Generalized Bernstein--Reznikov integrals

0.37 MB·

by  Jean-Louis Clerc

Generalized Bernstein–Reznikov integrals
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Jean–Louis Clerc, Toshiyuki Kobayashi, Bent Ørsted, Michael Pevzner
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n
a
J

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Abstract
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] We find a closed formula for the triple integral on spheres in
A R2n R2n R2n whose kernel is given by powers of the standard
C × ×
symplectic form. This gives a new proof to the Bernstein–Reznikov
.
h integral formula in the n = 1 case. Our method also applies for linear
at and conformal structures.
m
[
1 Triple product integral formula

4
v
4 We consider the symplectic form [ , ] on R2n = Rn Rn given by
7 ⊕
8
2 [(x,ξ),(y,η)] := x,η + y,ξ . (1.1)
−h i h i
.
6
0 In this paper we prove a closed formula for the following triple integral:
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0 Theorem 1.1. Let dσ be the Euclidean measure on the sphere S2n−1. Then,
:
v
Xi α−n β−n γ−n
[Y,Z] 2 [Z,X] 2 [X,Y] 2 dσ(X)dσ(Y)dσ(Z)
ar ZS2n−1×S2n−1×S2n−1
= 2πn−21 3Γ 2−n4(cid:12)(cid:12)+α Γ (cid:12)(cid:12)2−n4+(cid:12)(cid:12)β Γ 2−(cid:12)(cid:12)n4+γ(cid:12)(cid:12)Γ δ+4n(cid:12)(cid:12) .
Γ(n)Γ n−λ1 Γ n−λ2 Γ n−λ3
(cid:0) (cid:1) (cid:0)2 (cid:1) 2(cid:0) (cid:1)2(cid:0) (cid:1)
(cid:0) (cid:1)
Here, α = λ λ λ(cid:0), β =(cid:1) (cid:0)λ +(cid:1)λ(cid:0) λ(cid:1), γ = λ λ + λ , δ =
1 2 3 1 2 3 1 2 3
− − − − − −
λ λ λ = α+β +γ.
1 2 3
− − −
2000 Mathematics Subject Classifications: Primary 42C05; Secondary 11F67, 22E45,
33C20, 33C55.
1
The integral converges absolutely if and only if (λ ,λ ,λ ) C3 lies in
1 2 3
∈
the following non-empty open region (see Proposition 6.8) defined by:
Reα > n 2, Reβ > n 2, Reγ > n 2 (n 2),
− − − ≥
Reα > n 2, Reβ > n 2, Reγ > n 2, Reδ > 1 (n = 1).
− − − −
The integral under consideration extends as a meromorphic function of λ ,λ
1 2
and λ ([3, Theorem 2], [15], see also [7]). A special case (n = 1) of Theorem
3
1.1 was previously established by J. Bernstein and A. Reznikov [5].
The strategy of our proof is to interpret the triple product integral as
the trace of a certain integral operator, for which we will find an explicit
formula of eigenvalues and their multiplicities. For this, our approach uses
the Fourier transform in the ambient space, appeals to the classical Bochner
identity, and finally reduces it to the special value of the hypergeometric
function F . It gives a new proof even when n = 1.
5 4
Sections 2 and 3 are devoted to the proof of Theorem 1.1. In Section 4,
we discuss analogous integrals of the triple product kernels involving x y λ
| − |
or x,y λ instead of [x,y] λ.
|h i|
The underlying symmetries for Theorem 1.1 are given by the symplectic
(cid:12) (cid:12)
group Sp(n,R) of any(cid:12)rank(cid:12)n, whereas those of Theorem 4.2 correspond to
the rank one group SO (m+1,1). Even in rank one case, our methods give
0
a new and simple proof of the original results due to Deitmar [6] for the case
x y λ (see Theorem 4.2). Section 5 highlights some general perspectives
| − |
from representation theoretic point of view.
In Sections 2 and 6 we have made an effort, following questions of the
referee, to explain the role of meromorphic families of homogeneous distri-
butions and the precise condition for the absolute convergence of the triple
integral, respectively.
