VOL. 71 1996 NO. 2
SPECTRA FOR GELFAND PAIRS
ASSOCIATED WITH THE HEISENBERG GROUP
BY
CHAL B E N S O N (ST. LOUIS, MISSOURI), JOE J E N K I N S (ALBANY, NEW YORK), GAIL R A T C L I F F (ST. LOUIS, MISSOURI) AND
TEFERA W O R K U (ALBANY, NEW YORK)
Let K be a closed Lie subgroup of the unitary group U (n) acting by au- tomorphisms on the (2n+1)-dimensional Heisenberg group H n . We say that (K, H n ) is a Gelfand pair when the set L 1 K (H n ) of integrable K-invariant functions on H n is an abelian convolution algebra. In this case, the Gelfand space (or spectrum) for L 1 K (H n ) can be identified with the set ∆(K, H n ) of bounded K-spherical functions on H n . In this paper, we study the natural topology on ∆(K, H n ) given by uniform convergence on compact subsets in H n . We show that ∆(K, H n ) is a complete metric space and that the
“type 1” K-spherical functions are dense in ∆(K, H n ). Our main result shows that one can embed ∆(K, H n ) quite explicitly in a Euclidean space by mapping a spherical function to its eigenvalues with respect to a certain finite set of (K ⋉ H n )-invariant differential operators on H n . This viewpoint on the spectrum for ∆(K, H n ) was previously known for K = U (n) and is referred to as “the Heisenberg fan”.
1. Introduction. Given a locally compact group G and compact sub- group K ⊂ G, the pair (G, K) is called a Gelfand pair if L 1 (G//K), the space of integrable, K-bi-invariant functions on G, is commutative. Per- haps the best known examples are those defining symmetric spaces, that is, when G is a connected semisimple Lie group with finite center, and K is a maximal compact subgroup. The analysis associated with such pairs plays an important role in the representation theory of semisimple Lie groups and has been extensively developed in the last four decades (cf. e.g. [9], [11]). In sharp contrast to this case, one might begin by assuming that G is a solvable Lie group. But then, if G is simply connected for example, there may be no non-trivial compact subgroups. One can, however, consider pairs of the
1991 Mathematics Subject Classification: Primary 22E30, 43A55.
[305]
form (K ⋉ G, K) where K is a compact subgroup of Aut(G), the group of automorphisms of G.
The Heisenberg group H n of dimension 2n+1 is defined in Section 2. The unitary group U (n) is a maximal compact connected subgroup of Aut(H n ).
Throughout this paper, K will denote a compact Lie subgroup of U (n) for which (K ⋉ H n , K) is a Gelfand pair. We say, by abuse of terminology, that “(K, H n ) is a Gelfand pair”. In this setting, L 1 (K ⋉ H n //K) can be identified with the algebra L 1 K (H n ) of integrable functions on H n that are invariant under the action of K. In [1], it is shown that the Gelfand pairs of this sort play a central role in the study of Gelfand pairs associated with more general solvable Lie groups.
Under the conditions described above, the set D K (H n ) of differential operators on H n which are simultaneously left-H n -invariant and K-invariant forms an abelian algebra. In Section 2 we construct a canonical set of generators {L γ 1 , . . . , L γ d , T } for this algebra. Here T is the derivative in the central direction of H n , and each L γ j is a polynomial coefficient differential operator of degree at least 2 involving derivatives in non-central directions.
The operators L γ j arise from the invariant theory for the action of K on the ring C[z 1 , . . . , z n ].
The Gelfand space (or spectrum) of the commutative Banach ∗-algebra L 1 K (H n ) is the set of continuous non-zero algebra homomorphisms from L 1 K (H n ) to C. This can be identified, via integration, with the set ∆(K, H n ) of bounded K-spherical functions on H n . These are the smooth bounded K-invariant functions ψ : H n → C which are joint eigenfunctions for the operators D ∈ D K (H n ) and are normalized to take the value 1 at the iden- tity. A description of the bounded K-spherical functions can be found in [2] and is summarized below in Section 2. These functions are of two types.
The K-spherical functions of type 1 yield non-trivial characters when re- stricted to the center of H n , whereas the K-spherical functions of type 2 are constant on the center. The latter can be regarded as functions defined on the quotient of H n by its center and reflect the abelian component of analysis on H n . For K = U (n), these can be expressed in terms of Bessel functions (see equation (3.4)). The K-spherical functions of type 1 reflect the non-abelian component of analysis on H n . For the case K = U (n), these can be expressed in terms of Laguerre polynomials (see equation (3.3)).
Our focus here is on the topology of the Gelfand space, where the usual weak ∗ -topology coincides with the compact-open topology on ∆(K, H n ).
Our main result is stated as Theorem 4.1. It asserts that a sequence (ψ N ) ∞ N =1
in ∆(K, H n ) converges to ψ ∈ ∆(K, H n ) if and only if the sequences of
eigenvalues for ψ N with respect to each generator for D K (H n ) converge to
the corresponding eigenvalue for ψ. We require a careful analysis of the
behavior of such eigenvalues, and these results are described in Section 3.
As a corollary to Theorem 4.1, we find that the map which sends ψ ∈
∆(K, H n ) to its eigenvalues with respect to (L γ 1 , . . . , L γ d , T ) is a homeo- morphism of the space ∆(K, H n ) onto its image in C d+1 . In Section 3 we show that the eigenvalues for the L γ j ’s are real numbers with constant sign and that the T -eigenvalues are pure imaginary numbers. Thus we obtain a map to (R + ) d × R with image F(K, H n ) homeomorphic to ∆(K, H n ). We call F(K, H n ) the Heisenberg fan for the Gelfand pair (K, H n ) and describe this explicitly as the union of a d-dimensional algebraic set in (R + ) d × {0}
and a countable family of 1-dimensional polynomial curves. This picture is derived from the invariant theory for the action of K on the ring of polyno- mials C[z 1 , . . . , z n ].
We refine our description of the topology on the Gelfand space by proving two final results. Proposition 4.5 asserts that ∆(K, H n ) is complete. That is, if a sequence of bounded K-spherical functions converges to some func- tion in the compact-open topology, then the limit is necessarily a bounded K-spherical function. Proposition 4.4 asserts that the K-spherical functions of type 1 are dense in ∆(K, H n ). Thus, every bounded K-spherical function is a limit in the compact-open topology of some sequence of K-spherical functions of type 1. Our proof of Proposition 4.4 uses the fact that the K- spherical functions of type 2 form a set of measure zero with respect to the Godement–Plancherel measure on ∆(K, H n ). Section 5 contains a descrip- tion of the Godement–Plancherel measure and the K-spherical transform.
