Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials

0.33 MB·English

by  Michael Voit

Multidimensional Heisenberg convolutions
and product formulas for multivariate
2 Laguerre polynomials
1
0
2

n Michael Voit
a
J Fakult¨at Mathematik, Technische Universit¨at Dortmund

8 Vogelpothsweg 87
1
D-44221 Dortmund, Germany

]
A e-mail: [email protected]
C
.
h
t
a
m
[

Abstract
1
v
6 Letp,qpositiveintegers.ThegroupsUp(C)andUp(C) Uq(C)act
¯ ¯ × ¯
7 ontheHeisenberggroupH := M (C) Rcanonicallyasgroupsof
p,q p,q
7 ¯ ׯ
automorphismswhereM (C)isthevectorspaceofallcomplexp q-
3 p,q ¯ ×
. matrices. The associated orbit spaces may be identified with Πq R
1 × ¯
and Ξ R respectively with the cone Π of positive semidefinite
0 q q
× ¯
2 matrices and the Weyl chamber Ξ = x Rq : x ... x 0 .
q 1 q
1 { ∈ ¯ ≥ ≥ ≥ }
In this paper we compute the associated convolutions on Π R
: q × ¯
v and Ξ R explicitly depending on p. Moreover, we extend these
q
i × ¯
X convolutions by analytic continuation to series of convolution struc-
r tures for arbitrary parameters p 2q 1. This leads for q 2 to
a ≥ − ≥
continuous series of noncommutative hypergroups on Π R and
q
× ¯
commutative hypergroups on Ξ R. In the latter case, we describe
q
× ¯
thedualspaceintermsofmultivariate LaguerreandBesselfunctions
on Π and Ξ . In particular, we give a non-positive product formula
q q
for these Laguerre functions on Ξ .
q
The paper extends the known case q = 1 due to Koornwinder,
Trimeche, and others as well as the group case with integers p dueto
Faraut, Benson, Jenkins, Ratcliff, and others. Moreover, it is closely
related to product formulas for multivariate Bessel and other hyper-
geometric functions of Ro¨sler.
2010 Mathematics Subject Classification: Primary 43A62; Secondary 33C67. 33C52,
43A90, 43A20
Key words and phrases: Heisenberg convolution,matrix cones,Weyl chambers,mul-
tivariate Laguerre polynomials, multivariate Bessel functions, product formulas, hyper-
groups, hypergroup characters.
1
Multidimensional Heisenberg convolutions 2
1 Introduction
Forpositiveintegersp q considerthevector spaceM ofallp q matrices
p,q
≥ ×
over C. Consider the associated Heisenberg group H := M R with the
p,q p,q
¯ ׯ
product
(x,a) (y,b) = (x+y,a+b Imtr(x y))
∗
· −
where tr denotes the trace of the q q matrix x y. Clearly, the unitary
∗
×
groups K := U := U (C) and K := U U act on H via
p p p q p,q
¯ ×
u(x,a) := (ux,a) and (u,v)(x,a) := (uxv ,a)
∗
for u U , v U , x M , and a R respectively as groups of auto-
p q p,q
∈ ∈ ∈ ∈ ¯
morphisms. The associated orbit spaces may be identified with Π R and
q
× ¯
Ξ R respectively with the cone Π of complex positive semidefinite ma-
q q
× ¯
trices and the Weyl chamber Ξ = x Rq : x ... x 0 of type B.
q 1 q
{ ∈ ¯ ≥ ≥ ≥ }
It is well-known that the Banach- - algebras M (H ,K) of all K-invariant
b p,q
∗
bounded regular Borel measures with the convolution as multiplication are
commutative always for K := U U , and for K := U for q = 1 only (in
p q p
×
which case the cases K := U := U (C) and K := U U lead to the same
p p p 1
¯ ×
result). Moreover, in these Gelfand pair cases, the associated spherical func-
tions are well-known in terms of multivariate Laguerre and Bessel functions;
we refer to [BJR1], [BJR2], [BJRW], [C], [F], [Kac], [W] andreferences there
for this topic.
Inthispaperwecomputetheassociatedorbitconvolutions onΠ R and
q
ׯ
Ξ Rexplicitlydependingonthedimensionparameterp.Thiscomputation
q
ׯ
is similar to that of Ro¨sler [R2] where the action of U and U U on M
p p q p,q
×
is considered for the fields R,C,H, and where multivariate Bessel functions
¯ ¯ ¯