Notations: N = 0,1,2,... , R = x R : x > 0 .
+
{ } { ∈ }
2 Eigenvalues of integral transforms
µ
T
We introduce a family of linear operators that depend meromorphically on
µ C by
∈
: C∞(S2n−1) C∞(S2n−1)
µ
T →
defined by
−µ−n
( f)(η) := f(ω) [ω,η] dσ(ω). (2.1)
µ
T
ZS2n−1
(cid:12) (cid:12)
(cid:12) (cid:12)
2
The integral (2.1) converges absolutely if Reµ < n + 1, and has a
−
meromorphiccontinuationforµ C. Ifµisrealandsufficientlynegative(e.g.
∈
µ < n), then the kernel function K(ω,η) := [ω,η] −µ−n is square integrable
− | |
on S2n−1 S2n−1 and K(ω,η) = K(η,ω), and consequently, becomes
µ
× T
a self-adjoint, Hilbert–Schmidt operator on L2(S2n−1). In this section, we
determine all the eigenvalues of and the corresponding eigenspaces (see
µ
T
Theorem 2.1).
2.1 Harmonic polynomials on R2n and Cn
First, let us remind the classic theory of spherical harmonics on real and
complex vector spaces.
For k N, we denote by k(RN) the vector space consisting of homoge-
∈ H
N ∂2
neous polynomials p(x ,...,x ) of degree k such that p = 0.
1 N ∂x2
j=1 j
X
In the polar coordinates x = rω (r 0, ω SN−1), we have
≥ ∈
N ∂2 1 ∂ 2 2
∂x2 = r2 r∂r +(N −2) r∂r +∆SN−1 ,
Xj=1 j (cid:18)(cid:16) (cid:17) (cid:16) (cid:17) (cid:19)
where ∆SN−1 denotes the Laplace–Beltrami operator on the unit sphere en-
dowed with the standard Riemannian metric. In light that r ∂ p = kp for a
∂r
homogeneous function p of degree k, we see that the restriction p SN−1 for
|
p k(RN) belongs to the following eigenspace of ∆SN−1:
∈ H
Vk(SN−1) := ϕ C∞(SN−1) : ∆SN−1ϕ = k(k +N 2)ϕ . (2.2)
{ ∈ − − }
Since homogeneous functions on RN are determined uniquely by the re-
striction to SN−1, we get an injective map
k(RN) Vk(SN−1), p p SN−1
H → 7→ |
for each k N. This map is also surjective (see [9, Introduction, Theorem
∈
3.1] for example), and we shall identify k(RN) with V (SN−1). Thus, we
k
H
regard the following algebraic direct sum
∞
(RN) := k(RN)
H H
k=0
M
3
as a dense subspace of C∞(SN−1).
Analogously, we can define the space of harmonic polynomials on Cn. For
α,β N, we denote by α,β(Cn) the vector space consisting of polynomials
p(Z,∈Z¯) on Cn subject toHthe following two conditions:
(1) p(Z,Z¯) is homogeneous of degree α in Z = (z ,...,z ) and of degree β
1 n
in Z¯ = (z¯ ,...,z¯ ).
1 n
n ∂2
(2) p(Z,Z¯) = 0.
∂z ∂z¯
i i
j=1
X
Then, α,β(Cn) is a finite dimensional vector space. It is non-zero except for
H
the case where n = 1 and α,β 1.
≥
By definition, we have a natural linear isomorphism:
k(R2n) α,β(Cn). (2.3)
H ≃ H
α+β=k
M
We shall see that α,β(Cn) is an eigenspace of the operator for any
µ
H T
µ and for every α and β. To be more precise, we introduce a meromorphic
function of µ by
0 (k : odd),
Ak(µ,Cn) ≡ Ak(µ) := 2πn−21Γ(1−n2−µ)Γ(k+n2+µ) (k : even). (2.4)
 Γ(n+µ)Γ(k+n−µ)
2 2

We shall use the notation A (µ,Cn) when we emphasize the ambient
k
space Cn (see (4.6)).
Theorem 2.1. For α,β N,
∈
= ( 1)βA (µ)id.