Propositions 4.4 and 4.5 are clear for the case K = U (n). Theorem 4.1 is not obvious, but was also known in this case (cf. [4]). We have adopted the term “Heisenberg fan” from [7] and [17], which discuss the case when K = U (n). Our proof of Theorem 4.1 in Section 4 is, however, self-contained.
The main interest in the current paper is that our results encompass Gelfand pairs (K, H n ) for proper subgroups K of U (n).
2. Notation and preliminaries. We need to establish notation and recall some results concerning Gelfand pairs and spherical functions associ- ated with Heisenberg groups. For details concerning this preliminary mate- rial, we refer the reader to two earlier papers, [1] and [2], by the first three authors.
2.1. Heisenberg group. Given a complex vector space V of dimension n with Hermitian inner product h·, ·i, one forms the Heisenberg group H n = V × R with group law
(z, t)(z ′ , t ′ ) = z + z ′ , t + t ′ − 1 2 Imhz, z ′ i .
At times, it will be convenient to work in coordinates on V . One can use an
orthonormal basis to identify V with C n so that hz, z ′ i = z ·z ′ for z, z ′ ∈ C n .
The left-invariant vector fields generated by the one-parameter subgroups
through ((0, . . . , 0, 1 ± i, 0, . . . , 0) , 0) are written explicitly as Z j = 2 ∂
∂z j
+ i z j
2
∂
∂t , Z j = 2 ∂
∂z j − i z j
2
∂
∂t . In addition let
T := ∂
∂t ,
so that {Z 1 , . . . , Z n , Z 1 , . . . , Z n , T } is a basis for the Lie algebra h n of H n . With these conventions one has [Z j , Z j ] = −2iT .
2.2. Gelfand pairs. The group U (n) of unitary transformation of (V, h·, ·i) acts by automorphisms on H n via
k · (z, t) := (kz, t) for k ∈ U(n) and (z, t) ∈ H n .
This yields a maximal compact connected subgroup of Aut(H n ). If K is a compact Lie subgroup of U (n) then we say that (K, H n ) is a Gelfand pair when the algebra L 1 K (H n ) := {f ∈ L 1 (H n ) | f(kz, t) = f(z, t)} of integrable K-invariant functions on H n is commutative under convolution. The group U (n), and many proper subgroups K of U (n), yield Gelfand pairs. One can find a complete classification of all such subgroups in [14] and [3]. Under our identification of V with C n , U (n) can be regarded as the group of n × n unitary matrices.
2.3. Multiplicity free decomposition. An important ingredient in the theory of Gelfand pairs (K, H n ) is the representation of K on the space of polynomials C[V ] given by
k · p(z) := p(k −1 z).
One has the following result:
Theorem 2.1 (cf. [5], [1]). (K, H n ) is a Gelfand pair if and only if the representation of K on C[V ] is multiplicity free.
Throughout this paper, K will denote a closed Lie subgroup of U (n) for which (K, H n ) is a Gelfand pair. We decompose C[V ] into K-irreducible subspaces P α ,
(2.1) C [V ] = X
α∈Λ
P α ,
where Λ is some countably infinite index set. In view of Theorem 2.1,
this is a canonical decomposition. Since the representation of K on C[V ]
preserves the space P m (V ) of homogeneous polynomials of degree m, each
P α is a subspace of some P m (V ). For α ∈ Λ, we write |α| for the degree of
homogeneity of the polynomials in P α . Thus we have P α ⊂ P |α| (V ) and P m (V ) = X
|α|=m
P α .
2.4. Fock space. Fock space F consists of entire functions f : V → C which are square integrable with respect to e −|z| 2 /2 dz with Hilbert space structure
hf, gi F =
1 2π
n
\V
f (z)g(z)e −|z| 2 /2 dz.
Here “dz” denotes Lebesgue measure on the underlying real space V R ∼ = R 2n for V ∼ = C n . The holomorphic polynomials C[V ] form a dense subspace in F and one has the formula
hp, qi F = p
2 ∂
∂z
q(z)
z=0
for p, q ∈ C[V ],
where q(z) denotes the holomorphic polynomial obtained in coordinates by conjugating the coefficients of q. The monomials z a = z 1 a 1 . . . z n a n with |a| = a 1 + . . . + a n = m form an orthogonal basis for P m (V ) in F. One has kz a k F = √
2 |a| a!, where as usual a! := a 1 ! . . . a n !.
2.5. Invariant polynomials. Since the trivial representation of K occurs in C[V ] as P 0 (V ), and C[V ] is K-multiplicity free, there can be no non- constant K-invariant holomorphic polynomials. One does, however, have invariant polynomials on the underlying real vector space V R for V . We denote the set of these by C[V R ] K . More explicitly, given α ∈ Λ, let {v 1 , . . . . . . , v dim(P α ) } be any orthonormal basis for P α and define
(2.2) p α (z) :=
dim(P α )
X
j=1
v j (z)v j (z).
This definition of p α is independent of the basis chosen for P α . Further, p α is a K-invariant polynomial on V R homogeneous of degree 2|α|, and {p α | α ∈ Λ} is a vector space basis for C[V R ] K . Note that p α takes values in the set R + of non-negative reals.
A result due to Howe and Umeda (cf. [12]) shows that C[V R ] K is freely generated as an algebra. So there are polynomials γ 1 , . . . , γ d ∈ C[V R ] K so that
C [V R ] K = C[γ 1 , . . . , γ d ].
We call γ 1 , . . . , γ d the fundamental invariants. In fact, one can take, for each γ j , a p α for which P α contains a primitive highest weight vector. We let δ 1 , . . . , δ d ∈ Λ be the indices for which
γ j = p δ j .
One computes that
p m (z) := X
|α|=m
p α (z) = 1
2 m m! |z| 2m . Letting γ 0 (z) be defined as
γ 0 (z) = |z| 2 ,
this becomes p m = γ 0 m /(2 m m!). When K acts irreducibly on V , γ 0 will be one of the fundamental invariants. In general, one will have γ 0 = P
|α j |=1 γ j . 2.6. The value space. The invariants for a smooth linear action of a compact Lie group K on a real vector space separate K-orbits (cf. for example page 133 in [16]). Thus, the map
γ := γ 1 × . . . × γ d : V → (R + ) d
yields a bijection between the set V /K of K-orbits in V and the subset Γ K + := γ(V )
in (R + ) d . We call Γ K + the value space for K.
2.7. Invariant differential operators. The algebra D(H n ) of left-invariant differential operators on H n is generated by {Z 1 , . . . , Z n , Z 1 , . . . , Z n , T }. We denote the subalgebra of K-invariant differential operators by
D K (H n ) := {D ∈ D(H n ) | D(f ◦ k) = D(f) ◦ k for k ∈ K, f ∈ C ∞ (H n )}.
