriction to K. For U ∈ Ĝ(τ) let H_{U} be the Hilbert space where U acts, and let P_{τ} be the projector of H_{U} onto H_{τ}, the subspace of vectors which transform under K according to τ (see, e.g., [11]):

P_{τ} = d_{τ} ∫_{K} U(k)χ_{τ}(k^{−1})dk.  (9)

Since m(τ, U) = 1, we can identify H_{τ} with V_{τ}. Define the (operator valued) spherical function Φ_{τ}^{U} on G by

Φ_{τ}^{U}(g) ≡ P_{τ}U(g)P_{τ}, g ∈ G.  (10)

For each g ∈ G, Φ_{τ}^{U}(g) may be regarded as an element of End(V_{τ}) and satisfies Φ_{τ}^{U}(g)^{∗} = Φ_{τ}^{U}(g^{−1}), and

Φ_{τ}^{U}(k_{1}gk_{2}) = τ(k_{1})Φ_{τ}^{U}(g)τ(k_{2}), ∀g ∈ G, ∀k_{1}, k_{2} ∈ K.  (11)

We choose an orthonormal basis {v_{A}}_{A=1···∞} of H_{U} adapted to the decomposition of U|_{K} into different K -types, U|_{K} = ∑_{δ∈K̂} δ. We can always assume that for a = 1, …, d_{τ} the vectors {v_{a}} span H_{τ} ≃ V_{τ}.

The functions Φ_{τ}^{U} generalize to vector bundles the notion of positive definite spherical functions. It is easy to see that the traces

ϕ_{τ}^{U}(x) = Tr [P_{τ}U(x)P_{τ}] = Tr Φ_{τ}^{U}(x)

(known as spherical trace functions of type τ, see [28] vol.II), are positive definite functions on G in the usual sense. The functions Φ_{τ}^{U} and ϕ_{τ}^{U} are related as follows.

Lemma 3.1. Let the Haar measure on K be normalized by ∫_{K} dk = 1. Then ∀x ∈ G

∫_{K} Φ_{τ}^{U}(kxk^{−1})dk = \frac{1}{d_{τ}} ϕ_{τ}^{U}(x)1,  (12)
d_{τ} ∫_{K} ϕ_{τ}^{U}(xk^{−1})τ(k)dk = Φ_{τ}^{U}(x),  (13)

where 1 in (12) denotes the identity operator in V_{τ}.

Proof. Let Φ_{τ,K}^{U}(x) denote the left-hand side of (12). Then Φ_{τ,K}^{U}(x) is in End_{K}(V_{τ}), ∀x ∈ G (this follows from (11)). Since τ is irreducible, Φ_{τ,K}^{U}(x) = φ(x)1, where φ is a function on G. Taking the trace gives d_{τ}φ(x) = Tr Φ_{τ,K}^{U}(x) = Tr Φ_{τ}^{U}(x) = ϕ_{τ}^{U}(x). This proves (12). Eq. (13), rewritten as

d_{τ} ∫_{K} Tr [Φ_{τ}^{U}(x)τ(k^{−1})] τ(k)dk = Φ_{τ}^{U}(x),

follows from the Schur orthogonality relations.

Taking the trace of Eq. (13) gives the well known relation ϕ_{τ}^{U} ∗ d_{τ}χ_{τ} = ϕ_{τ}^{U}, see [11].

For f ∈ C_{0}^{∞}(G) and U ∈ Ĝ(τ), let U(f) denote the operator
U(f) = ∫_{G} f(x)U(x)dx.

Then, as is well known, U(f)P_{τ} = U(f ∗ d_{τ} χ̄_{τ}), P_{τ}U(f) = U(d_{τ} χ̄_{τ} ∗ f), and

P_{τ}U(f)P_{τ} = U(d_{τ} χ̄_{τ} ∗ f ∗ d_{τ} χ̄_{τ}).

In particular for f ∈ I_{0,τ}(G) we have the following result.

Proposition 3.2. Let f ∈ I_{0,τ}(G) and U ∈ Ĝ(τ). Then

P_{τ}U(f)P_{τ} = U(f),
U(f)U(k) = U(k)U(f),  k ∈ K.

Let U_{τ}(f) denote the restriction of U(f) to H_{τ}. Then the set of operators U_{τ}(f), f ∈ I_{0,τ}(G), is the centralizer of the representation k → τ(k) of K on H_{τ}.

Proof. See [28] vol.I p.307, and Prop. 4.5.1.7 p.310.

This proposition and (12) imply that for f ∈ I_{0,τ}(G)

U_{τ}(f) = ∫_{G} f(x)Φ_{τ}^{U}(x)dx = ∫_{G} ∫_{K} f(kxk^{−1})Φ_{τ}^{U}(x)dkdx

= ∫_{G} ∫_{K} f(y)Φ_{τ}^{U}(k^{−1}yk)dkdy = \frac{1}{d_{τ}} 1 ∫_{G} f(x)ϕ_{τ}^{U}(x)dx,

and

Θ_{U}(f) ≡ Tr U(f) = Tr U_{τ}(f) = ∫_{G} f(x)ϕ_{τ}^{U}(x)dx.  (14)

Since

U(f_{1} ∗ f_{2}) = U(f_{1})U(f_{2}),

we see that the map f → f̂(U) ∈ ℂ, where

f̂(U) ≡ \frac{1}{d_{τ}} ∫_{G} f(x)ϕ_{τ}^{U}(x)dx,  (15)

is a continuous homomorphism of the convolution algebra I_{0,τ}(G) into ℂ,

ˆ (f_{1} ∗ f_{2})(U) = f̂_{1}(U)f̂_{2}(U), ∀f_{1}, f_{2} ∈ I_{0,τ}(G).  (16)

The map f̂: Ĝ(τ) → ℂ, given in (15), is called the spherical Fourier transform of f ∈ I_{0,τ}(G).

For F ∈ C_{0}^{∞}(G, τ, τ), consider the operator T = ∫_{G} Φ_{τ}^{U}(x)F (x)dx. It is easy to see that T ∈ End_{K}(V_{τ}). Since τ is irreducible, T = c1, where c = \frac{1}{d_{τ}} ∫_{G} Tr [Φ_{τ}^{U}(x)F (x)]dx.

Definition 3.3. The spherical Fourier transform of F ∈ C_{0}^{∞}(G, τ, τ) is the function F̂ : Ĝ(τ) → ℂ defined by

F̂(U) ≡ \frac{1}{d_{τ}} ∫_{G} Tr [Φ_{τ}^{U}(x)F (x)]dx,  U ∈ Ĝ(τ).  (17)

When τ is the trivial representation of K, the function F̂, defined on the set of equivalence classes of irreducible unitary spherical representations of G, or equivalently, on the set of positive definite zonal spherical functions on G, reduces to the usual spherical Fourier transform of F, see [28] vol.II p.337.

Lemma 3.4. In the notations of Proposition 2.1 we have for all U ∈ Ĝ(τ)

F̂(U) = f̂_{F} (U), f̂(U) = F̂_{f}(U).  (18)

Proof. Using (13) we have

F̂(U) = \frac{1}{d_{τ}} ∫_{G} Tr [Φ_{τ}^{U}(x)F (x)]dx
= ∫_{G} ∫_{K} ϕ_{τ}^{U}(xk^{−1})Tr [τ(k)F (x)]dkdx
= ∫_{K} ∫_{G} ϕ_{τ}^{U}(xk^{−1})Tr [F (xk^{−1})]dxdk
= ∫_{G} ϕ_{τ}^{U}(x)Tr [F (x)]dx
= \frac{1}{d_{τ}} ∫_{G} ϕ_{τ}^{U}(x)f_{F} (x)dx = f̂_{F} (U).

Reading the argument backwards proves the second equality in (18).

Putting together eqs. (8), (16) and (18), we see that the map F → F̂(U) is a continuous homomorphism of C_{0}^{∞}(G, τ, τ) into ℂ,

ˆ (F_{1} ∗ F_{2})(U) = F̂_{1}(U)F̂_{2}(U), ∀F_{1}, F_{2} ∈ C_{0}^{∞}(G, τ, τ).

This leads to the following definition of spherical function of type τ and spherical transform in the commutative case.

Definition 3.5. Assume C_{0}^{∞}(G, τ, τ) commutative. A function Φ : G → End(V_{τ}) is called a spherical function of type τ on G if it satisfies (11) and if the map

F → F̂(Φ) = \frac{1}{d_{τ}} ∫_{G} Tr [Φ(x)F (x)]dx  (19)

is a homomorphism of C_{0}^{∞}(G, τ, τ) into ℂ. The map F̂, defined on the set of spherical functions of type τ by (19), is called the spherical Gelfand transform of F (spherical transform, for short). For U ∈ Ĝ(τ) we write F̂(Φ_{τ}^{U}) = F̂(U) (consistently with (17)).

Let Φ be a spherical function of type τ on G, and let ϕ(x) = Tr Φ(x). It is clear that Φ and ϕ are related as Φ_{τ}^{U} and ϕ_{τ}^{U} in (12)-(13) (same proof as in Lemma 3.1). Notice that ϕ is K -central and satisfies ϕ ∗ d_{τ}χ_{τ} = ϕ. Moreover, defining the spherical transform of f ∈ I_{0,τ}(G) by

f̂(ϕ) = \frac{1}{d_{τ}} ∫_{G} f(x)ϕ(x)dx,  (20)

we have

F̂(Φ) = f̂_{F} (ϕ), f̂(ϕ) = F̂_{f}(Φ)  (21)

(same proof as in Lemma 3.4). Therefore the map f → f̂(ϕ) is a homomorphism of I_{0,τ}(G) into ℂ. We could define a spherical function of type τ to be a scalar valued function ϕ on G satisfying the three conditions (i) ϕ is K -central; (ii) ϕ ∗ d_{τ}χ_{τ} = ϕ; (iii) the map f → f̂(ϕ) is a homomorphism of I_{0,τ}(G) into ℂ (see [28] vol.II p.14). This definition would then be equivalent to Definition 3.5. Indeed given ϕ satisfying (i)-(iii) above, the function Φ(x) given by the left hand side of (13) (with ϕ_{τ}^{U} → ϕ) satisfies the conditions of Definition 3.5.