µ α+β
T Hα,β(Cn) −
(cid:12)
(cid:12)
The rest of this sectio(cid:12)n is devoted to the proof of Theorem 2.1.
4
2.2 Preliminary results on homogeneous distributions
In this section we collect some basic concepts and results on distributions
in a way that we shall use later. See [7, Chapters 1 and 2], and also [12,
Appendix].
A distribution F depending on a complex parameter ν is defined to be
ν
meromorphic if for every test function ϕ, F ,ϕ is a meromorphic function
ν
h i
in ν. We say F has a pole at ν = ν if F ,ϕ has a pole at ν = ν for some
ν 0 ν 0
h i
ϕ. Then, taking its residue at ν , we get a new distribution
0
ϕ res F ,ϕ ,
ν
7→ ν=ν0h i
which we denote by resF .
ν
ν=ν0
Suppose F is a distribution defined in a conic open subset in RN. We say
F is homogeneous of degree λ if
N
∂
x F = λF (2.5)
j
∂x
j
j=1
X
in the sense of distributions, or equivalently, F,ϕ 1 = aλ+N F,ϕ for any
h a· i h i
test function ϕ and a > 0.
(cid:0) (cid:1)
Globally defined homogeneous distributions on RN are determined by
their restrictions to RN 0 for generic degree:
\{ }
Lemma 2.2. Suppose f is a distribution on RN which is homogeneous of
degree λ. If f = 0 and λ / N, N 1, N 2,... , then f = 0
RN\{0}
| ∈ {− − − − − }
as distribution on RN.
Proof. By the general structural theory on distributions, if suppf 0
⊂ { }
then f must be a finite linear combination of the Dirac delta function δ(x)
and its derivatives. On the other hand, the degree of the delta function and
its derivatives is one of N, N 1, N 2,.... By our assumption on f,
− − − − −
this does not happen. Hence, we conclude f = 0 as distribution on RN.
For a given function p C∞(SN−1), we define its extension into a homo-
∈
geneous function of degree λ by
p (rω) := rλp(ω), (r > 0, ω SN−1).
λ
∈
We regard the locally integrable functions as distributions by multiplying the
Lebesgue measure.
5
Lemma 2.3. Let p C∞(SN−1), then
∈
1) p is locally integrable on RN if Reλ > N.
λ
−
2) p extends to a tempered distribution which depends meromorphically on
λ
λ C. Its poles are simple and contained in the set N, N 1,... .
∈ {− − − }
3) The distribution p is homogeneous of degree λ in the sense of (2.5) if λ
λ
is not a pole.
Proof. 1)Clear fromtheformulaoftheLebesgue measure dx = rN−1drdσ(ω)
in the polar coordinates x = rω (r > 0, ω SN−1).
∈
2) Take a test function ϕ (RN). Suppose first Reλ > N. Then, we
∈ S −
can decompose p ,ϕ as
λ
h i
p ,ϕ = p (x)ϕ(x)dx+ p (x)ϕ(x)dx.
λ λ λ
h i
Z|x|≤1 Z|x|>1
The second term extends holomorphically in the entire complex plane. Let
us prove that the first term extends meromorphically in C. For this, we fix
k N, and consider the Taylor expansion of ϕ:
∈
ϕ(α)(0)
ϕ(x) = xα +ϕ (x),
k
α!
|α|≤k
X
whereα = (α ,...,α )isamulti-index, xα = xα1 xαN, α = α + +α ,
1 N 1 ··· N | | 1 ··· N
and ϕ (x) = O( x k+1). Accordingly, we have
k
| |
p (x)ϕ(x)dx
λ
Z|x|≤1
ϕ(α)(0) 1
= p(ω)ωαdσ(ω) rλ+|α|+N−1dr + p (x)ϕ (x)dx
λ k
α!
|α|≤k ZSN−1 Z0 Z|x|≤1
X
1 ϕ(α)(0)
= p(ω)ωαdσ(ω)+ p ϕ (x)dx.