Since (K, H n ) is a Gelfand pair, D K (H n ) is an abelian algebra. A result of Thomas (cf. [18]) shows that the converse is also true, at least when K is connected.
Each K-invariant polynomial p ∈ C[V R ] K gives rise to a differential op- erator
L p := e S(p) ∈ D K (H n ).
Here e S = S ◦ j ◦ Ω, where Ω : C[V R ] → C[V R ∗ ] is the algebra isomorphism induced by the symplectic pairing − Imh·, ·i on V R , and j : C[V R ∗ ] ֒→ C[h ∗ n ] is inclusion and S : C[h ∗ n ] → D(H n ) is the symmetrization map. We have (2.3) (L p f )(z, t) = p
2 ∂
∂ζ , 2 ∂
∂ζ
f
z + ζ, t − 1
2 Imhz, ζi
ζ=0
for f ∈ C ∞ (H n ).
The operators L p can be written explicitly using coordinates as follows:
Let p(z) = P
c a ,b z a z b , where a, b are multi-indices and z a = z 1 a 1 . . . z n a n ,
z b = z b 1 1 . . . z b n n . Since p is K-invariant, it is easy to see that the operator
p(Z, Z) defined by p(Z, Z) := X
c a ,b Z a Z b = X
c a ,b Z 1 a 1 . . . Z n a n Z b 1 1 . . . Z b n n
belongs to D K (H n ). Note that the operator L p is intrinsically defined, whereas p(Z, Z) depends on the basis used to identify V with C n . One has
L p = Sym(p(Z, Z)), where Sym is the linear map characterized by
Sym(Z a Z b ) = 1 (|a| + |b|)!
X
σ∈S |a|+|b|
σ(Z a Z b ).
Here, as usual, |a| = a 1 +. . .+a n and σ(Z a Z b ) denotes the result of applying the permutation σ to the |a| + |b| terms in Z a Z b .
For a d-multi-index a, let γ a := γ 1 a 1 . . . γ d a d , kak := a 1 |δ 1 | + . . . + a d |δ d | (the homogeneous degree of γ a ), and L a γ := L a γ 1 1 . . . L a γ d
d . Using the definition of the map e S, together with the fact that [Z j , Z j ] = −2iT , one sees that
(2.4) L γ a = L a γ + X
kbk<kak
c a ,b L b γ T kak−kbk
for some coefficients c a ,b ∈ C. Since γ 1 , . . . , γ d generate C[V R ], it follows easily that {L γ 1 , . . . , L γ d , T } generates the algebra D K (H n ).
2.8. Spherical functions. A smooth function ψ : H n → C is called K-spherical if
(1) ψ is K-invariant,
(2) ψ is an eigenfunction for every D ∈ D K (H n ), and (3) ψ(0, 0) = 1.
Since L γ 1 , . . . , L γ d , T generate D K (H n ), (2) holds if and only if ψ is an eigenfunction for each of L γ 1 , . . . , L γ d , T . We write b D(ψ) for the eigenvalue of D ∈ D K (H n ) on a K-spherical function ψ, that is, D(ψ) = b D(ψ)ψ. Note that since ψ(0, 0) = 1, one has
(2.5) D(ψ) = D(ψ)(0, 0). b
We denote the set of positive definite K-spherical functions on H n by
∆(K, H n ). In [1], it is shown that every bounded K-spherical function
is positive definite, so ∆(K, H n ) is also the set of bounded K-spherical
functions. We remark that this result is not true for more general Gelfand
pairs (G, K) and contrasts with the situation for symmetric spaces. The
positive definite K-spherical functions ψ are the matrix coefficients obtained
using unit K-fixed vectors in the representation spaces for the K-spherical
representations of the semidirect product group K ⋉ H n . In [2], it is shown
that these functions can be described directly in terms of the representation theory for H n and the action of K on V .
One has an irreducible unitary representation π of H n on F defined as (π(z, t)f )(w) = e it−hw,zi/2−|z| 2 /4
f (w + z).
For α ∈ Λ let
(2.6) φ α (z, t) := 1 dim(P α )
dim(P α )
X
j=1
hπ(z, t)v j , v j i F ,
where {v 1 , . . . , v dim(P α ) } is an orthonormal basis for P α . This description of φ α does not depend on our choice of basis {v j }. Define φ λ,α for λ ∈ R × and α ∈ Λ by
(2.7) φ λ,α (z, t) := φ α
p |λ|z, λt ,
so that φ α = φ 1,α . The φ λ,α ’s are distinct bounded K-spherical functions.
We refer to these elements of ∆(K, H n ) as the spherical functions of type 1.
From equation (2.6) one can show that φ α has the general form φ α (z, t) = e it q α (z)e −|z| 2 /4 ,
where q α is a K-invariant polynomial on V R with homogeneous component of highest degree given by (−1) |α| p α /dim(P α ).
In addition to the K-spherical functions of type 1, there are K-spherical functions which arise from the one-dimensional representations of H n . For w ∈ V , let
(2.8) η w (z, t) :=
\
K
e i Rehw,kzi dk =
\
K
e i Rehz,kwi dk,
where “dk” denotes normalized Haar measure on K. The η w are the bounded K-spherical functions of type 2. Note that η 0 is the constant function 1 and η w = η w ′ if and only if Kw = Kw ′ . We have one K-spherical function for each K-orbit in V and sometimes write “η Kw ” in place of “η w ”.
It is shown in [2] that every bounded K-spherical function is of type 1 or type 2. Thus we have:
Theorem 2.2. The bounded K-spherical functions on H n are parame- terized by the set (R × × Λ) ∪ (V/K) via
∆(K, H n ) = {φ λ,α | λ ∈ R × , α ∈ Λ} ∪ {η Kw | w ∈ V }.
Note that, for ψ ∈ ∆(K, H n ), one has (2.9) ψ(z, t) = e iλt ψ(z, 0),
where λ = −i b T (ψ) ∈ R. We observe that ψ is of type 2 if and only if λ = 0.
3. Eigenvalues b D(ψ). The eigenvalue b T (ψ) of T on ψ ∈ ∆(K, H n ) is easily understood via equation (2.9). Eigenvalues of the form b L p (ψ) are much more subtle. In this section, we present a number of results concerning these eigenvalues. The first of these shows that b L p (ψ) can be computed by replacing L p with its “highest order terms”.