The connection between spherical functions of type τ on G and representations of G is as follows. Let U be a topologically completely irreducible (TCI) Banach representation of G (see [28] vol.I p.228 for the definition of TCI). Suppose that τ occurs in U|_{K} with multiplicity one, and define Φ_{τ}^{U}(g) by means of eq. (10). Then Φ_{τ}^{U} is a spherical function of type τ on G. Conversely, if Φ is a spherical function of type τ on G, there exists a TCI Banach representation U of G such that Φ = Φ_{τ}^{U}. [See [28] vol.II p.15.] We say that a spherical function Φ of type τ is of positive type if the scalar valued function ϕ(x) = Tr Φ(x) is positive definite on G. Then the spherical functions of type τ of positive type are precisely the functions Φ_{τ}^{U} with U ∈ Ĝ(τ) (see [28] vol.II p.15, the remark).

As in the scalar case, spherical functions of type τ on G are characterized by certain functional equations.

Theorem 3.6. Let Φ be a spherical function of type τ on G, and let ϕ(x) = Tr Φ(x). Then ∀x, y ∈ G

d_{τ} ∫_{K} ϕ(xkyk^{−1})dk = ϕ(x)ϕ(y),  (22)

d_{τ} ∫_{K} Φ(xkyk^{−1})dk = Φ(x)ϕ(y),  (23)

d_{τ} ∫_{K} Φ(xky)χ_{τ}(k^{−1})dk = Φ(x)Φ(y).  (24)

Conversely, let Φ be a nonzero continuous function on G with values in End(V_{τ}) which satisfies (11) and either (23) or (24). Then Φ is a spherical function of type τ on G.

Proof. Eq. (22) is the functional equation for the spherical trace functions of height one, see [11]. The proof given here is similar to [10] Proposition I.3 p.319.

For a function f ∈ C_{0}^{∞}(G) put

f_{K}(x) = ∫_{K} f(kxk^{−1})dk,

and f^{τ} = d_{τ} χ̄_{τ} ∗ f ∗ d_{τ} χ̄_{τ}, i.e., explicitly

f^{τ}(x) = d_{τ}^{2} ∫_{K×K} f(k_{1}xk_{2})χ_{τ}(k_{1})χ_{τ}(k_{2})dk_{1}dk_{2}.

It is easy to see that (f_{K})^{τ} = (f^{τ})_{K}. Put f^{#} = (f^{τ})_{K}. Then the map f → f^{#} is a projection of C_{0}^{∞}(G) onto I_{0,τ}(G).

Let Φ be a spherical function of type τ on G, and let ϕ(x) = Tr Φ(x). The map f → f̂(ϕ) (cf. (20)) is a homomorphism of I_{0,τ}(G) into ℂ. Put ϕ(f) ≡ ∫_{G} f(x)ϕ(x)dx. Let f_{1}, f_{2} be in C_{0}^{∞}(G). Then

0 = \frac{1}{d_{τ}} ϕ(f_{1}^{#} ∗ f_{2}^{#}) − \frac{1}{d_{τ}^{2}} ϕ(f_{1}^{#})ϕ(f_{2}^{#})
= \frac{1}{d_{τ}} ∫_{G}(f_{1}^{#} ∗ f_{2}^{#})(z)ϕ(z)dz − \frac{1}{d_{τ}^{2}} ∫_{G} f_{1}^{#}(x)ϕ(x)dx ∫_{G} f_{2}^{#}(y)ϕ(y)dy
= \frac{1}{d_{τ}} ∫_{G×G} f_{1}^{#}(zy^{−1})f_{2}^{#}(y)ϕ(z)dydz − \frac{1}{d_{τ}^{2}} ∫_{G×G} f_{1}^{#}(x)f_{2}^{#}(y)ϕ(x)ϕ(y)dxdy
= \frac{1}{d_{τ}^{2}} ∫_{G×G} [d_{τ}ϕ(xy) − ϕ(x)ϕ(y)] f_{1}^{#}(x)f_{2}^{#}(y)dxdy.

Using the definition of f^{#} and the fact that ϕ(x) is K -central and satisfies ϕ ∗ d_{τ}χ_{τ} = ϕ, it is easy to show that the latter expression equals

\frac{1}{d_{τ}^{2}} ∫_{G×G} [d_{τ} ∫_{K} ϕ(xkyk^{−1})dk − ϕ(x)ϕ(y)] f_{1}(x)f_{2}(y)dxdy.

This proves (22), since f_{1}, f_{2} are arbitrary in C_{0}^{∞}(G). To prove (23) we use (13), to write

d_{τ} ∫_{K} Φ(xkyk^{−1})dk = d_{τ}^{2} ∫_{K×K} ϕ(xkyk^{−1}k_{1}^{−1})τ(k_{1})dk_{1}dk
= d_{τ}^{2} ∫_{K×K} ϕ(xk_{1}^{−1}k_{2}yk_{2}^{−1})τ(k_{1})dk_{1}dk_{2}
= d_{τ} ∫_{K} ϕ(xk_{1}^{−1})ϕ(y)τ(k_{1})dk_{1} = Φ(x)ϕ(y).

Finally to prove (24) we write

d_{τ} ∫_{K} Φ(xky)χ_{τ}(k^{−1})dk = d_{τ}^{2} ∫_{K} Φ(xky) [∫_{K} τ(k_{1}k^{−1}k_{1}^{−1})dk_{1}] dk
= d_{τ}^{2} ∫_{K×K} Φ(xkyk_{1}k^{−1})τ(k_{1}^{−1})dk_{1}dk
= d_{τ}^{2} ∫_{K} [∫_{K} Φ(xkyk_{1}k^{−1})dk] τ(k_{1}^{−1})dk_{1}
= d_{τ} ∫_{K} Φ(x)ϕ(yk_{1})τ(k_{1}^{−1})dk_{1} = Φ(x)Φ(y),

as claimed.

Conversely, suppose that Φ : G → End(V_{τ}) satisfies (11) and (23). Then ϕ(x) ≡ Tr Φ(x) satisfies (22). Clearly Φ and ϕ are related as Φ_{τ}^{U} and ϕ_{τ}^{U} in (12)- (13) (same proof as in Lemma 3.1). For F ∈ C_{0}^{∞}(G, τ, τ) and f ∈ I_{0,τ}(G), define F̂(Φ) and f̂(ϕ) by (19) and (20). Then (21) holds (same proof as in Lemma 3.4). Now ϕ is K -central, and ϕ ∗ d_{τ}χ_{τ} = ϕ. Let f_{1}, f_{2} ∈ I_{0,τ}(G). Then

ˆ (f_{1} ∗ f_{2})(ϕ) = \frac{1}{d_{τ}} ∫_{G} ∫_{G} f_{1}(xy^{−1})f_{2}(y)ϕ(x)dydx
= \frac{1}{d_{τ}} ∫_{G} ∫_{G} f_{1}(x)f_{2}(y)ϕ(xy)dxdy
= \frac{1}{d_{τ}} ∫_{G} ∫_{G} ∫_{K} f_{1}(x)f_{2}(kyk^{−1})ϕ(xy)dkdxdy
= \frac{1}{d_{τ}} ∫_{G} ∫_{G} f_{1}(x)f_{2}(y) [∫_{K} ϕ(xkyk^{−1})dk] dxdy
= \frac{1}{d_{τ}^{2}} ∫_{G} ∫_{G} f_{1}(x)f_{2}(y)ϕ(x)ϕ(y)dxdy = f̂_{1}(ϕ)f̂_{2}(ϕ).

The relation ˆ (F_{1} ∗ F_{2})(Φ) = F̂_{1}(Φ)F̂_{2}(Φ) ∀F_{1}, F_{2} ∈ C_{0}^{∞}(G, τ, τ), follows immediately from (21) and (8). Thus Φ is a spherical function of type τ on G.

Finally let Φ : G → End(V_{τ}) satisfy (11) and (24). Then Φ satisfies (23). Indeed

d_{τ} ∫_{K} Φ(xkyk^{−1})dk = d_{τ}^{2} ∫_{K} Φ(xkyk^{−1}) [∫_{K} τ(k_{1})χ_{τ}(k_{1}^{−1})dk_{1}] dk
= d_{τ}^{2} ∫_{K×K} Φ(xkyk^{−1}k_{1})χ_{τ}(k_{1}^{−1})dk_{1}dk
= d_{τ}^{2} ∫_{K} [∫_{K} Φ(xk_{1}k_{2}yk_{2}^{−1})χ_{τ}(k_{1}^{−1})dk_{1}] dk_{2}
= d_{τ} ∫_{K} Φ(x)Φ(k_{2}yk_{2}^{−1})dk_{2} = Φ(x)ϕ(y).

This completes the proof of the theorem.

Theorem 3.7. Let Φ be a nonzero spherical function of type τ on G, and let ϕ(x) = Tr Φ(x). Then Φ(e) = 1, and ∀x ∈ G

∫_{G} Φ(xy)F (y)dy = Φ(x)F̂(Φ),  ∀F ∈ C_{0}^{∞}(G, τ, τ),  (25)

∫_{G} ϕ(xy)f(y)dy = ϕ(x)f̂(ϕ),  ∀f ∈ I_{0,τ}(G).  (26)

Conversely, let Φ be a nonzero continuous function on G with values in End(V_{τ}) satisfying (11) and Φ(e) = 1. Suppose that for any F ∈ C_{0}^{∞}(G, τ, τ) there exists a complex number F̂(Φ) such that (25) holds. Then F̂(Φ) = \frac{1}{d_{τ}} ∫_{G} Tr [Φ(x)F (x)]dx, and Φ is a spherical function of type τ on G.

Proof. Let Φ be a spherical function of type τ on G, not identically zero. Letting g = e and k_{1} = k = k_{2}^{−1} in Φ(k_{1}gk_{2}) = τ(k_{1})Φ(g)τ(k_{2}), gives Φ(e)τ(k) = τ(k)Φ(e) ∀k ∈ K. Then Φ(e) = c1. Moreover Φ(k) = Φ(ke) = τ(k)Φ(e) = c τ(k). Letting x = y = e in (24) gives c^{2} = c. The function ϕ(x) = Tr Φ(x) can not vanish identically, because d_{τ} ∫_{K} ϕ(xk^{−1})τ(k)dk = Φ(x). Let x_{0} ∈ G be such that ϕ(x_{0}) ≠ 0. Letting x = x_{0} and y = e in (22) gives ϕ(e) = d_{τ}, whence c = 1.