appear as spherical functions. Moreover, following [R2], we extend these
convolutions by analytic continuation to series of convolution structures for
arbitrary parameters p 2q 1 by using the famous theorem of Carleson.
≥ −
This extension leads for q 2 to continuous series of noncommutative
≥
hypergroups on Π R and continuous series of commutative hypergroups
q
× ¯
on Ξ R. In the latter case, we shall describe the dual spaces in terms
q
× ¯
of multivariate Laguerre and Bessel functions on Π and Ξ . Moreover, we
q q
determine further data of these hypergroups like the Haar measures, the
Plancherel measures, automorphisms and subhypergroups.
The main results will be as follows: For the (for q 2 noncommutative)
≥
hypergroup structures on Π R we shall derive in Section 2:
q
× ¯
1.1 Theorem. Let q 1 be an integer and p ]2q 1, [. Define a con-
≥ ∈ − ∞
volution of point measures on Π R by
q
ׯ
(δ δ )(f) = (1.1)
(r,a) p,q (s,b)
∗
= κ f r2 +s2 +rws+(rws) ,a+b Imtr(rws)
p,q ∗
−
ZBq
(cid:0)p∆(I w w)p 2qdw (cid:1)
q ∗ −
· −
Multidimensional Heisenberg convolutions 3
for f C (Π R), r,s Π , a,b R, where
b q q
∈ ׯ ∈ ∈¯
B := w M : w w < I , i.e.,I w w is positive definite ,
q q,q ∗ q q ∗
{ ∈ − }
∆ is the determinant of a q q matrix, and κ > 0 is a suitable constant.
p,q
×
Then this formula establishes by unique bilinear, weakly continuous exten-
sion an associative convolution on M (Π R). More precisely, (Π R, )
b q q p,q
ׯ ׯ ∗
is a hypergroup in the sense of Jewett (see [BH], [J]) with (0,0) as identity
and with the involution (r,a) := (r, a). Moreover,
−
ω (f) = f(√r,a)∆(r)p qdrda
p,q −
ZΠqׯR
defines a left and right Haar measure.
For the commutative hypergroup structures on Ξ R we shall derive in
q
× ¯
Section 3:
1.2 Theorem. Let q 1 be an integer and p ]2q 1, [. Then Ξ R
q
≥ ∈ − ∞ ׯ
carries a commutative hypergroup structure with convolution
(δ δ )(f) (1.2)
(ξ,a) p,q (η,b)
◦
= κ f σ( ξ2 +uη2u +ξwuηu +uηu w ξ),
p,q ∗ ∗ ∗ ∗
ZBq ZUq
(cid:16) p
a+b Imtr(ξwuηu ) ∆(I w w)p 2qdudw
∗ q ∗ −
− · −
(cid:17)
for f C (Ξ R), (ξ,a),(η,b) Ξ R where du means integration with
b q q
∈ ׯ ∈ ׯ
respectto the normalizedHaarmeasure onU andξ Ξ is identifiedwith the
q
∈
associated diagonal matrix in Π . The neutral element of this hypergroup is
q
given by (0,0) Ξ R, and the involution by (ξ,a) := (ξ, a). Moreover,
q
∈ ׯ −
a Haar measure is given by dω˜ (ξ,a) := h (ξ) dξ da with the Lebesgue
p,q p,q
density
q
hp,q(ξ) := ξi2p−2q+1 (ξi2 −ξj2)2. (1.3)
i=1 i<j
Y Y
Moreover, the dual spaces of these commutative hypergroups, i.e., the
sets of all bounded continuous multiplicative functions, will be described
precisely as a Heisenberg fan consisting of multivariate Laguerre and Bessel
functions which were studied in [BF], [F], [FK], [He] and many others. As
already noticed above, this description is well known for the group cases
with integer p by [BJRW], [F].
In Section 4 we shall use the product formula on Ξ R of Section 3 in
q
× ¯
order to derive a product formula for the normalized Laguerre functions
lp (x2/2) Lp (x2)
ϕ˜pm(x) := mlmp (0) = e−(x21+...+x2q)/2 Lmpm(0) (x ∈ Ξq)
Multidimensional Heisenberg convolutions 4
for p > 2q 1 which are introduced, for instance, in [FK]. We shall show
−
that for all partitions m and all ξ,η Ξ ,
q
∈
ϕ˜p (ξ) ϕ˜p (η) = κ ϕ˜p σ( ξ2 +uη2u +ξwuηu +uηu w ξ)
m m p,q m ∗ ∗ ∗ ∗
·
ZBq ZeUiqImtr(cid:16)(ξwupηu∗)∆(I w w)p 2qdudw. (1(cid:17).4)
−· q ∗ −
· −
For q = 1, this formula was derived by Koornwinder [Ko] who also gave
another version of this product formula using Bessel functions.
We here notice that on all three levels discussed above also degenerated
product formulas are available for the limit case p = 2q 1. We do not