Lemma 3.1. For p ∈ C[V R ] K and ψ ∈ ∆(K, H n ), one has L b p (ψ) = ∂ p (ψ)(0, 0),
where ∂ p := p 2 ∂z ∂ , 2 ∂z ∂
. That is, ∂ p is the operator obtained by replacing each occurrence of z j in p by 2 ∂z ∂
j and each z j by 2 ∂z ∂
j . P r o o f. Using equations (2.5) and (2.3) we see that
L b p (ψ) = L p (ψ)(0, 0) = p
2 ∂
∂ζ , 2 ∂
∂ζ
ψ
0 + ζ, 0 − 1
2 Imh0, ζi
ζ=0
=
p
2 ∂
∂z , 2 ∂
∂z
ψ
(0, 0).
The eigenvalue for an operator of the form L p α on a K-spherical function of type 2 can be computed as follows. Recall that P α ⊂ P |α| , so that p α is homogeneous of degree 2|α|.
Lemma 3.2. b L p α (η w ) = (−1) |α| p α (w).
P r o o f. Using equation (2.8) together with Lemma 3.1 yields L b p α (η w ) = ∂ p α (η w )(0, 0) = ∂ p α
h
\K
e i Rehz,kwi dk i z=0 . One computes
∂ p α (e i Rehz,kwi )(z) = i 2|α| p α (kw)e i Rehz,kwi = (−1) |α| p α (w)e i Rehz,kwi . Thus, b L p α (η w ) = (−1) |α| p α (w)η w (0) = (−1) |α| p α (w).
Lemmas 3.3, 3.4 and 3.5 concern eigenvalues on K-spherical functions of type 1.
Lemma 3.3. For D ∈ D K (H n ), α ∈ Λ and all v ∈ P α one has π(D)v = b D(φ α )v.
P r o o f. This is Proposition 3.20 in [2]. For the reader’s convenience, we
outline the proof here. One observes that the operator π(D) on F preserves
C [V ] and commutes with the action of K. Since the decomposition given
by equation (2.1) is multiplicity free, π(D) must preserve each P α , and by
Schur’s Lemma, π(D)| P α is a scalar operator, c α I P α say. Equations (2.5)
and (2.6) now show that
D(φ b α ) = D(φ α )(0, 0) = 1 dim(P α )
dim(P α )
X
j=1
hπ(0, 0)π(D)v j , v j i F = c α .
Lemma 3.4. b L p α (φ λ,β ) = |λ| |α| L b p α (φ β ) for α, β ∈ Λ, λ ∈ R × .
P r o o f. Using Lemma 3.1 together with equation (2.7), we compute L b p α (φ λ,β ) = ∂ p α (φ λ,β )(0, 0) = p α
2 ∂
∂z , 2 ∂
∂z
φ β p
|λ|z, 0
z=0
= p
|λ| 2|α|
p α
2 ∂
∂z , 2 ∂
∂z
[φ β (z, 0)]
z=0
,
since p α is homogeneous of degree 2|α|. Thus, b L p α (φ λ,β ) = |λ| |α| ∂ p α (φ β )(0, 0)
= |λ| |α| L b p α (φ β ) as desired.
Lemma 3.5. b L p α (ψ) is a real number with sign (−1) |α| for all α ∈ Λ and ψ ∈ ∆(K, H n ).
P r o o f. In view of Lemmas 3.2 and 3.4, it suffices to show that b L p α (φ β ) is a real number with sign (−1) |α| for α, β ∈ Λ. We have
L p α = Sym(p α (Z, Z)) = 1 (2|α|)!
h p α (Z, Z) + X
|δ|<|α|
c α,δ p δ (Z, Z)(2iT ) |α|−|δ| i , for some coefficients c α,δ . The value of c α,δ is obtained by collecting terms after reordering the monomials arising from symmetrization, so that the Z j ’s precede the Z j ’s. Since Z j Z j = Z j Z j + 2iT , we see that c α,δ ≥ 0.
Thus we have (3.1) L b p α (φ β )
= 1
(2|α|)!
h p α (Z, Z) ∧ (φ β ) + X
|δ|<|α|
c α,δ p δ (Z, Z) ∧ (φ β )(−2) |α|−|δ| i .
Let {v 1 , . . . , v dim(P δ ) } be an orthonormal basis for P δ and let u β be a unit vector in P β . Using Lemma 3.3 and equation (2.2) we see that
p δ (Z, Z) ∧ (φ β ) = hπ(p δ (Z, Z))u β , u β i F = X
j
hπ(v j (Z))π(v j (Z))u β , u β i F
= X
j
hπ(v j (Z))u β , π(v j (−Z))u β i F
= (−1) |δ| X
j
kπ(v j (Z))u β )k 2 F .
Here we have used the fact that π(−Z j ) is the adjoint operator for π(Z j ) on F. This is equation (4.13) in [2] and follows easily from the formulas (3.2) π(Z j )f (z) = −z j f (z), π(Z j )f (z) = 2 ∂f
∂z j (z) for f ∈ F.
Thus, in equation (3.1), p α (Z, Z) ∧ (φ β ) is a real number with sign (−1) |α|
and each of the p δ (Z, Z) ∧ (φ β )’s are real numbers with sign (−1) |δ| . As the c α,δ ’s are non-negative real numbers, we conclude that b L p α (φ β ) is a real number with sign (−1) |α| .
The Gelfand pair (U (n), H n ) plays a special role in our proof of Theo- rem 4.1. The U (n)-spherical functions on H n are well known and have been computed independently by many authors. We refer the reader to [2] for one treatment and additional references. The U (n)-spherical functions of type 1 associated with P r (V ) (r = 0, 1, . . .) can be written as
(3.3) φ λ,r (z, t) = e iλt L (n−1) r
|λ|
2 |z| 2
e −|λkz| 2 /4 ,
where L (n−1) r is the Laguerre polynomial of order n − 1 and degree r nor- malized to have value 1 at z = 0. The U (n)-spherical functions of type 2 are η 0 (z, t) ≡ 1 and
(3.4) η w (z, t) = 2 n−1 (n − 1)!
(|wkz|) n−1 J n−1 (|wkz|),
for w 6= 0. Here J n−1 is the Bessel function of order n − 1 of the first kind.
We have L γ 0 = 1 2 P
j (Z j Z j + Z j Z j ), and using equations (3.2) one com- putes that π(L γ 0 ) = −2E − n, where E = P
j z j ∂
∂z j is the degree operator.
Thus, π(L γ 0 )| P r (V ) is the scalar operator −(2r + n). From Lemmas 3.3 and 3.4 we obtain the following result, which is, in any case, very well known.
Lemma 3.6. The eigenvalues for L γ 0 on the U (n)-spherical functions of type 1 are b L γ 0 (φ λ,r ) = −|λ|(2r + n).
The eigenvalues b L p (φ λ,r ) for operators L p arising from polynomials p ∈ C [V R ] U (n) (= C[γ 0 ]) of degree greater than 2 are more difficult to compute.