λ k
λ+ α +N α!
|α|≤k | | ZSN−1 Z|x|≤1
X
The last term extends holomorphically to the open set λ C : Reλ >
{ ∈
N k . Since k is arbitrary we see that p ,ϕ extends meromorphically
λ
− − } h i
to the entire complex plane, and all its poles are simple and contained in the
set N, N 1, N 2,... . Thus, the second statement is proved.
{− − − − − }
6
N
∂
3) The differential equation x p (x) = λp (x) holds in the sense
j λ λ
∂x
j
j=1
X
of distributions for Reλ 0. This equation extends to all complex λ except
≫
for poles because the distribution p depends meromorphically on λ.
λ
Example 2.4 (k = 0 case). For k = 0, k(RN) is one-dimensional, spanned
H
by the constant function 1. We denote by rλ the corresponding homogeneous
distribution 1 .
λ
As we saw in the proof of Lemma 2.3, the distribution rλ has a simple
pole at λ = N and its residue is given by
−
N
2π2
res rλ = vol(SN−1)δ(x) = δ(x). (2.6)
λ=−N Γ(N)
2
Example 2.5 (N = 1 case). In the one dimensional case, SN−1 consists of
two points, 1 and 1, and consequently, the homogeneous distribution p is
λ
−
determined by the values p (1) and p ( 1). From this viewpoint, we give a
λ λ
−
list of classical homogeneous distributions on R.
p xλ xλ x λ x λsgnx (x+i0)λ (x i0)λ
λ + − | | | | −
p(1) 1 0 1 1 1 1
p( 1) 0 1 1 1 eiπλ e−iπλ
− −
Table 2.5.1: Homogeneous distributions on R
The notation (x i0)λ indicates that these distributions are obtained as the
±
boundary values of holomorphic functions in the upper (or lower) half plane.
For Reλ > 1,
−
lim(x iε)λ = (x i0)λ
ε↓0 ± ±
holds both in the ordinary sense and in distribution sense. The distribu-
tions (x i0)λ extend holomorphically to all complex λ, whereas the poles
±
of xλ, x λ, x λsgnx are located at 1, 2, 3,... , 1, 3, 5,... ,
± | | | | {− − − } {− − − }
2, 4, 6,... , respectively.
{− − − }
7
For λ = 1, 2,..., any two in Table 2.5.1 form a basis in the space of
6 − −
homogeneous distributions of degree λ. For example, by a simple basis change
one gets:
π π
(x i0)λ = e−iπ2λ(cos λ x λ +isin λ x λsgnx). (2.7)
− 2 | | 2 | |
2.3 Application of the Bochner identity
Let , be the standard inner product on RN. We consider the Fourier
h i
transform on RN normalized by
RN
F ≡ F
( f)(Y) := f(X)e−2πihX,YidX,
F
ZRN
and we extend to the space ′(RN) of tempered distributions.
F S
If f ′(RN) is homogeneous of degree λ, then its Fourier transform f
∈ S F
is homogeneous of degree λ N.
− −
Example 2.6 (N = 1 case).
e−i2π(λ+1)Γ(λ+1)
1) (xλ)(y) = (y i0)−λ−1.
F + (2π)λ+1 −
Γ(λ+1)
2) ( x λ)(y) = 2 y −λ−1, and
F | | πλ+12Γ(−λ)| |
2
iΓ(λ+2)
( x λsgnx)(y) = − 2 y −λ−1sgny.
F | | πλ+21Γ(1−λ)| |
2
These formulas may be found for instance in [7, Chapter II, 2.3], how-
§
ever, we shall give a brief proof because its intermediate step (e.g. (2.8)
below) will be used later (see the proof of Proposition 2.13).
Proof of Example 2.6. 1) Suppose Reλ > 1. Then xλ is locally integrable
− +
on R, and we have
lime−2πεxxλ = xλ
+ +
ε↓0
both in the ordinary sense and in the sense of distributions. Then, by
Cauchy’s integral formula and by the definition of the Gamma function, we
get
e−π2i(λ+1)Γ(λ+1)
(e−2πεxxλ)(y) = (y iε)−λ−1, (2.8)
F + (2π)λ+1 −
8
forε > 0. Takingthelimitasε 0wegetthedesiredidentityforReλ > 1.