Now using Proposition 2.1, (23) and (21) we have

∫_{G} Φ(xy)F (y)dy = ∫_{G} ∫_{K} Φ(xy)τ(k)f_{F} (ky)dkdy
= ∫_{K} ∫_{G} Φ(xk^{−1}y)τ(k)f_{F} (y)dydk
= ∫_{G} [∫_{K} Φ(xk^{−1}yk)dk] f_{F} (y)dy
= \frac{1}{d_{τ}} Φ(x) ∫_{G} ϕ(y)f_{F} (y)dy
= Φ(x)f̂_{F} (ϕ) = Φ(x)F̂(Φ),

which is (25). Eq. (26) can be proved in a similar way, using the K -centrality of f and (22).

Conversely, let Φ : G → End(V_{τ}) satisfy (11), Φ(e) = 1, and (25), for some F̂(Φ) ∈ ℂ. Setting x = e in (25) gives F̂(Φ)1 = ∫_{G} Φ(y)F (y)dy. Taking the trace of this equation gives F̂(Φ) = \frac{1}{d_{τ}} ∫_{G} Tr [Φ(x)F (x)]dx.

Given F_{1}, F_{2} ∈ C_{0}^{∞}(G, τ, τ), it is easy to see that

ˆ (F_{1} ∗ F_{2})(Φ) = \frac{1}{d_{τ}} Tr ∫_{G} ∫_{G} Φ(xy)F_{1}(y)F_{2}(x)dxdy.

Then (25) implies ˆ (F_{1} ∗ F_{2})(Φ) = F̂_{1}(Φ)F̂_{2}(Φ), i.e., Φ is a spherical function of type τ on G.

From the differential point of view, the spherical functions of type τ are the radial joint eigenfunctions of D(G, τ), the algebra of G-invariant differential operators mapping sections of E^{τ} to sections of E^{τ}. [See [20] for a Lie-algebraic description of D(G, τ).] By methods similar to those of [25], one can show that D(G, τ) is commutative if and only if C_{0}^{∞}(G, τ, τ) is commutative (see [8]). Thus D(G, τ) is commutative in our case.

Let C^{∞}(G, τ) (resp. C^{∞}(G, τ, τ)) be the space of C^{∞} maps from G to V_{τ} (resp. End(V_{τ})) satisfying (1) (resp. (3)). D(G, τ) acts naturally on C^{∞}(G, τ), in view of the identification of C^{∞}(G, τ) with Γ(E^{τ}), the space of C^{∞} cross sections of E^{τ}. For F ∈ C^{∞}(G, τ, τ) and v ∈ V_{τ}, the function f(g) = F (g)v is in C^{∞}(G, τ). Therefore we can let D(G, τ) act on C^{∞}(G, τ, τ) by

(DF)(·)v ≡ D(F (·)v), ∀D ∈ D(G, τ), ∀F ∈ C^{∞}(G, τ, τ), ∀v ∈ V_{τ}.

Clearly DF ∈ C^{∞}(G, τ, τ). Let η be the map on G defined by sending g to g^{−1}. Let Φ be a spherical function of type τ on G. Then Φ ∘ η ∈ C^{∞}(G, τ, τ). We shall now prove that Φ ∘ η is an eigenfunction of each D ∈ D(G, τ), using the functional equation (24).

Theorem 3.8. Let Φ : G → End(V_{τ}) be a spherical function of type τ on G. Let η : g → g^{−1}, and let Φ̃ = Φ ∘ η ∈ C^{∞}(G, τ, τ). Then for each D ∈ D(G, τ) there exists a complex number μ_{Φ}(D) such that

DΦ̃ = μ_{Φ}(D)Φ̃.  (27)

Moreover (DΦ̃)(e) = μ_{Φ}(D)1, and the map D → μ_{Φ}(D) is a homomorphism of D(G, τ) into ℂ. Conversely, let T ∈ C^{∞}(G, τ, τ), and suppose there is a homomorphism, D → μ_{T} (D), of D(G, τ) into ℂ such that

DT = μ_{T} (D)T, ∀D ∈ D(G, τ),

with DT (e) = μ_{T} (D)1. Then there is a spherical function Φ of type τ on G such that T = Φ ∘ η.

Proof. Let Φ be a spherical function of type τ on G. Using (24), it is easy to derive the following functional equation for Φ̃:

d_{τ} ∫_{K} Φ̃(xky) χ_{τ}(k)dk = Φ̃(y)Φ̃(x), x, y ∈ G.  (28)

Acting with D ∈ D(G, τ) on the y variable in (28), using the fact that D is left-invariant, letting y = e, and observing that DΦ̃ ∈ C^{∞}(G, τ, τ), we obtain

[DΦ̃](e)Φ̃(x) = d_{τ} ∫_{K} χ_{τ}(k)[DΦ̃](xk)dk
= d_{τ} ∫_{K} χ_{τ}(k)τ(k^{−1})dk [DΦ̃](x) = [DΦ̃](x).  (29)

Now notice that if T ∈ C^{∞}(G, τ, τ), then T (e) ∈ End_{K}(V_{τ}), so that T (e) = c1, with c = \frac{1}{d_{τ}} Tr T (e) ∈ ℂ. Applying this to T = DΦ̃ ∈ C^{∞}(G, τ, τ), gives

DΦ̃(e) = μ_{Φ}(D)1,

with μ_{Φ}(D) = \frac{1}{d_{τ}} Tr [DΦ̃(e)] ∈ ℂ. Using this in (29) proves (27).

Finally, we have

μ_{Φ}(D_{1}D_{2}) = \frac{1}{d_{τ}} Tr [(D_{1}D_{2}Φ̃)(e)]
= \frac{1}{d_{τ}} Tr [μ_{Φ}(D_{2})(D_{1}Φ̃)(e)]
= \frac{1}{d_{τ}} Tr [μ_{Φ}(D_{2})μ_{Φ}(D_{1})1]
= μ_{Φ}(D_{1})μ_{Φ}(D_{2}).

The proof of the converse is analogous to [28] Theorem 6.1.2.3. One proves first that T is analytic, using the fact that D(G, τ) has an elliptic element. Then using Taylor’s formula, one proves that T satisfies the functional equation (28). The result follows then from Theorem 3.6.

We shall now use the isomorphism between C_{0}^{∞}(G, τ, τ) and I_{0,τ}(G) to derive an inversion formula on C_{0}^{∞}(G, τ, τ) from the Plancherel (inversion) formula on the group G.

Theorem 3.9. The spherical transform (19) is inverted by

F (g) = \frac{1}{d_{τ}} ∫_{Ĝ(τ)} Φ_{τ}^{U}(g^{−1})F̂(U)dμ(U),  F ∈ C_{0}^{∞}(G, τ, τ),  (30)

where dμ(U) is the Plancherel measure on Ĝ (suitably normalized).

Proof. Let U ∈ Ĝ, and let Θ_{U} denote the global character of U. Consider the Fourier component Θ_{U,τ} of Θ_{U}, defined by the rule

Θ_{U,τ}(f) ≡ Θ_{U}(f ∗ d_{τ} χ̄_{τ}), f ∈ C_{0}^{∞}(G),

the convolution being over K. As proved in [28] vol.II p.18 (the remark), the distribution Θ_{U,τ} coincides with the spherical trace function ϕ_{τ}^{U}, i.e., for f ∈ C_{0}^{∞}(G),

Θ_{U,τ}(f) = ∫_{G} ϕ_{τ}^{U}(x)f(x)dx.  (31)

Clearly if U is not in Ĝ(τ), then

Θ_{U,τ}(f) = 0.

Let F ∈ C_{0}^{∞}(G, τ, τ), and consider the function f_{F} ∘L_{g}, where f_{F} ∈ I_{0,τ}(G) is defined in (5), and L_{g} denotes left-translation on G, L_{g}(x) = gx. This function satisfies

(f_{F} ∘ L_{g}) ∗ d_{τ} χ̄_{τ} = f_{F} ∘ L_{g},

as follows immediately from the definition of I_{0,τ}(G). Therefore

Θ_{U}(f_{F} ∘ L_{g}) = Θ_{U}((f_{F} ∘ L_{g}) ∗ d_{τ} χ̄_{τ}) = Θ_{U,τ}(f_{F} ∘ L_{g}),

and if U ∉ Ĝ(τ)

Θ_{U}(f_{F} ∘ L_{g}) = Θ_{U,τ}(f_{F} ∘ L_{g}) = 0.

We now use these relations and (31) in the Plancherel (inversion) formula for f_{F} ∈ I_{0,τ}(G):

f_{F} (g) = ∫Ĝ Θ_{U}(f_{F} ∘ L_{g})dμ(U)
= ∫_{Ĝ(τ)} Θ_{U,τ}(f_{F} ∘ L_{g})dμ(U)
= ∫_{Ĝ(τ)} ∫_{G} ϕ_{τ}^{U}(x)f_{F} (gx)dxdμ(U)
= ∫_{Ĝ(τ)} ∫_{G} ϕ_{τ}^{U}(g^{−1}y)f_{F} (y)dydμ(U)
= ∫_{Ĝ(τ)} ϕ_{τ}^{U}(g^{−1})f̂_{F} (U)dμ(U) = ∫_{Ĝ(τ)} ϕ_{τ}^{U}(g^{−1})F̂(U)dμ(U),

where we have used (26) and Lemma 3.4. To pass from f_{F} to F we apply the map f → F_{f} (cf. (6)), and observe that F_{f_{F}} = F. Then

F (g) = ∫_{K} τ(k)f_{F} (kg)dk
= ∫_{K} ∫_{Ĝ(τ)} τ(k)ϕ_{τ}^{U}(g^{−1}k^{−1})F̂(U)dμ(U)dk
= ∫_{Ĝ(τ)} (∫_{K} ϕ_{τ}^{U}(g^{−1}k^{−1})τ(k)dk) F̂(U)dμ(U)
= \frac{1}{d_{τ}} ∫_{Ĝ(τ)} Φ_{τ}^{U}(g^{−1})F̂(U)dμ(U),

which is the inversion formula (30).

The inversion formula for the spherical transform f̂ of f ∈ I_{0,τ}(G) (cf. (20)) is

f(g) = ∫_{Ĝ(τ)} ϕ_{τ}^{U}(g^{−1})f̂(U)dμ(U), f ∈ I_{0,τ}(G).

The Plancherel theorem for the spherical transform follows now from Theorem 3.9 by well known standard arguments.

Corollary 3.10. The spherical Fourier transform F → F̂, defined in (17), extends to an isometry of the Hilbert space L^{2}(G, τ, τ) onto the Hilbert space L^{2}(Ĝ(τ), dμ(U)).

Remark 3.11. The map F → F̂ given in (19) is the Gelfand transform on the commutative convolution algebra C_{0}^{∞}(G, τ, τ). We see from (30) that the Gelfand measure in the inversion formula on C_{0}^{∞}(G, τ, τ) may be identified with the Plancherel measure on Ĝ, restricted to those representations of G which contain τ.