We provide the following estimate for b L p m (φ r ). As explained in Section 2.5, p m = γ 0 m /(2 m m!) is the U (n)-invariant polynomial obtained from P m (V ) via equation (2.2).
Lemma 3.7. |b L p m (φ r )| ≤ n+r+m−1 m
.
P r o o f. Using Lemma 3.3, we have b L γ 0 m (φ r ) = hπ(L γ 0 m )u r , u r i F , where u r is any unit vector in P r (V ). We take u r (z) := z r 1 /(2 r r!). One has
L γ 0 m = X
|a|=m
m!
a! Sym(Z a Z a ).
Applying π and using equations (3.2) we obtain L b γ 0 m (φ r ) = X
|a|=m
m!
a!
Sym
(−z) a
2 ∂
∂z
a u r , u r
F
= X
|a|=m
(−2) m m!
a!
Sym
z a
∂
∂z
a u r , u r
F
. One sees that
Sym
z a
∂
∂z
a u r , u r
F
≤
∂
∂z
a
z a u r , u r
F
= (a 1 + r)!
r! a 2 ! . . . a n !, where the quantity on the left is a positive real number. We obtain
|b L γ m 0 (φ r )| ≤ X
|a|=m
2 m m!
a!
(a 1 + r)!
r! a 2 ! . . . a n !
= m!2 m X
|a|=m
a 1 + r r
= m!2 m X m a 1 =0
a 1 + r r
m − a 1 + n − 2 n − 2
,
where the second binomial coefficient in the last expression counts the num- ber of terms (a 2 , . . . , a n ) with a 2 + . . . + a n = m − a 1 .
We have a sum of the general form P m
a 1 =0 A a 1 B m−a 1 , which is the coef- ficient of the mth term in the product of the following two series:
X ∞ j=0
A j x j = X ∞ j=0
r + j j
x j =
1 1 − x
r+1
, X ∞
j=0
B j x j = X ∞ j=0
j + n − 2 j
x j =
1 1 − x
n−1
. This product is
1
1 − x
n+r
= X ∞ j=0
n + r + j − 1 j
x j
and the mth term has coefficient n+r+m−1 m
. Thus we have shown that
|b L γ m
0 (φ r )| ≤ m!2 m
n + r + m − 1 m
. This completes the proof, since p m = γ 0 m /(m!2 m ).
The following lemma shows that the eigenvalues b L p α (φ λ,β ) for a Gelfand pair (K, H n ) are controlled by eigenvalues for the pair (U (n), H n ).
Lemma 3.8. For α, β ∈ Λ one has
|b L p α (φ β )| ≤ |b L p |α| (φ |β| )|.
Here the eigenvalue on the right hand side of this inequality is for the pair (U (n), H n ).
P r o o f. Letting m := |α|, we have p m = P
|δ|=m p δ and thus L b p m (φ β ) = X
|δ|=m
L b p δ (φ β ).
Lemma 3.5 shows that the b L p δ (φ β )’s are real numbers with common sign.
We conclude that
|b L p α (φ β )| ≤ |b L p m (φ β )|.
Since P β ⊂ P |β| (V ), Lemma 3.3 shows that both b L p m (φ β ) and b L p m (φ |β| ) are equal to hπ(L p m )u β , u β i F , where u β is any unit vector in P β . Thus,
|b L p α (φ β )| ≤ |b L p m (φ β )| = |b L p m (φ |β| )|.
The eigenvalues b L p α (ψ) for a K-spherical function ψ are related to the coefficients in a Taylor series expansion for ψ. One can find results closely related to Proposition 3.9 in [19].
Proposition 3.9. For ψ ∈ ∆(K, H n ), one has ψ(z, 0) = X
δ∈Λ
L b p δ (ψ) dim(P δ ) p δ (z),
where the series converges absolutely and uniformly on compact subsets in V . Thus we have the following series expansions for the K-spherical functions of types 1 and 2 respectively:
φ λ,α (z, t) = e iλt X
δ∈Λ
|λ| |δ| L b p δ (φ δ ) dim(P δ ) p δ (z), η w (z, t) = X
δ∈Λ
(−1) |δ| p δ (w) dim(P δ ) p δ (z).
Here , convergence is absolute and uniform on compact subsets in H n .
P r o o f. The expansions for φ λ,α (z, t) and η w (z, t) follow immediately from that for ψ(z, 0) together with Lemmas 3.4 and 3.2. It is a general fact that the spherical functions for a Gelfand pair (G, K) are real analytic (cf. Proposition 1.5.15 in [9]). For pairs of the form (K, H n ), one can see this directly from the functional forms of the two types of K-spherical functions. Write the Taylor series expansion of ψ(z, 0) centered at z = 0 as ψ(z, 0) = P ∞
m=0 h m (z), where h m (z) is a homogeneous polynomial of degree m on V R (i.e. in the variables (z, z)). Since ψ is K-invariant, one sees by K-averaging this expression that each h m is K-invariant. As {p δ | δ ∈ Λ} is a basis for C[V R ] K , we can rewrite the Taylor series as
ψ(z, 0) = X
δ∈Λ
c δ p δ (z)
for some coefficients c δ . Note that since Taylor series converge absolutely, it is not necessary to specify an ordering on the set Λ of indices for this sum.
We use Lemma 3.1 and perform term-wise differentiation of this Taylor series to obtain
L b p α (ψ) = ∂ p α (ψ)(0, 0) = X
δ∈Λ
c δ ∂ p α (p δ )(0).
Let {v 1 , . . . , v dim(P α ) } be an orthonormal basis for P α and {u 1 , . . . , u dim(P δ ) } be an orthonormal basis for P δ . Thus p α = P
v j (z)v j (z), p δ = P
u i (z)u i (z) and
∂ p α (p δ )(0) = X
i,j
v j
2 ∂
∂z
u i (z)v j
2 ∂
∂z
u i (z)
z=0
= X
i,j
|hv j , u i i F | 2 =
0 if δ 6= α, dim(P α ) if δ = α.
Hence b L p α (ψ) = c α dim(P α ) and c δ = b L p δ (ψ)/dim(P δ ). Since this is a Taylor series, the convergence is absolute and uniform on compact sets.
It is well known that a spherical function is completely determined by its spectrum of eigenvalues. Proposition 3.9 shows explicitly how ψ ∈ ∆(K, H n ) is determined by { b D(ψ) | D ∈ D K (H n )}. Since L γ 1 , . . . , L γ d , T generate D K (H n ), it follows that ψ is in fact determined by (b L γ 1 (ψ), . . . , b L γ d (ψ), T (ψ)). Indeed, equation (2.4) shows that eigenvalues of the form b b L γ a (ψ) are determined by the eigenvalues of the sort (L b γ ) ∧ (ψ) = b L γ 1 (ψ) b 1 . . . b L γ d (ψ) b d together with b T (ψ). Since each p δ is a linear combination of γ a ’s, we obtain the following corollary from Proposition 3.9.