−
consider the case p < 2q 1.
−
It is a pleasure to thank Margit Ro¨sler for many valuable discussions
about their multivariate Bessel convolutions and multivariate special func-
tions.
2 Heisenberg convolutions associated with ma-
trix cones
For positive integers p,q consider the vector space M of all p q matrices
p,q
×
over C. Consider the associated Heisenberg group H := M R with the
p,q p,q
¯ ׯ
product
(x,a) (y,b) = (x+y,a+b Imtr(x y))
∗
· −
where tr denotes the trace of the q q matrix x y. Clearly, the unitary
∗
×
group U := U (C) acts on H via
p p p,q
¯
u(x,a) := (ux,a) for u U , x M , a R
p p,q
∈ ∈ ∈ ¯
as a group of automorphisms. We regard U as a compact subgroup of the
p
associated semidirect product G := U ⋉ H in the natural way and
p,q p p,q
consider the double coset space G //U which may be also regarded as
p,q p
the space of all orbits of the action of U on H in the canonical way.
p p,q
Moreover, using uniqueness of polar decomposition of p q matrices, we see
×
immediately that we may identify this space of orbits also with the space
Π R with
q
× ¯
Π := z M : z Hermitian and positive semidefinite
q q,q
{ ∈ }
via
U ((x,a)) ( x ,a),
p
≃ | |
where x := √x x Π stands for the unique positive semidefinite square
∗ q
| | ∈
root of x x Π .
∗ q
∈
Now consider the Banach- -algebra M (G U ) of all U -biinvariant
b p,q p p
∗ ||
bounded signed Borel measures on G with the usual convolution of mea-
p,q
sures as multiplication. If we extend the canonical projection P : G
p,q
→
Multidimensional Heisenberg convolutions 5
G //U Π R to measures by taking images of measures w.r.t. P, this
p,q p q
≃ × ¯
extension becomes an isometric isomorphism between the Banach spaces
M (G U ) and M (Π R). We thus may transfer the Banach- -algebra
b p,q p b q
|| × ¯ ∗
structure on M (G U ) to M (Π R) by this isomorphism and obtain a
b p,q p b q
|| × ¯
probability preserving, weakly continuous convolution on M (Π R)
p,q b q
∗ × ¯
in this way. The pair (Π R, ) forms a so-called hypergroup; for general
q p,q
ׯ ∗
details on hypergroups the the construction above via double cosets and
orbits we refer to [BH] and [J].
Clearly, this Heisenberg-type convolution on (measures on) Π
p,q q
∗ ×
R is commutative iff so is M (G U ) , i.e., iff (G ,U ) is a Gelfand
b p,q p p,q p
¯ ||
pair. As Gelfand pairs associated with Heisenberg groups were classified
completely (see [BJR2], [C], [Kac], [W]), it turns out that the convolution
iscommutativeprecisely forq = 1.Moreover, forq = 1,theconvolutions
p,q
∗
on Π R = [0, [ R and the associated hypergroup structures were
p,q 1
∗ × ¯ ∞ ׯ
investigated by several authors; see [Ko], and the monographs [T], [BH] as
well as references therein.
We next compute the convolution for arbitrary positive integers q
p,q
∗
under the technical restriction p 2q which will become clear below. We
≥
do this by using the approach for the Gelfand pair (U ⋉M ,U ) in [R2]
p p,q p
where the double coset space (U ⋉ M //U ) is identified with Π , and
p p,q p q
where the same restriction appears. The computation here is only slightly
more involved, and we obtain:
2.1 Proposition. Let p 2q 1 be integers. Then the convolution of
p,q
≥ ≥ ∗
point measures is given by
(δ δ )(f) (2.1)
(r,a) p,q (s,b)
∗
= κ f r2 +s2 +rws+(rws) ,a+b Imtr(rws)
p,q ∗
−
ZBq
∆(cid:0)(pI w w)p 2qdw (cid:1)
q ∗ −
· −
for f C (Π R), r,s Π , a,b R with I M the identity matrix,
b q q q q,q
∈ ׯ ∈ ∈¯ ∈
B := w M : w w < I , i.e.,I w w is positive definite ,
q q,q ∗ q q ∗
{ ∈ − }
dw denoting integration w.r.t. Lebesgue measure onM ,
q,q
∆ denoting the determinant of a q q matrix, and
×
1
−