We now derive an “inversion formula” for f ∈ C_{0}^{∞}(G, τ), i.e., for arbitrary (smooth compactly supported) sections of E^{τ} (not necessarily radial). We can proceed in two ways. We can either imitate the argument in the proof of Theorem 3.9, by observing that if f satisfies (1) then f ∗ d_{τ} χ̄_{τ} = f, and similarly (f ∘ L_{g}) ∗ d_{τ} χ̄_{τ} = f ∘ L_{g}. Then we can apply the Plancherel formula to each component f_{a} of f(g) = ∑_{a=1}^{d_{τ}} f_{a}(g)v_{a} and obtain (in index-free notation)

f(g) = ∫_{Ĝ(τ)} ∫_{G} ϕ_{τ}^{U}(g^{−1}x)f(x)dxdμ(U)
= ∫_{Ĝ(τ)} ∫_{G} ∫_{K} ϕ_{τ}^{U}(g^{−1}x)τ(k)f(xk)dkdxdμ(U)
= ∫_{Ĝ(τ)} ∫_{G} ∫_{K} ϕ_{τ}^{U}(g^{−1}yk^{−1})τ(k)f(y)dkdydμ(U)
= \frac{1}{d_{τ}} ∫_{Ĝ(τ)} ∫_{G} Φ_{τ}^{U}(g^{−1}y)f(y)dydμ(U).  (32)

Alternatively, we can use (30). Observe that any f ∈ C_{0}^{∞}(G, τ) can be written as

f(g) = F (g)v, g ∈ G,

where v is an arbitrary (fixed) unit vector in V_{τ}, and F (g) ∈ End(V_{τ}) is defined by

F (g)w ≡ f(g)⟨w, v⟩, w ∈ V_{τ}.

Notice that F satisfies

F (gk) = τ(k^{−1})F (g), ∀g ∈ G, ∀k ∈ K,

and ∀O ∈ End(V_{τ}),

Tr [OF (g)] = ⟨Of(g), v⟩.  (33)

Put

F_{g}(x) ≡ ∫_{K} F (gkx)τ(k)dk.  (34)

It is easy to see that F_{g} ∈ C_{0}^{∞}(G, τ, τ). Hence, we can expand it according to (30),

F_{g}(x) = \frac{1}{d_{τ}} ∫_{Ĝ(τ)} Φ_{τ}^{U}(x^{−1})F̂_{g}(U)dμ(U).  (35)

The spherical Fourier transform of F_{g} is given by

d_{τ} F̂_{g}(U) = ∫_{G} Tr [Φ_{τ}^{U}(y)F_{g}(y)] dy
= ∫_{G} Tr [Φ_{τ}^{U}(y) ∫_{K} F (gky)τ(k)dk] dy
= Tr ∫_{K} ∫_{G} Φ_{τ}^{U}(k^{−1}g^{−1}x)F (x)τ(k)dxdk
= ∫_{G} Tr [Φ_{τ}^{U}(g^{−1}x)F (x)] dx = ∫_{G}⟨Φ_{τ}^{U}(g^{−1}x)f(x), v⟩dx, (36)

where we have used (33). Now set x = e in (34) and take the trace to get

Tr F_{g}(e) = Tr ∫_{K} F (gk)τ(k)dk
= Tr ∫_{K} τ(k^{−1})F (g)τ(k)dk
= Tr F (g) = ⟨f(g), v⟩,  (37)

where we have used (33) with O = 1. Setting x = e in (35) and taking the trace gives

Tr F_{g}(e) = ∫_{Ĝ(τ)} F̂_{g}(U)dμ(U).

Using here (36) for F̂_{g}(U) and equating to (37), we obtain, since v is arbitrary,

f(g) = \frac{1}{d_{τ}} ∫_{Ĝ(τ)} ∫_{G} Φ_{τ}^{U}(g^{−1}x)f(x)dxdμ(U),  (38)

in agreement with (32).

Notice that for a radial section f [i.e., f(g) = F (g)v, with v ∈ V_{τ}, and F ∈ C_{0}^{∞}(G, τ, τ)], eq. (38) reduces to (30) upon using (25).

We define the convolution of Φ_{τ}^{U} and f by

(Φ_{τ}^{U} ∗ f)(g) = ∫_{G} Φ_{τ}^{U}(g^{−1}x)f(x)dx, f ∈ C_{0}^{∞}(G, τ), g ∈ G,

so that eq. (38) becomes

f(g) = \frac{1}{d_{τ}} ∫_{Ĝ(τ)}(Φ_{τ}^{U} ∗ f)(g)dμ(U).

The space Φ_{τ}^{U} ∗ C_{0}^{∞}(G, τ) can be given the positive definite inner product

⟨Φ_{τ}^{U} ∗f_{1}, Φ_{τ}^{U} ∗f_{2}⟩ = \frac{1}{d_{τ}} ∫_{G×G}⟨Φ_{τ}^{U}(g^{−1}x)f_{1}(x), f_{2}(g)⟩dx dg
= \frac{1}{d_{τ}} ∫_{G×G}⟨f_{1}(x), Φ_{τ}^{U}(x^{−1}g)f_{2}(g)⟩dx dg.

Denoting its completion by L_{U}^{2}(G, τ), we have the direct integral decompositions

L^{2}(G, τ) = ∫_{Ĝ(τ)}^{⊕} L_{U}^{2}(G, τ)dμ(U),
∥f∥^{2} = ∫_{Ĝ(τ)} ∥Φ_{τ}^{U} ∗f∥^{2}dμ(U), ∀f ∈ C_{0}^{∞}(G, τ),  (39)

where L^{2}(G, τ) is the completion of C_{0}^{∞}(G, τ) with respect to the scalar product (2). From the representation theoretic point of view, this result gives the direct integral decomposition over Ĝ of the induced representation ind_{K}^{G}(τ), according to

ind_{K}^{G}(τ) = ∫_{Ĝ(τ)}^{⊕} m(τ, U)Udμ(U)  (40)

(see [21] Lemma 1, or [18] p.58). This result may be regarded as a generalization to the noncompact case of the classical Frobenius Reciprocity Theorem. That is, the multiplicity with which U occurs in ind_{K}^{G}(τ) coincides with the multiplicity of τ in U|_{K}. Of course the notion of multiplicity in a direct integral requires a little care. For U in the discrete series, dμ(U) is discrete and (40) takes the same form as in the classical (compact) case, see [18] p.58.

There are two important differences between the compact and the noncompact case: 1) in the noncompact case the Plancherel measure may have both a discrete and a continuous part, rather than only discrete; 2) in the noncompact case a representation in Ĝ may contain τ and still not appear in the decomposition of ind_{K}^{G}(τ) (if it is not tempered). In fact the support of dμ(U) is, in general, a proper subset of Ĝ. For example the so called complementary series of SL(2, ℝ) have zero Plancherel measure.

Remark 3.12. In this section we have fixed τ ∈ K̂ so that m(τ, U) = 1, ∀U ∈ Ĝ(τ). However for τ generic in K̂, the multiplicity ξ_{U} ≡ m(τ, U) will exceed 1. (It is known that m(τ, U) ≤ d_{τ}, giving an estimate independent of U, see [11].) In this more general case, the results of this section will be modified as follows: 1) the algebras I_{0,τ}(G) and C_{0}^{∞}(G, τ, τ) will no longer be commutative; 2) the spherical functions Ψ_{τ}^{U}(g) = P_{τ}U(g)P_{τ} will take values in End(V_{τ} ⊗ ℂ^{ξ_{U}}); 3) the spherical transform F̂(U) will no longer be scalar valued, but will take values in End(ℂ^{ξ_{U}}). However the inversion formula on C_{0}^{∞}(G, τ, τ) can still be derived from the Plancherel formula on C_{0}^{∞}(G), and is similar to (30). The noncommutative case will be discussed in another paper.

Remark 3.13. The direct integral decomposition (39) does not involve any Fourier transform concept for general f ∈ C_{0}^{∞}(G, τ). [The spherical transform applies only to radial sections.] However in the scalar case, Helgason has defined a Fourier transform for general functions f ∈ C_{0}^{∞}(G/K), see [13]. The Helgason Fourier transform on Riemannian symmetric spaces G/K has recently been generalized to homogeneous vector bundles, see [7].

We have not used, so far, the structure theory of semisimple Lie groups (Cartan or Iwasawa decomposition). In fact, the theory of spherical functions of type τ can be formulated for any pair (G, K) of a locally compact unimodular Lie group G and a compact subgroup K ⊂ G, provided that every U ∈ Ĝ is K -finite (see [11] section 3). As mentioned in the introduction, if K is uniformly large in G (see [28] vol.I p. 305), then the results obtained up to this point apply (for I_{0,τ}(G) commutative).

We now consider the semisimple case in more detail. It is well known (see, e.g., [15, 27, 28]) that for a noncompact semisimple Lie group with finite center, the irreducible unitary representations that appear in the Plancherel formula (the so-called tempered spectrum of G) are the (generalized) principal series (constructed from a complete set of cuspidal parabolic proper subgroups by the method of induced representation), and the discrete series, which exist if and only if G has a compact Cartan subgroup, i.e., if and only if rank G = rank K. The Plancherel measure has been obtained by Harish-Chandra. Using Harish Chandra’s Plancherel Theorem and Subquotient Theorem, we shall write down in a more precise way the spherical transform and the inversion formula on C_{0}^{∞}(G, τ, τ) in the commutative case.

Let G = KAN be an Iwasawa decomposition of G, and write

x = k(x) exp[H(x)] n(x), ∀x ∈ G,

where H(x) ∈ 𝔞 (the Lie algebra of A), k(x) ∈ K and n(x) ∈ N. Let M be the centralizer of A in K, and let P = MAN be the corresponding minimal parabolic subgroup of G. Given σ ∈ M̂ and a linear function λ : 𝔞 → ℂ, let U^{σλ} denote the representation ind_{P}^{G}(σ ⊗ e^{iλ} ⊗ 1) in the minimal principal series of G with parameters σ and λ. U^{σλ} is unitary if and only if λ is real valued.

The Subquotient Theorem of Harish Chandra (see, e.g., [28] Theorem 5.5.1.5) implies that each U ∈ Ĝ (more generally every TCI Banach representation of G) is infinitesimally equivalent to a subquotient representation of a nonunitary principal series U^{σλ}, for suitable σ ∈ M̂ and λ ∈ 𝔞_{ℂ}^{∗} (the complexification of 𝔞^{∗}).