Corollary 3.10. A K-spherical function ψ ∈ ∆(K, H n ) is completely determined by the d + 1 eigenvalues
(b L γ 1 (ψ), . . . , b L γ d (ψ), b T (ψ)).
4. The topology on ∆(K, H n ). Given a Gelfand pair (K, H n ), one can consider three topological spaces:
• The set ∆(K, H n ) of bounded K-spherical functions on H n endowed with the compact-open topology. That is, we give ∆(K, H n ) the topol- ogy of uniform convergence on compact sets.
• The Gelfand space (or spectrum) ∆(L 1 K (H n )) of the commutative Ba- nach algebra L 1 K (H n ). This is the set of continuous non-zero alge- bra homomorphisms from L 1 K (H n ) to C regarded as a subspace of L 1 K (H n ) ∗ with the weak ∗ -topology.
• The set b G K of K-spherical representations of G := K ⋉ H n . This is the set of irreducible unitary representions of G that possess cyclic K-fixed vectors. We give b G K the Fell topology.
Since bounded K-spherical functions are necessarily positive definite, we have a natural bijection between b G K and ∆(K, H n ). Moreover, inte- gration produces a bijection between ∆(K, H n ) and ∆(L 1 K (H n )). That is, ψ ∈ ∆(K, H n ) yields a continuous algebra homomorphism L 1 K (H n ) → C via
f 7→
\
H n
f (z, t)ψ(z, t) dz dt.
It is standard that these set bijections are homeomorphisms when
∆(K, H n ), ∆(L 1 K (H n )) and b G K are endowed with the natural topologies de- scribed above. Moreover, it is known that these spaces are locally compact, second countable and metrizable. The study of these topologies amounts to the following analytic question:
Under what conditions does a given sequence (ψ N ) ∞ N =1 of K-spherical functions converge uniformly on compact sets to a given K-spherical func- tion ψ?
The following theorem answers this question and is our main result.
Theorem 4.1. Let (ψ N ) ∞ N =1 be a sequence of K-spherical functions and ψ ∈ ∆(K, H n ). Then ψ N converges to ψ in the topology of ∆(K, H n ) (i.e. uniformly on compact sets) if and only if b L γ j (ψ N ) → b L γ j (ψ) for j = 1, . . . , d, and b T (ψ N ) → b T (ψ).
P r o o f. Suppose that (ψ N ) ∞ N =1 converges uniformly to ψ on compact
subsets of H n . Since ψ N (0, t) = eb T (ψ N )t and ψ(0, t) = eb T (ψ)t , we must
have b T (ψ N ) → b T (ψ). The series expansions for ψ N (z, 0) and ψ(z, 0) given
in Proposition 3.9 ensure that b L p δ (ψ N ) → b L p δ (ψ) for each δ ∈ Λ. Since
{γ 1 , . . . , γ d } ⊂ {p δ | δ ∈ Λ}, this shows in particular that b L γ j (ψ N ) → b L γ j (ψ)
for j = 1, . . . , d.
Conversely, suppose that b L γ j (ψ N ) → b L γ j (ψ) for j = 1, . . . , d and b T (ψ N )
→ b T (ψ). It follows that b L p (ψ N ) → b L p (ψ) for every p ∈ C[V R ] K . Indeed, each p ∈ C[V R ] K is a linear combination of monomials γ a in the fundamental invariants, and from equation (2.4) one obtains
N →∞ lim (L γ a ) ∧ (ψ N ) = lim
N →∞
h L b a γ (ψ N ) + X
kbk<kak
c a ,b L b b γ (ψ N ) b T (ψ N ) kak−kbk i
= b L a γ (ψ) + X
kbk<kak
c a ,b L b b γ (ψ) b T (ψ) kak−kbk = (L γ a ) ∧ (ψ).
It suffices to consider two cases:
(1) Each ψ N is a K-spherical function of type 1.
(2) Each ψ N is a K-spherical function of type 2.
Indeed, if (ψ N ) ∞ N =1 contains infinitely many terms of both types, then one reduces the situation to these cases by forming the subsequences consisting of the terms of each type.
We begin with the easier case (2) and write ψ N = η w N . Since b T (ψ) = lim b T (ψ N ) = 0, ψ must be a K-spherical function of type 2, ψ = η w
say. Lemma 3.2 shows that b L γ j (η w N ) = (−1) |δ j | γ j (w N ) and b L γ j (η w ) = (−1) |δ j | γ j (w). We conclude that γ(w N ) → γ(w) in the value space Γ K + and hence Kw N → Kw in V/K. Equation (2.8) shows that η w N and η w can be obtained via integration over the K-orbits Kw N and Kw. It follows easily that η w N converges to η w uniformly on compact sets.
Next consider case (1). Let ψ N = φ λ N ,α N and let S be a given compact subset of H n . In case (1), ψ can be a K-spherical function of either type.
Suppose first that ψ is of type 2 and write ψ = η w . (Later we will also treat the situation where ψ is a K-spherical function of type 1.)
Choose a constant c 1 large enough so that γ 0 (z) = |z| 2 ≤ c 1 for all (z, t) ∈ S. We write r N for |α N | and consider the tail of the series expansion for φ λ N ,α N consisting of the terms with indices δ satisfying |δ| ≥ M. For M ≥ 2 and all (z, t) ∈ S, we obtain the following bound:
X
δ∈Λ
|δ|≥M
|b L p δ (φ α N )|
dim(P δ ) p δ (z)|λ N | |δ| ≤ X ∞ m=M
|b L p m (φ r N )| X
|δ|=m
p δ (z)|λ N | m
≤ X ∞ m=M
m + n + r N − 1 m
1
2 m m! γ m 0 (z)|λ N | m
≤ 1
2 M M ! X ∞ m=M
m + n + r N − 1 m
|c 1 λ N | m
≤ 1 2 M M !
X ∞ m=2
m + n + r N − 1 m
|c 1 λ N | m
= R 2 (|c 1 λ N |)
2 M M ! say.
We have used Lemmas 3.7 and 3.8 here, together with the identity P
|δ|=m p δ
= γ 0 m /(2 m m!).