→ −
By the meromorphic continuation on λ, the first statement is proved.
2) Similarly to 1), we can obtain a closed formula for (xλ)(y). Then the
F −
second statement follows readily from the base change matrix for the three
bases xλ,xλ , x λ, x λsgnx , and (x+i0)λ,(x i0)λ for homogeneous
{ + −} {| | | | } { − }
distributions on R. (We also use the duplication formula of the Gamma
function.)
We are ready to state the main result of this subsection. Let us define
the following meromorphic function of λ by
Γ(k+λ+N)
BN(λ,k) := π−λ−N2 i−k Γ(k−2λ) .
2
Lemma 2.7. For any p k(RN), we have the following identity
∈ H
p = B (λ,k)p (2.9)
λ N −λ−N
F
as distributions on RN that depend meromorphically on λ.
Example 2.8. Since (2.9) is an identity for meromorphic distributions, we
can pass to the limit, or compute residues at special values whenever it makes
sense. For instance, let k = 0. Then, by (2.6), the special value of (2.9) at
λ = 0 yields
(1) = limB (λ,0)r−λ−N = δ(y).
N
F λ→0
In view of the identity B (λ,0)B ( λ N,0) = 1, the residue of (2.9) at
N N
− −
λ = N yields
−
(δ(x)) = 1.
F
This, of course, is in agreement with the inversion formula for the Fourier
transform.
Proof of Lemma 2.7. For N = 1, k equals either 0 or 1, and correspondingly,
p is a scalar multiple of x λ or x λsgnx, respectively. Hence, Lemma 2.7
λ
| | | |
in the case N = 1 is equivalent to Example 2.6 2).
Let us prove (2.9) for N 2 as the identity of distributions on RN. We
≥
shall first prove the identity (2.9) on RN 0 in the non-empty domain:
\{ }
1
N < Reλ < (N +1). (2.10)
− −2
9
Since the both sides of (2.9) are homogeneous distributions of the same
degree, this will imply that the identity (2.9) holds on RN by Lemma 2.2.
Further,sincethebothsidesof (2.9)dependmeromorphicallyonλbyLemma
2.3, the identity (2.9) holds for all λ in the sense of distributions that depend
meromorphically on λ.
The rest of this proof is devoted to show (2.9) on RN 0 in the domain
\{ }
(2.10). For this, it is sufficient to prove that
p ,gq = B (λ,k) p ,gq ,
λ N −λ−N
hF i h i
for any compactly supported function g C∞(R ) and any q l(RN)
∈ c + ∈ H
(l N) because the linear spans of such functions form a dense subspace in
∈
C∞(R 0 ). Here, gq stands for a function on RN 0 defined by
c \{ } \{ }
(gq)(sη) = g(s)q(η) (s > 0,η SN−1).
∈
BydefinitionoftheFouriertransformon ′(RN), p ,gq = p , (gq) .
λ λ
S hF i h F i
Hence, what we need to prove is
p , (gq) = B (λ,k) p ,gq . (2.11)
λ N −λ−N
h F i h i
Wenotethat bothp andp arelocallyintegrable functions onRN under
λ −λ−N
the assumption (2.10). To calculate the left-hand side of (2.11), we use the
Bochner identity for q l(RN):
∈ H
q(ω)e−iνhω,ηidω = (2π)N2 i−lν1−N2 Jl+N−1(ν)q(η),
ZSN−1 2
where J (ν) denotes the Bessel function of the first kind. Then, we get the
µ
following formula after a change of variables x = 2πrs:
∞
F(gq)(rω) = 2πi−lr1−N2 q(ω) sN2 g(s)Jl+N2−1(2πrs)ds.
Z0
Hence, we have
∞ ∞
p , (gq) = I(r,s)ds p(ω)q(ω)dσ(ω)dr, (2.12)
λ
h F i
Z0 ZSN−1(cid:16)Z0 (cid:17)
where we set
I(r,s) := 2πi−lrλ+N2 sN2 g(s)Jl+N−1(2πrs).
2
At this point, we prepare the following:
10