In the commutative case we have, for the given τ, m(τ, U^{σλ}) ≤ 1 ∀σ ∈ M̂, ∀λ ∈ 𝔞_{ℂ}^{∗}. By Frobenius Reciprocity, the multiplicity m(σ, τ) of σ in τ|_{M} is also ≤ 1 ∀σ ∈ M̂. Then the Subquotient Theorem implies that every (nonzero) spherical function of type τ on G can be written as Φ_{τ}^{σλ} ≡ P_{τ}U^{σλ}P_{τ}, for suitable σ ⊂ τ|_{M}, and λ ∈ 𝔞_{ℂ}^{∗}. Let ϕ_{τ}^{σλ}(x) = Tr Φ_{τ}^{σλ}(x) be the corresponding spherical trace function of type τ. By [28] Corollary 6.2.2.3 one has

ϕ_{τ}^{σλ}(x) = d_{τ} ∫_{K}(χ_{τ} ∗ χ_{σ})(k(k^{−1}xk)) e^{(iλ−ρ)(H(xk))}dk,  (41)

where ρ ∈ 𝔞^{∗} is half the sum of the positive restricted roots, χ_{σ} is the character of σ, and the convolution is over M. One can easily calculate Φ_{τ}^{σλ}(x) using (41), (13), and the Schur orthogonality relations. The result is the following integral representation for Φ_{τ}^{σλ}:

Φ_{τ}^{σλ}(x) = \frac{d_{τ}}{d_{σ}} ∫_{K} τ (k(xk)) P_{σ}τ(k^{−1}) e^{(iλ−ρ)(H(xk))}dk,  (42)

where

P_{σ} = d_{σ} ∫_{M} τ(m^{−1})χ_{σ}(m)dm

is the projector of V_{τ} onto V_{σ} ⊂ V_{τ} (the subspace of vectors of V_{τ} which transform under M according to σ), and d_{σ} is the dimension of σ.

The spherical transform of F ∈ C_{0}^{∞}(G, τ, τ) can then be defined as the set of functions F̂_{σ} : 𝔞_{ℂ}^{∗} → ℂ (where σ ⊂ τ|_{M}), given by

F̂_{σ}(λ) = \frac{1}{d_{τ}} ∫_{G} Tr [Φ_{τ}^{σλ}(x)F (x)]dx,  λ ∈ 𝔞_{ℂ}^{∗}.  (43)

The inversion formula (30) can be written more explicitly as follows. Let P′ be a cuspidal parabolic subgroup of G such that P′ ⊇ P and A′ ⊆ A, where P′ = M′A′N′ is a Langlands decomposition of P′. Given σ′ in the discrete series of M′ and ν′ ∈ 𝔞^{′∗} (the real dual of the Lie algebra of A′), let U^{σ′ν′} denote the representation ind_{P′}^{G}(σ′⊗e^{iν′}⊗1) in the unitary principal P′-series with parameters σ′ and ν′. If τ ⊂ U^{σ′ν′}|_{K}, put as usual Φ_{τ}^{U^{σ′ν′}}(x) = P_{τ}U^{σ′ν′}(x)P_{τ}, and

F̂(U^{σ′ν′}) = \frac{1}{d_{τ}} ∫_{G} Tr [Φ_{τ}^{U^{σ′ν′}}(x)F (x)]dx, F ∈ C_{0}^{∞}(G, τ, τ).

(For U^{σ′ν′} irreducible, this is the spherical Fourier transform (17) with U(= U^{σ′ν′}) in the tempered spectrum of G). By the Subquotient Theorem we can identify U^{σ′ν′} with a subquotient of a nonunitary (minimal) principal series U^{σ̃′λ}, for suitable σ̃′ ∈ M̂ and λ ∈ 𝔞_{ℂ}^{∗}.

More precisely, let K′ = K ⋂ M′ be maximal compact in M′, and let M′ = K′A_{1}N_{1} be an Iwasawa decomposition of M′ such that A = A′A_{1} and N = N′N_{1}. We choose parameters σ̃′ ∈ M̂ and μ_{1} ∈ (𝔞_{1})_{ℂ}^{∗} (𝔞_{1} the Lie algebra of A_{1}) by the Subquotient Theorem, so that σ′ is infinitesimally equivalent to a subquotient of the nonunitary (minimal) principal series of M′ given by

ω_{σ̃′μ_{1}} = ind_{MA_{1}N_{1}}^{M′}(σ̃′ ⊗ e^{μ_{1}} ⊗ 1).  (44)

Then the (generalized) principal series U^{σ′ν′} is infinitesimally equivalent to a subquotient of the nonunitary (minimal) principal series of G

U^{σ̃′,ν′−iμ_{1}} = ind_{MAN}^{G}(σ̃′ ⊗ e^{iν′+μ_{1}} ⊗ 1).  (45)

This follows by a double induction formula, as in [15] pp. 171, 240.

Then the parameter λ above equals ν′−iμ_{1}, and Φ_{τ}^{U^{σ′ν′}}(x), for τ ⊂ U^{σ′ν′}|_{K}, is given by

Φ_{τ}^{U^{σ′ν′}}(x) = Φ_{τ}^{σ̃′,ν′−iμ_{1}}(x) = \frac{d_{τ}}{d_{}_{σ̃′}} ∫_{K} τ(k(xk))P_{}_{σ̃′}τ(k^{−1})e^{(iν′+μ_{1}−ρ)(H(xk))}dk

(cf. (42)). Thus the spherical Fourier transform F̂(U^{σ′ν′}) of F ∈ C_{0}^{∞}(G, τ, τ) relative to U^{σ′ν′} equals F̂_{}_{σ̃′}(ν′−iμ_{1}), given by (43).

Finally, let dμ(U^{σ′ν′}) = p_{σ′}(ν′)dν′ be the Plancherel measure associated with U^{σ′ν′}, where dν′ is a properly normalized Euclidean measure on 𝔞^{′∗}. Then, for a suitable normalization of the relevant Haar measures, and for suitable constants c_{P′} > 0, we have the following inversion formula for the spherical transform (43) of F ∈ C_{0}^{∞}(G, τ, τ):

F (x) = \frac{1}{d_{τ}} ∑_{P′} c_{P′} ∑_{σ′} ∫_{𝔞^{′∗}} Φ_{τ}^{σ̃′,ν′−iμ_{1}}(x^{−1})F̂_{}_{σ̃′}(ν′−iμ_{1})p_{σ′}(ν′)dν′.  (46)

Here the sum ∑_{P′} is over all cuspidal parabolic subgroups P′ of G such that P′ ⊇ P and A′ ⊆ A, and the sum ∑_{σ′} is over all discrete series σ′ of M′ such that U^{σ′ν′}|_{K} ⊃ τ. [These are in finite number only.] This result follows from (30) and from Harish Chandra’s Plancherel Theorem for semisimple Lie groups (see, e.g., [27] Theorem 13.4.1, or [15] Theorem 13.11). If rank G = rank K, then G itself is cuspidal parabolic, and the contribution of P′ = G in (46) is the discrete series contribution. In this case the parameter ν′ is trivial, and the integral over 𝔞^{′∗} drops out. For P′ = P (the minimal parabolic subgroup), the parameter μ_{1} is trivial. The corresponding contribution in (46) is

F_{minimal}(x) = \frac{1}{d_{τ}} c_{P} ∑_{σ} ∫_{𝔞^{∗}} Φ_{τ}^{σλ}(x^{−1})F̂_{σ}(λ)p_{σ}(λ)dλ,

where the sum is over all M -types contained in τ|_{M}.

Now due to the Cartan decomposition G = KAK, the spherical functions Φ_{τ}^{σλ} are determined by their restriction to A. Notice that Φ_{τ}^{σλ}(a) ∈ End_{M}(V_{τ}), ∀a ∈ A. Write τ|_{M} = ∑_{j=1}^{n} σ_{j}, with σ_{j} ∈ M̂. Then σ_{i} ≁ σ_{j} for i ≠ j, and σ is just one of the σ_{j}, say σ = σ_{k}. We have a direct sum decomposition V_{τ} = ∑_{j} V_{j}, where each V_{j} may be identified with the representation space of σ_{j}. Let P_{j} be the orthogonal projection of V_{τ} onto V_{j}. Schur’s Lemma implies that

Φ_{τ}^{σ_{k}λ}(a) = ∑_{j} ϕ_{τj}^{kλ}(a)1_{j}, ∀a ∈ A

(direct sum of linear operators), where 1_{j} is the identity in V_{j}. The scalar functions ϕ_{τj}^{kλ}(a) = \frac{1}{d_{j}} Tr [P_{j}Φ_{τ}^{σ_{k}λ}(a)P_{j}] (d_{j} ≡ d_{σ_{j}}) admit the integral representation (by (42))

ϕ_{τj}^{κλ}(a) = \frac{d_{τ}}{d_{j}d_{κ}} ∫_{K} Tr [P_{j}τ (k(ak)) P_{κ}τ(k^{−1})P_{j}] e^{(iλ−ρ)(H(ak))}dk.

In a similar way, each F ∈ C_{0}^{∞}(G, τ, τ) is determined by restriction to A. Since F (a) ∈ End_{M}(V_{τ}), ∀a ∈ A, then F (a) = ∑_{j} F_{j}(a)1_{j}, where F_{j}(a) are scalar functions on A. Let us write the integral formula for the Cartan decomposition as (see, e.g., [15] Prop.5.28 p.141):

∫_{G} f(g)dg = ∫_{K×A^{+}×K} f(k_{1}ak_{2})δ(a)dk_{1}dadk_{2},  (47)

where A^{+} = exp 𝔞^{+} (𝔞^{+} the positive Weyl chamber in a), and

δ(a) ≡ ∏_{+}(sinh α(log a))^{m_{α}},  (48)

where the product is over the positive restricted roots of the symmetric space (m_{α} is the multiplicity of the root α). The Haar measure on K is normalized by ∫_{K} dk = 1.

The spherical transform (43) can then be written as

F̂_{k}(λ) ≡ F̂_{σ_{k}}(λ) = \frac{1}{d_{τ}} ∫_{A^{+}} Tr [Φ_{τ}^{σ_{k}λ}(a)F (a)]δ(a)da

= \frac{1}{d_{τ}} ∑_{j} d_{j} ∫_{A^{+}} ϕ_{τj}^{kλ}(a)F_{j}(a)δ(a)da.  (49)

Thus the spherical transform can be described as follows in the commutative case. Let n be the number of M -types contained in τ|_{M}. Then each F ∈ C_{0}^{∞}(G, τ, τ) is determined by n scalar functions F_{j}(a) on A (j = 1, …, n). The spherical transform associates to this set of functions the n scalar functions F̂_{k}(λ) on 𝔞_{ℂ}^{∗} (k = 1, …, n) given by (49).