We now regard R 2 (x) as a remainder term for the Taylor series for (1 − x) −(n+r N ) . Computing the integral form for this remainder term yields
R 2 (x) =
x
\
0
(n + r N )(n + r N + 1)(x − t)
(1 + t) n+r N +1 dt ≤ (n + r N )(n + r N + 1)x 2 , provided 0 < x < 1. Since b T (φ λ N ,α N ) = iλ N converges to b T (η w ) = 0, we have λ N → 0. Thus for N sufficiently large, N ≥ N 0 say, we have 0 < |c 1 λ N | < 1 and hence
R 2 (|c 1 λ N |) ≤ (n + r N )(n + r N + 1)|c 1 λ N | 2 .
Moreover, we have b L γ 0 (φ λ N ,α N ) = b L γ 0 (φ λ N ,r N ) = −|λ N |(2r N + n) by Lem- ma 3.6 and since b L γ 0 (φ λ N ,α N ) converges to b L γ 0 (η w ), we conclude that (|λ N |r N ) is convergent. We now see, in particular, that both (λ N ) and (|λ N |r N ) are bounded sequences and hence
(n + r N )(n + r N + 1)|c 1 λ N | 2 ≤ c 2
for some constant c 2 and all N . Thus, for all N ≥ N 0 and all (z, t) ∈ S we have
X
δ∈Λ
|δ|≥M
|b L p δ (φ λ N ,α N )|
dim(P δ ) p δ (z) ≤ c 2
2 M M ! .
This bound ensures that the series converges uniformly to η w on compact subsets of H n .
Finally, suppose that ψ N = φ λ N ,α N and ψ N → ψ, where ψ = φ λ,α is a K-spherical function of type 1. Since b T (φ λ N ,α N ) = iλ N converges to T (φ b λ,α ) = iλ, we have λ N → λ. Since b L γ 0 (φ λ N ,α N ) = |λ N |b L γ 0 (φ |α N | ) =
|λ N |(2|α N | + n) converges to b L γ 0 (φ λ,α ) = |λ|(2|α| + n), also r N := |α N | must converge to |α|. Thus, both (λ N ) and (r N ) are bounded sequences and we choose constants c 1 , c 2 with
|λ| ≤ c 1 , 0 ≤ r N ≤ c 2
for all N . Choose constants c 3 and c 4 with γ 0 (z) = |z| 2 ≤ c 3 for all (z, t) ∈ S and
γ 0 (z) m |λ N | m
m! ≤ (c 3 c 1 ) m m! ≤ c 4
2 m
for all m, N and all (z, t) ∈ S. As before, we obtain, for (z, t) ∈ S,
e iλ N t X
δ∈Λ
|δ|≥M
L b p δ (φ λ N ,α N ) dim(P δ ) p δ (z)
≤ X ∞ m=M
m + n + r N − 1 m
1
2 m m! γ 0 m (z)|λ N | m
≤ c 4
2 M X ∞ m=M
m + n + r N − 1 m
1 2
m
≤ c 4
2 M X ∞ m=0
m + n + r N − 1 m
1 2
m
= c 4
2 M 2 n+r N ≤ c 5
2 M ,
where c 5 := c 4 2 n+c 2 . From this, we conclude that φ λ N ,α N → φ λ,α uniformly on S by reasoning as in the case where φ N = φ λ N ,α N and ψ = η w .
For the Gelfand pair (U (n), H n ), one has the single fundamental invari- ant γ 0 (z) = |z| 2 . Theorem 4.1, together with Lemma 3.6, shows that a sequence (φ λ N ,r N ) ∞ N =1 of U (n)-spherical functions of type 1 converges in
∆(U (n), H n ) to a U (n)-spherical function η w of type 2 if and only if λ N → 0 and |λ N |(2r N + n) → |w| 2 .
These spherical functions are given explicitly by equations (3.3) and (3.4).
From this we obtain, for example, the following corollary to Theorem 4.1.
Corollary 4.2. The sequence x n−1
2 n−1 (n − 1)! L (n−1) m
x 2
2(2m + n)
e − 4(2m+n) x2
converges to J n−1 (x) uniformly on compact subsets of R + = {x ∈ R | x ≥ 0}.
The reader will find classical viewpoints on Corollary 4.2 discussed on page 92 in [7].
One can obtain a clearer understanding of the topology on ∆(K, H n ) by recasting Theorem 4.1 as follows. Consider the map E : ∆(K, H n ) → C d+1 defined by
E(ψ) := ((−1) |δ 1 | L b γ 1 (ψ), . . . , (−1) |δ d | L b γ d (ψ), −i b T (ψ)).
Since b L γ j (ψ) is a real number with sign (−1) |δ j | and b T (ψ) is pure imaginary,
E takes values in (R + ) d × R. We denote the image of E by F(K, H n ) := E(∆(K, H n )) ⊂ (R + ) d × R
and give this the subspace topology from (R + ) d × R. We call F(K, H n ) the Heisenberg fan for (K, H n ). In view of Corollary 3.10, the map E is a bijection between the sets ∆(K, H n ) and F(K, H n ). Since both ∆(K, H n ) and F(K, H n ) are metrizable spaces, Theorem 4.1 shows that both E and E −1 are continuous maps. Thus we have proved the following theorem.
Theorem 4.3. The map E is a homeomorphism between ∆(K, H n ) and the Heisenberg fan F(K, H n ).
Lemma 3.2 shows that the image under E of the K-spherical functions of type 2 is precisely the value space Γ K + in (R + ) d :
E({η w | w ∈ V }) = Γ K + × {0}.
Lemma 3.4 shows that for fixed α ∈ Λ we have
E({φ λ,α | λ ∈ R × }) = {(κ α,1 |λ| |δ 1 | , . . . , κ α,d |λ| |δ d | , λ) | λ ∈ R × }, where κ α,j := |b L γ j (φ α )|. This is the disjoint union of two algebraic curves,
E({φ λ,α | λ ∈ R × }) = L + α ∐ L − α , where
L ± α = {(κ α,1 λ |δ 1 | , . . . , κ α,d λ |δ d | , ±λ) | λ > 0}.
Thus we see that F(K, H n ) is the algebraic set in (R + ) d × R consisting of the value space Γ K + together with a countable family of curves:
F(K, H n ) = (Γ K + × {0}) ∐ a
α∈Λ
L + α
∐ a
α∈Λ
L − α
. F(U (n), H n ) is depicted in Figure 1.
Fig. 1. The Heisenberg fan for (U (n), H n )
One has
F(U(n), H n ) = (R + × {0}) ∐ a ∞
m=0
L + m
∐ a ∞
m=0
L − m
, where L ± m are the lines
L ± m = {((2m + n)λ, ±λ) | λ > 0}
of slope ±1/(2m + n) in R + × R. The Heisenberg fan for (U(n), H n ) is described in [7] and in [17], although the latter paper does not explicitly discuss the topology on ∆(U (n), H n ). A proof that E : ∆(U (n), H n ) → F(U (n), H n ) is a homeomorphism is implicit in Bougerol’s paper [4], which includes a description of a system of open neighborhoods for η 0 ≡ 1 in
∆(U (n), H n ). Thus Theorems 4.1 and 4.3 are known results for the case K = U (n).