If G/K has rank one the functions ϕ_{τj}^{kλ}(a) can be calculated explicitly in terms of Jacobi functions (using the radial part of the Casimir operator of G). Examples of this will be seen in the next section.

4. The case of real hyperbolic spaces

In this section we apply the above considerations to G = Spin(N, 1) and K = Spin(N), the double covers of the Lorentz and orthogonal groups SO_{0}(N, 1) and SO(N), respectively. (For N > 2 these are also the universal covering groups.) In this way, we can include in our discussion the spinor bundles over the real hyperbolic spaces H^{N}(ℝ) ≃ G/K. Using Theorem 2.3, one can prove that Spin(N) is multiplicity free in Spin(N, 1), so that the algebras C_{0}^{∞}(G, τ, τ) are commutative ∀τ ∈ K̂.

For N odd all Cartan subgroups of G are conjugate, and there are no discrete series. For N even there are two conjugacy classes of Cartan subgroups, one for a compact Cartan subgroup contained in K, the other for a noncompact Cartan subgroup with one generator in G/K. In this case there are discrete series. If the irreducible representation τ of Spin(N) is contained in some discrete series, then the vector bundle E^{τ} has square-integrable eigensections of the Laplacian, corresponding to the discrete spectrum. (An example of this will be seen later.) Otherwise the L^{2}-spectrum of the Laplacian on E^{τ} is purely continuous (e.g., for τ the trivial representation).

The Plancherel measure for the Lorentz group has been given by Hirai [14]. We shall now summarize his results, and write down the Plancherel (inversion) formula for the double cover Spin(N, 1). As specific examples we shall consider the homogeneous vector bundles of Dirac spinors, differential forms, and symmetric traceless tensor fields over H^{N}(ℝ).

Case 1. N = 2k + 2 (k = 1, 2, …): The unitary principal series representations are denoted U^{σλ}, where λ is a real number, and σ = (n_{1}, n_{2}, …, n_{k}) is a row of numbers that are either all integers or all half-odd-integers and satisfy

0 ≤ n_{1} ≤ n_{2} ≤ ⋯ ≤ n_{k}.

In the standard Cartan-Weyl labeling scheme, σ is the highest weight of an irreducible representation of M = Spin(N −1) (see [2] Th.2 p.221). In fact, U^{σλ} is nothing but the representation ind_{P}^{G}(σ ⊗e^{iλ} ⊗1) unitarily induced from a minimal parabolic subgroup P = MAN, where G = KAN is an Iwasawa decomposition of G. Thus σ ∈ M̂, and it is well known that M (the centralizer of A in K) can be identified with Spin(N −1) in this case. We define l_{j} = n_{j} +j − \frac{1}{2} (j = 1, 2, …, k), and denote the global character of U^{σλ} by Θ_{σλ}.

There are two sets of representations in the discrete series denoted U_{+}^{σn_{0}} and U_{−}^{σn_{0}} , where σ = (n_{j}) is as above, n_{0} ∈ ℤ (resp. ℤ+ \frac{1}{2} ) ⟺ n_{j} ∈ ℤ (resp. ℤ+ \frac{1}{2}), and

\frac{1}{2} < n_{0} ≤ n_{1} ≤ n_{2} ≤ ⋯ ≤ n_{k}.

(For n_{0} = \frac{1}{2} we have the two “limits of discrete series”, which are irreducible unitary representations, but are not square-integrable.) Define l_{j} (j = 0, …, k) as above, and denote the sum of the characters of U_{+}^{σn_{0}} and U_{−}^{σn_{0}} by Θ_{σn_{0}} . Then the inversion formula on C_{0}^{∞}(Spin(N, 1)) takes the following form (we have corrected the continuous part by a factor of 2; this makes the formula consistent with ref. [22]):

c f(e) = ∑_{0<l_{1}<···<l_{k}} ∫_{0}^{∞} iP (−iλ, l_{1}, …, l_{k}) g(λ) Θ_{σλ}(f) dλ

+ ∑_{0<l_{0}<l_{1}<···<l_{k}} P (l_{0}, l_{1}, …, l_{k}) Θ_{σn_{0}}(f),  (50)

where c > 0 is a normalization constant (to be determined later),

g(λ) = \{ \begin{matrix}tanh(πλ), l_{j} half-odd-integers, \\ coth(πλ), l_{j} integers, \end{matrix}

and P is the following polynomial, corresponding to the product over the positive roots of the Spin(N, 1) Lie algebra:

P (x_{1}, x_{2}, …, x_{k+1}) = x_{1}x_{2} ⋯ x_{k+1} ∏_{1≤s<r≤k+1}(x_{r}^{2} − x_{s}^{2}).

The case of N = 2 (k = 0) corresponds to G = SL(2, ℝ) ≃ Spin(2, 1). In this case, the continuous part is the sum of two terms, one with g(λ) = tanh(πλ), and the other with g(λ) = coth(πλ), see [15] p. 42.

For σ fixed, the continuous part of the Plancherel measure dμ(U^{σλ}) = p_{σ}(λ)dλ has the following λ-dependence:

p_{σ}(λ) ∝ λ(λ^{2} + l_{1}^{2})(λ^{2} + l_{2}^{2}) ⋯ (λ^{2} + l_{k}^{2}) g(λ),

and the proportionality constant depends on l_{j}.

Case 2. N = 2k + 1 (k = 1, 2, …): The principal series representations are denoted U^{σλ}, with λ ∈ ℝ and σ = (n_{1}, n_{2}, …, n_{k}), where the numbers n_{j} are either all integers or all half-odd-integers and satisfy

|n_{1}| ≤ n_{2} ≤ ⋯ ≤ n_{k}.

The number n_{1} can be negative, and again σ defines a representation of Spin(N − 1) (see [2] Th.2 p.221). Let l_{j} = n_{j} + j − 1 (j = 1, 2, …, k). Then the inversion formula on C_{0}^{∞}(Spin(N, 1)) reads

c f(e) = ∑_{|l_{1}|<l_{2}<···<l_{k}} ∫_{0}^{∞} P (iλ, l_{1}, …, l_{k}) Θ_{σλ}(f) dλ,  (51)

where c > 0 is a normalization constant, and P is the polynomial corresponding to the product over the positive roots of the Spin(N, 1) Lie algebra:

P (x_{1}, x_{2}, …, x_{k+1}) = ∏_{1≤s<r≤k+1}(x_{r}^{2} − x_{s}^{2}).

For σ fixed, the Plancherel density is just a polynomial in λ^{2}:

p_{σ}(λ) ∝ (λ^{2} + l_{1}^{2})(λ^{2} + l_{2}^{2}) ⋯ (λ^{2} + l_{k}^{2}).

Now let τ be an irreducible representation (irrep) of Spin(N), and let E^{τ} be the corresponding homogeneous vector bundle over H^{N}(ℝ). In order to find the Plancherel measure for the cross sections of E^{τ}, we simply have to identify the irreducible unitary representations in the Plancherel formula on Spin(N, 1) which contain τ upon restriction to Spin(N). Thus we need the branching rule for Spin(N, 1) ⊃ Spin(N) for principal and discrete series (see, e.g., [24]). Again we distinguish the cases with N odd and N even.

Let N = 2k + 2, and let τ be the irrep of Spin(N) labelled by (f_{1}, f_{2}, …, f_{k+1}), where f_{j} are either all integers or all half-odd-integers satisfying (see [2] p.221)

|f_{1}| ≤ f_{2} ≤ ⋯ ≤ f_{k+1},

and f_{1} can be negative. Then the principal series representation U^{σλ} (σ = {n_{j}}) contains τ if and only if 1) n_{j} ∈ ℤ (resp. ℤ+\frac{1}{2} ) ⟺ f_{j} ∈ ℤ (resp. ℤ+\frac{1}{2}), and 2)

|f_{1}| ≤ n_{1} ≤ f_{2} ≤ n_{2} ≤ ⋯ ≤ f_{k} ≤ n_{k} ≤ f_{k+1}.  (52)

Using the branching rule for Spin(N) ⊃ Spin(N − 1), it is easy to see that (52) is equivalent to the condition that τ contain the irrep σ of Spin(N − 1) (see [2] Th.2 p.228). Therefore U^{σλ}|_{K} contains τ if and only if τ|_{M} contains σ, i.e., in symbols

τ|_{Spin(N−1)} = ⊕_{j=1}^{n}σ_{j} ⟺ U^{σ_{j}λ}|_{Spin(N)} ⊃ τ, ∀j = 1, …, n.  (53)

(Of course this follows by Frobenius Reciprocity, since U^{σλ}|_{K} = ind_{M}^{K}(σ).)

A geometric interpretation of (53) may be given as follows. According to the so-called polar coordinates decomposition [12], a (noncompact) Riemannian symmetric space G/K is diffeomorphic to A^{+} × K/M (up to a zero-measure set), where A^{+} is a fundamental domain of the Weyl group in A. The coset space K/M is diffeomorphic to the orbits of K in G/K. When a field on G/K (i.e., a cross section of E^{τ}, τ ∈ K̂) is restricted to the K -orbits, we obtain a set of fields on K/M. These fields are cross sections of homogeneous vector bundles E^{σ_{j}} over K/M, defined by irreps σ_{j} of M, with σ_{j} ⊂ τ|_{M}. In our case the above decomposition reads H^{N}(ℝ) ≃ ℝ^{+} × S^{N−1}, and the representations σ_{j} are all different. For example if τ is the defining vector representation of SO(N) in ℝ^{N}, then E^{τ} is the tangent bundle over H^{N}(ℝ). When we restrict a vector in H^{N}(ℝ) to S^{N−1}, we get a vector and a scalar, i.e., τ|_{M} = σ_{1} ⊕ σ_{2}, where σ_{1} is the trivial representation of SO(N), and σ_{2} is the vector one.

Now according to (53), the principal series that contain τ and enter in the decomposition of the induced representation ind_{K}^{G}(τ) (i.e., in the right hand side of (40)), are precisely the U^{σ_{j}λ}. In the previous example we have that the principal series contributing to the harmonic analysis of vector fields over H^{N}(ℝ) are U^{σ_{1}λ} and U^{σ_{2}λ}. It is possible to show that the vector valued functions on G given by f_{σ_{1}λ}(g) = P_{τ}U^{σ_{1}λ}(g^{−1})v (v any vector of H_{U}) correspond to pure gradients, whereas the functions f_{σ_{2}λ}(g) = P_{τ}U^{σ_{2}λ}(g^{−1})v correspond to divergencefree vectors (see [1] Prop. 4.1 and 4.2).

Concerning the discrete series, we have the following branching rule. A representation in the discrete series U_{±}^{σn_{0}} contains τ = (f_{1}, …, f_{k+1}) if and only if, in addition to (52), the following condition is satisfied (as before n_{j} ∈ ℤ (or ℤ+ \frac{1}{2} ) ⟺ f_{j} ∈ ℤ (or ℤ+ \frac{1}{2})):

\frac{1}{2} < n_{0} ≤ f_{1} ≤ n_{1}, for U_{+}^{σn_{0}},  (54)
\frac{1}{2} < n_{0} ≤ −f_{1} ≤ n_{1}, for U_{−}^{σn_{0}}.  (55)

Thus f_{1} must be nonzero (positive for U_{+}^{σn_{0}} , and negative for U_{−}^{σn_{0}} ). [See, e.g., ref. [24] Th.5 p.36, and Th.2 p.28, Case 2; in our notation |ν| + \frac{1}{2} = n_{0}, and Λ(ω)_{m} = f_{1}.]

Let N = 2k + 1. The branching rule for U^{σλ}|_{K} ⊃ τ = (f_{1}, …, f_{k}) is
|n_{1}| ≤ f_{1} ≤ n_{2} ≤ f_{2} ≤ … ≤ n_{k} ≤ f_{k}.  (56)

This is equivalent to τ|_{M} ⊃ σ, and (53) is again true (see [2] Th.2 p.228).

Example 4.1. Let τ be the trivial representation of Spin(N). Then E^{τ} is the trivial bundle, whose sections are the scalar functions on H^{N}(ℝ). In this case the spherical transform reduces to the Jacobi transform [17].

Since G has real rank one, 𝔞^{+} ≃ ℝ^{+}, and each a ∈ A can be written as a_{t} = exp(tH), where H is the element of 𝔞 such that β(H) = 1 (β the positive restricted root). In this normalization δ(a_{t}) = [sinh(t)]^{N−1} in (48). Moreover C_{0}^{∞}(G, τ, τ) and I_{0,τ}(G) both coincide with C_{0}^{#}(G), the set of compactly supported smooth functions on G which are biinvariant under K. The spherical transform (49) reduces to

f̂(λ) = ∫_{0}^{∞} f(a_{t})ϕ_{λ}(a_{t})(sinh t)^{N−1}dt, f ∈ C_{0}^{#}(G),  (57)

and we have the inversion formula [17]

f(a_{t}) = \frac{2^{N−2}}{π} ∫_{0}^{∞} f̂(λ)ϕ_{λ}(a_{t})|C(λ)|^{−2}dλ.  (58)

Here ϕ_{λ}(a_{t}) = F (iλ + ρ, −iλ + ρ, \frac{N}{2} , − sinh^{2} \frac{t}{2}) are the spherical functions (ρ ≡ (N − 1)/2, F is the hypergeometric function), and C(λ) is the Harish-Chandra function

C(λ) = \frac{2^{N−2}Γ(N/2)}{\sqrt{π}} \frac{Γ(iλ)}{Γ(iλ + ρ)}.  (59)

It is easy to see that (50) and (51) agree with (58) at t = 0. Indeed for any N, the highest weight of τ is (f_{j}) = (0, 0, …, 0). Thus there is no discrete series containing τ (a well known result, see [15] p.455). Since τ|_{M} is just the trivial representation of Spin(N − 1), we have σ = (n_{j}) = (0, 0, …, 0) ∀N. Thus the only representations in Ĝ which contain τ and have nonzero Plancherel measure are the spherical principal series U^{σλ}. The only term that survives in (50) and (51) (for f K -biinvariant) is the term with (l_{j}) = (\frac{1}{2}, \frac{3}{2}, …, k − \frac{1}{2}) for N = 2k +2, and (l_{j}) = (0, 1, …, k−1) for N = 2k+1. Using the given values of l_{j} in (50) and (51) we verify that the Plancherel measure as a function of λ is indeed proportional to |C(λ)|^{−2}. Moreover Θ_{σλ}(f) = f̂(λ) for f K -biinvariant [use (47) in (14), and compare with (57)]. By writing dμ(U^{σλ}) = p_{0}(λ)dλ, with scalar Plancherel density p_{0}(λ) = \frac{2^{N−2}}{π} |C(λ)|^{−2}, we can match eqs. (50) and (51) with eq. (58), to obtain the value of the normalization constant c of Hirai (which is independent of τ!). The result is

c = k! ∏_{s=1}^{k}(2s)!, N = 2k + 2,

c = 2^{k}(Γ(k + \frac{1}{2}))^{2} ∏_{s=1}^{k−1}(2s)!, N = 2k + 1.

Example 4.2. Consider Dirac spinors on H^{N}(ℝ). The spinor bundle E^{τ} is defined by

τ = τ_{+} ⊕ τ_{−} = (\frac{1}{2}, \frac{1}{2}, …, \frac{1}{2}) ⊕ (−\frac{1}{2}, \frac{1}{2}, …, \frac{1}{2}), N even,

τ = (\frac{1}{2}, \frac{1}{2}, … \frac{1}{2}), N odd,

i.e., τ is the fundamental spinor representation of Spin(N), for N odd, and it is the direct sum of the two fundamental spinor representations of Spin(N), for N even (see [2] p.222-224). Notice that for N even the bundle is reducible.

Using the above values of (f_{j}) in (54)-(55), we see that no discrete series contain τ. Using these values in (52) and (56), we find that τ is contained in the principal series U^{σλ} with

σ = (\frac{1}{2}, \frac{1}{2}, …, \frac{1}{2}), N even,

σ = σ_{+}, σ_{−}, σ_{±} = (±\frac{1}{2}, \frac{1}{2}, …, \frac{1}{2}), N odd.

By Frobenius Reciprocity, this is simply the statement that for N even one has τ_{±}|_{Spin(N−1)} = σ, whereas for N odd τ|_{Spin(N−1)} = σ_{+} ⊕ σ_{−}.

Thus for f ∈ I_{0,τ}(G) only the terms corresponding to the above principal series survive in (50) and (51). The Plancherel measure so obtained agrees with the analytic calculation performed in [3]. Let σ denote either σ_{+} or σ_{−} in the case of N odd. Then we find, for any N, dμ(U^{σλ}) = p_{σ}(λ)dλ, with the spinor Plancherel density

p_{σ}(λ) = \frac{2^{N−2}}{π} d_{σ}|C_{σ}(λ)|^{−2},

where d_{σ} = 2^{[N/2]−1} is the dimension of σ, and

C_{σ}(λ) = \frac{2^{N−2}Γ(N/2)}{\sqrt{π}} \frac{Γ(iλ + \frac{1}{2})}{Γ(iλ + \frac{N}{2} )}.

The spinor spherical functions for the principal series [i.e., Φ_{+}^{λ} = P_{τ_{+}}U^{σλ}P_{τ_{+}} and Φ_{−}^{λ} = P_{τ_{−}}U^{σλ}P_{τ_{−}} (for N even), and Φ^{λ±} = P_{τ}U^{σ_{±}λ}P_{τ} (for N odd)] have been calculated in [6]. As in the scalar case, the spinor Harish-Chandra function C_{σ}(λ) is determined by the asymptotic form at infinity of the (normalized) spherical trace functions ϕ_{λ}, given by (see [3, 6])

ϕ_{λ}(a_{t}) = cosh \frac{t}{2} F (iλ + \frac{N}{2} , −iλ + \frac{N}{2} , \frac{N}{2} , − sinh^{2} \frac{t}{2}).

Example 4.3. Let τ be the representation of Spin(N) defined by the highest weight (f_{j}) = (0, …, 0, s), where s is a nonnegative integer. Then E^{τ} is the bundle of totally symmetric traceless tensor fields of rank s over H^{N}(ℝ) (see [2] Th.6 p.301). Again no discrete series contain τ. From (52) and (56) we find that τ is contained in s+1 different principal series representations, namely in U^{σ_{r}λ}, where σ_{r} = (0, …, 0, r), and r = 0, …, s. The representations U^{σ_{s}λ} correspond to the symmetric traceless and transverse (i.e., divergence-free) (STT) tensor fields of rank s (see [4]).

The Plancherel measure for STT tensor fields has been calculated analytically in [4], and the two methods give the same result. From (50) and (51) we find dμ(U^{σ_{s}λ}) = p_{s}(λ)dλ, with density p_{s}(λ) = \frac{2^{N−2}}{π} d_{s}|C_{s}(λ)|^{−2}, where

d_{s} = \frac{(2s + N − 3)(s + N − 4)!}{s!(N − 3)!}

is the dimension of σ_{s}, and

C_{s}(λ) = \frac{2^{N−2}Γ(N/2)}{\sqrt{π}} \frac{Γ(iλ)}{(iλ+s+ρ−1)Γ(iλ+ρ−1)}.  (60)

For s = 0 this reduces to the scalar result (59). The spherical functions Φ_{s}^{σ_{r}λ} = P_{τ}U^{σ_{r}λ}P_{τ} (r = 0, …, s) can be obtained from the results of ref. [4].

Example 4.4. We now consider the example of p-forms (i.e., totally antisymmetric tensor fields of rank p) over H^{N}(ℝ) (see also [5]).

Let N = 2k + 2. The bundle of p-forms on H^{N}(ℝ) is defined by the following irreps τ of Spin(N) (see [2] Th.5 p.299):

p = 0, 2k + 2 : τ = (0, …, 0);
p = 1, 2k + 1 : τ = (0, …, 0, 1);
p = 2, 2k :  τ = (0, …, 0, 1, 1);
⋮
p = k, k + 2 :  τ = (0, 1, …, 1)
p = k + 1 :  τ = (1, …, 1) ⊕ (−1, 1, …, 1) ≡ τ_{+} ⊕ τ_{−}.

The bundles of p-forms and (N − p)-forms correspond to the same τ as a consequence of duality. Notice that for p = k + 1 = N/2 the bundle is reducible.