The following result ensures that each K-spherical function η w of type 2 is the limit in ∆(K, H n ) of some sequence (φ λ N ,α N ) ∞ N =1 of K-spherical functions of type 1. Proposition 4.4 is clear for the case K = U (n) but is rather surprising when K 6= U(n). In these cases, Γ K + has dimension d > 1 but the L ± α ’s are one-dimensional. The result reflects the manner in which the countable family of L ± α ’s sit over the value space in F(K, H n ).
Proposition 4.4. The K-spherical functions of type 1 are dense in the space ∆(K, H n ).
Our proof for Proposition 4.4 requires the spherical transform and will be given in Section 5. Proposition 4.5 complements Proposition 4.4. If a sequence in ∆(K, H n ) converges uniformly on compact subsets of H n then the limit is necessarily a bounded K-spherical function.
Proposition 4.5. ∆(K, H n ) is a complete metric space. That is, if (ψ N ) ∞ N =1 is a sequence of bounded K-spherical functions that converges uni- formly to ψ on compact subsets in H n then ψ is a bounded K-spherical function.
P r o o f. It is clear that ψ is continuous, K-invariant and bounded and that ψ(0, 0) = 1. Moreover, if f, g ∈ L 1 K (H n ) have compact support then
\
H n
ψ(z, t)(f ∗ g)(z, t) dz dt = lim N →∞
\
H n
ψ N (z, t)(f ∗ g)(z, t) dz dt
= lim
N →∞
\
H n
ψ N (z, t)f (z, t) dz dt
\
H n
ψ N (z, t)g(z, t) dz dt
=
\
H n
ψ(z, t)f (z, t) dz dt
\
H n
ψ(z, t)g(z, t) dz dt.
Thus f 7→
T
H n ψ(z, t)f (z, t) dz dt defines a continuous non-zero algebra ho- momorphism L 1 K (H n ) → C. It follows that ψ ∈ ∆(K, H n ).
5. The K-spherical transform. The K-spherical transform for f ∈ L 1 K (H n ) is the function
f : ∆(K, H b n ) → C, f (ψ) := b
\
H n
f (z, t)ψ(z, t) dz dt.
Here “dzdt” denotes Haar measure for the group H n , which is simply Eu- clidean measure on V R × R. One has
(5.1) (f ∗ g) ∧ (ψ) = b f (ψ)bg(ψ) and (f ∗ ) ∧ (ψ) = b f (ψ) for f, g ∈ L 1 K (H n ), ψ ∈ ∆(K, H n ), where f ∗ (z, t) := f (−z, −t).
The compact-open topology on ∆(K, H n ) is the smallest topology that makes all of the maps { b f | f ∈ L 1 K (H n )} continuous. Since L 1 K (H n ) is a Banach ∗-algebra with respect to the involution f 7→ f ∗ , it follows that b f belongs to the space C 0 (∆(K, H n )) of continuous functions on ∆(K, H n ) that vanish at infinity. Moreover, we have
k b f k ∞ ≤ kfk 1
for f ∈ L 1 K (H n ). This follows immediately from the fact that for ψ ∈
∆(K, H n ) one has |ψ(z, t)| ≤ ψ(0, 0) = 1, since ψ is positive definite.
Godement’s Plancherel Theory for Gelfand pairs (G, K) (cf. [10], or Section 1.6 in [9]) ensures that there exists a unique positive Borel measure dµ on the space ∆(K, H n ) for which
(5.2)
\
H n
|f(z, t)| 2 dz dt =
\
∆(K,H n )
| b f (ψ)| 2 dµ(ψ)
for all continuous functions f ∈ L 1 K (H n ) ∩ L 2 K (H n ). If f ∈ L 1 K (H n ) ∩ L 2 K (H n ) is continuous and b f is integrable with respect to dµ then one has the Inversion Formula
(5.3) f (z, t) =
\
∆(K,H n )
f (ψ)ψ(z, t) dµ(ψ). b
In particular, this formula holds when f is continuous, positive definite and K-invariant. Moreover, the spherical transform f 7→ b f extends uniquely to an isomorphism between L 2 K (H n ) and L 2 (∆(K, H n ), dµ).
The following result makes the Godement–Plancherel measure on
∆(K, H n ) explicit. Given F : ∆(K, H n ) → C, we write F (w) and F (λ, α) in place of F (η w ) and F (φ λ,α ) respectively. The reader can find another proof of Theorem 5.1 in [19]. The result for K = U (n) is also discussed in [7]
and [17].
Theorem 5.1. The Godement–Plancherel measure dµ on ∆(K, H n ) is given by
\
∆(K,H n )
F (ψ) dµ(ψ) =
\
R ×
X
α∈Λ
dim(P α )F (λ, α)|λ| n dλ.
P r o o f. For λ ∈ R × let π λ be the irreducible unitary representation of H n on F defined by π λ (z, t) := π( p
|λ|, λt). One has the formula (cf. [2]) φ λ,α (z, t) =
\
K
hπ λ (kz, t)u α , u α i F dk, where u α is any unit vector in P α .
Since existence and uniqueness of the Godement–Plancherel measure is guaranteed, we need only verify that equation (5.2) holds for dµ as in the statement of the theorem and all f in a dense self-adjoint subalgebra of C(H n ) ∩ L 1 K (H n ) ∩ L 2 K (H n ). For f continuous, K-invariant and of compact suppport, we compute
hπ λ (f ∗ f ∗ )u α , u α i F =
\
H n
(f ∗ f ∗ )(z, t)hπ λ (z, t)u α , u α i F dz dt
=
\
H n
\
K
(f ∗ f ∗ )(k −1 z, t)hπ λ (z, t)u α , u α i F dk dz dt
=
\
H n
\
K
(f ∗ f ∗ )(z, t)hπ λ (kz, t)u α , u α i F dk dz dt
=
\
H n
(f ∗ f ∗ )(z, t)φ λ,α (z, t) dz dt
= (f ∗ f ∗ ) ∧ (λ, α) = | b f (λ, α)| 2 . Thus we see that tr(π λ (f ∗ f ∗ )) = P
α∈Λ dim(P α )| b f (λ, α)| 2 . The Plancherel formula for H n (cf. pages 37–39 in [8]) now gives
\
H n
|f(z, t)| 2 dz dt =
\
R ×
tr(π λ (f ∗ f ∗ ))|λ| n dλ
=
\