From (54)-(55) we see that the only p-forms contained in the discrete series are for p = k + 1, namely τ_{+} ⊂ U_{+}^{σn_{0}}|_{K}, and τ_{−} ⊂ U_{−}^{σn_{0}}|_{K}, where (σ, n_{0}) = (1, …, 1). This identifies in group theoretic terms the square-integrable harmonic k-forms on H^{2k} which may be found by a delicate spectral analysis of the Hodgede Rham operator Δ (see [9, 5]). Thus for N even and p = N/2 (and only in that case), Δ has discrete spectrum. The discrete part of the Plancherel measure (i.e., \frac{1}{c}P (l_{0}, …, l_{k}) in eq. (50)) is essentially the formal degree of the discrete series (see ref. [28] vol.II p. 407). In our case l_{j} = j + \frac{1}{2} , and a simple calculation gives

P (\frac{1}{2}, \frac{3}{2}, …, k + \frac{1}{2}) = \frac{(2k + 2)!}{(k + 1)!2^{2k+2}} ∏_{s=1}^{k}(2s)!

Concerning the principal series we obtain [using (52)] the following list of irreps σ = (n_{j}) of M = Spin(N − 1) such that U^{σλ}|_{K} ⊃ τ:

p = 0, 2k + 2 : σ = (0, …, 0);
p = 1, 2k + 1 : σ = \{ \begin{matrix}(0, …, 0), \\ (0, …, 0, 1); \end{matrix}

p = 2, 2k :  σ = \{ \begin{matrix}(0, …, 0, 1), \\ (0, …, 0, 1, 1); \end{matrix}
⋮
p = k, k + 2 :  σ = \{ \begin{matrix}(0, 1, …, 1), \\ (1, 1, …, 1); \end{matrix}
p = k + 1 :  σ = (1, 1, …, 1) for both τ_{±}.

For p = 0, 2k + 2 and p = k + 1 we have a “singlet”, for the other values of p we get a “doublet”. It is possible to show that for p = 1, …, k and for σ equal to the second member of each doublet, the vector valued functions on G given by f_{σλ}(g) = P_{τ}U^{σλ}(g^{−1})v (v ∈ H_{U}) correspond to coexact p-forms. When σ equals the first member of the doublet we get exact p-forms instead. For p = k + 2, …, 2k + 1, the role of the two members of a doublet is reversed, i.e., the first corresponds to coexact forms, and the second to exact ones.

Let N = 2k + 1. The discussion proceeds as before. For p = 0, 1, …, k, the irreps of Spin(N) defining p-forms are given by f_{j} = 0 (j = 1, …, |k − p|), and f_{j} = 1 (j = |k − p| + 1, …, k). For p = k + 1, …, 2k + 1, τ is the same as for (N − p)-forms. All bundles are now irreducible. Applying (56) for p = 0, …, k − 1 (and the corresponding (N − p)-forms), we get the same list of σ’s we had before. For p = k, k + 1 we obtain a “triplet”, namely σ = (ϵ, 1, …, 1), where ϵ = 0, ±1.

Again for p = 1, …, k−1 the second member of each “doublet” corresponds to coexact p-forms, while the first member of each doublet corresponds to exact p-forms. For p = k + 2, …, 2k the role of the members of each doublet gets reversed. For p = k the terms of the triplet with ϵ = ±1 (ϵ = 0) correspond to coexact (exact) k-forms, while for p = k + 1 it is the opposite.

The continuous part of the Plancherel measure may be written in unified form for any N, as follows. For p = 0, 1, …, [\frac{N−1}{2}], let σ_{p} be the irrep of Spin(N − 1) with highest weight

σ_{p} = (0, …, 0, \underbrace{1, …, 1}_{p−times}).  (61)

Then the continuous part of the Plancherel measure for coexact p-forms is given by dμ(U^{σ_{p}λ}) = μ_{p}^{CE}(λ)dλ, with density

μ_{p}^{CE}(λ) = \frac{2^{N−2}}{π} d_{p}|C_{p}^{CE}(λ)|^{−2},

where d_{p} is the dimension of σ_{p} [i.e., d_{p} = \binom{N−1}{p} for p ≠ \frac{N−1}{2} , and d_{p} = \frac{1}{2} \binom{N−1}{p} for p = \frac{N−1}{2} ], and

C_{p}^{CE}(λ) = \frac{2^{N−2}Γ(N/2)}{\sqrt{π}} \frac{(iλ+ρ−p) Γ(iλ)}{Γ(iλ+ρ+1)}.  (62)

For p = 0 we reobtain (59). For p = 1 (i.e., for divergence-free vector fields), (62) gives the same result as (60) for s = 1.

In all cases we verify the equality of the Plancherel measures for exact pforms and for coexact (p−1)-forms, and of those for coexact p-forms and coexact (N −p−1)-forms:

μ_{p}^{E}(λ) = μ_{p−1}^{CE}(λ), μ_{p}^{CE}(λ) = μ_{N−p−1}^{CE}(λ).

[This follows also by duality.] In particular, for N even and p = N/2, exact p-forms and coexact p-forms have the same Plancherel measure.

The spherical functions for p-forms can be deduced from the results of ref. [5], where the eigenfunctions (spherical or not) of the Hodge-de Rham Laplacian have been calculated by working in geodesic polar coordinates. The spherical functions have also been calculated by Pedon [23]. We give here their expression for p ≠ (N − 1)/2.

First let p ≠ \frac{N}{2} , \frac{N−1}{2} . Let τ_{p} be the irrep of K = Spin(N) with highest weight vector given by the right hand side of (61). Then τ_{p}|_{M} = σ_{p−1} ⊕ σ_{p}. This induces a direct sum decomposition V_{τ_{p}} = V_{σ_{p−1}} ⊕ V_{σ_{p}}. Define the τ_{p}-spherical functions

Φ_{p}^{σ_{p}λ}(g) = P_{τ_{p}}U^{σ_{p}λ}(g)P_{τ_{p}},
Φ_{p}^{σ_{p−1}λ}(g) = P_{τ_{p}}U^{σ_{p−1}λ}(g)P_{τ_{p}}.

As M centralizes A in K, we get from Schur’s lemma

Φ_{p}^{σ_{p}λ}(a_{t}) = γ_{λp}(t)1_{p−1} ⊕ δ_{λp}(t)1_{p},
Φ_{p}^{σ_{p−1}λ}(a_{t}) = α_{λp}(t)1_{p−1} ⊕ β_{λp}(t)1_{p},

where 1_{p−1} and 1_{p} denote the identity operators in V_{σ_{p−1}} and V_{σ_{p}}, respectively. Using the radial part of the Casimir operator one obtains a system of differential equations for the scalar functions γ_{λp}, δ_{λp} (or α_{λp}, β_{λp}). The solution is as follows:

γ_{λp}(t) = β_{λp}(t) = F (iλ + \frac{N + 1}{2}, −iλ + \frac{N + 1}{2}, 1 + \frac{N}{2} , − sinh^{2} \frac{t}{2}),

δ_{λp}(t) = α_{λ,N−p}(t) = \frac{1}{N − p}sinh t \frac{dγ_{λp}}{dt} + cosh t γ_{λp}(t),

where F is the hypergeometric function. Notice that γ_{λp} and β_{λp} are independent of p.

Now let p = N/2 (thus N is even). In this case the bundle of p-forms over H^{N}(ℝ) is reducible, namely we have τ = τ_{+} ⊕ τ_{−} and E^{τ} = E^{τ_{+}} ⊕ E^{τ_{−}}, where τ_{+} and τ_{−} are the representations of Spin(N) with highest weights

τ_{±} = (±1, 1, 1, …, 1),

and dimensions

d_{τ_{+}} = d_{τ_{−}} = \frac{1}{2} ( \begin{matrix}N \\ N/2 \end{matrix} ) .

We have the branching rule

τ_{+}|_{M} = τ_{−}|_{M} = σ = (1, …, 1).

Consider the spherical functions

Φ_{+}^{λ}(g) = P_{τ_{+}}U^{σλ}(g)P_{τ_{+}},
Φ_{−}^{λ}(g) = P_{τ_{−}}U^{σλ}(g)P_{τ_{−}}.

Schur’s Lemma gives

Φ_{±}^{λ}(a_{t}) = f_{λ±}(t)1_{±},

where f_{λ±} are scalar functions. Using the radial part of the Casimir operator one finds

f_{λ+}(t) = f_{λ−}(t) = cosh^{2}(\frac{t}{2})F (iλ + \frac{N + 1}{2}, −iλ + \frac{N + 1}{2}, \frac{N}{2} , − sinh^{2} \frac{t}{2}). (63)

The spherical functions for the discrete series are Φ_{+}(g) = P_{τ_{+}}U_{+}^{σn_{0}}(g)P_{τ_{+}} and Φ_{−}(g) = P_{τ_{−}}U_{−}^{σn_{0}}(g)P_{τ_{−}}, where (σ, n_{0}) = (1, …, 1). From Schur’s Lemma Φ_{±}(a_{t}) = f_{±}(t)1_{±}. The scalar functions f_{±}(t) can be obtained from f_{λ±}(t) in (63) by analytic continuation in λ, by letting λ = i/2 (or −i/2).

Indeed let us recall that the Casimir operator of G induces minus the Hodge-de Rham Laplacian, Ω_{G} = −Δ. For general p, the eigenvalues of Δ acting on the coclosed p-forms f_{σ_{p}λ}(g) = P_{τ_{p}}U^{σ_{p}λ}(g^{−1})v (v ∈ H_{U}) are given by ω_{λp} = λ^{2} + (ρ − p)^{2}, where ρ = (N − 1)/2 [5, 9]. Now for p = N/2, it is well known that the square-integrable p-forms corresponding to the discrete series are harmonic. [Moreover they are both closed and coclosed, in agreement with a general theorem of Andreotti and Vesentini, see [5].]

By the Subquotient Theorem, the discrete series U_{+}^{σn_{0}} is infinitesimally equivalent with a subquotient representation of a nonunitary principal series U^{σλ}, for suitable λ ∈ ℂ. [A similar statement holds for U_{−}^{σn_{0}} .] The condition ω_{λ,N/2} = 0 gives λ = ±i/2. Using this value in (63), we find

f_{+}(t) = f_{−}(t) = cosh^{2}(t/2)F (\frac{N}{2} + 1, \frac{N}{2} , \frac{N}{2} , − sinh^{2} \frac{t}{2}) = (cosh \frac{t}{2})^{−N}.

The case of p = (N − 1)/2 (N odd) is slightly more complicated, and will not be given here.

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Politecnico di Torino Corso Duca degli Abruzzi 24 10129 Torino, Italy Camporesi@polito.it

Received April 18, 1996