κ := ∆(I w w)p 2qdw > 0.
p,q q ∗ −
−
ZBq !
Proof. The canonical projection ϕ : H HUp Π R from the Heisen-
p,q p,q q
→ ≃ × ¯
berg group onto its orbit space is given explicitly by ϕ(x,a) = ( x ,a) with
| |
x := √x x. Moreover, if we define the block matrix
∗
| |
I
σ := q M ,
0 0 ∈ p,q
(cid:18) (cid:19)
Multidimensional Heisenberg convolutions 6
an “orbit” (r,a) Π R has the representative (σ r,a) H . By the
q 0 p,q
∈ × ¯ ∈
general definition of the orbit convolution (see Section 8.2 of [J] or [R2])
p,q
∗
we have
(δ δ )(f) = (δ δ )(f)
(r,a) ∗p,q (s,b) Up(σ0r,a) ∗p,q Up(σ0s,b)
= f ϕ (σ r,a) u((σ s,b) du
0 0
·
ZUp
(cid:0) (cid:0) (cid:1)(cid:1)
= f σ r +uσ s ,a+b Imtr(rσ uσ s) du (2.2)
| 0 0 | − 0∗ 0
ZUp
(cid:0) (cid:1)
where du denotes integration w.r.t. the normalized Haar measure on U .
p
Using the definition of the absolute value of a matrix above and denoting
the upper q q block of u by u := σ uσ M , we readily obtain
× q 0∗ 0 ∈ q,q
(δ δ )(f) = f r2 +s2 +ru s+(ru s) ,a+b Imtr(ru s) du.
(r,a) p,q (s,b) q q ∗ q
∗ −
ZUp q
(cid:0) (cid:1)
The truncation lemma 2.1 of [R3] now implies the proposition.
2.2 Remarks. (1) Theintegral inEq. (2.1)exists precisely forexponents
p 2q > 1 which shows that a formula for of the above kind is
p,q
− − ∗
available precisely for p 2q.
≥
(2) Let p 2q 1 be integers, and let f C (Π R), r,s Π , a,b R.
b q q
≥ ≥ ∈ ׯ ∈ ∈ ¯
Formula (2.1) and a straightforward computation yield that
(δ δ )(f)
(s,b) p,q (r,a)
∗
= κ f r2 +s2 +rws+(rws) ,a+b+Imtr(rws)
p,q ∗
ZBq
∆(I(cid:0)pw w)p 2qdw. (cid:1)
q ∗ −
· −
If one compares this with Eq. (2.1), the reader can check directly the
known fact that is non-commutative precisely for q 2. For this,
p,q
∗ ≥
1 0 0 0
take for instance, a = b = 0, r = , and s = with
0 0 0 1
(cid:18) (cid:19) (cid:18) (cid:19)
the zero matrix 0 M .
q 1,q 1
∈ − −
We next extend the definition of the Heisenberg convolution in Eq. (2.1)
to noninteger exponents p ]2q 1, [ for a fixed dimension parameter q
∈ − ∞
by Carlson’s theorem on analytic continuation. For the convenience of the
reader we recapitulate this result from [Ti], p.186:
2.3 Theorem. Let f(z) be holomorphic in a neighbourhood of z C :
{ ∈ ¯
Rez 0 satisfying f(z) = O ecz on Rez 0 for some c < π. If f(z) =
| |
≥ } ≥
0 for all nonnegative integers z, then f is identically zero for Rez > 0.
(cid:0) (cid:1)
This theorem will lead to the following extended convolution:
Multidimensional Heisenberg convolutions 7
2.4 Theorem. Let q 1 be an integer and p ]2q 1, [. Define the
≥ ∈ − ∞
convolution of point measures on Π R by
q
ׯ
(δ δ )(f) (2.3)
(r,a) p,q (s,b)
∗
= κ f r2 +s2 +rws+(rws) ,a+b Imtr(rws)
p,q ∗
−
ZBq
∆(I(cid:0)p w w)p 2qdw (cid:1)
q ∗ −
· −
for f C (Π R), r,s Π , a,b R where κ , dw, ∆ and other data
b q q p,q
∈ ׯ ∈ ∈ ¯
are defined as in Proposition 2.1 above. Then this formula defines a weakly
continuous convolution of point measures on Π R which can be extended
q
ׯ
uniquely in a bilinear, weakly continuous way to a probability preserving,
weakly continuous, and associative convolution on M (Π R). More pre-
b q
ׯ
cisely, (Π R, ) is a hypergroup with (0,0) as identity and with the
q p,q
ׯ ∗
involution (r,a) := (r, a).
−
Proof. It is clear from Eq. (2.3) that the mapping
(Π R) (Π R) M (Π R), ((r,a),(s,b)) δ δ
q q b q (r,a) p,q (s,b)
× ¯ × × ¯ → × ¯ 7→ ∗
is probability preserving and weakly continuous. It is now standard (see [J])
to extend this convolution uniquely in a bilinear and weakly continuous way
to a probability preserving convolution on M (Π R).
b q
× ¯
To prove associativity, it suffices to consider point measures. So let
r,s,t Π , a,b,c R, and f C (Π ). Then
q b q
∈ ∈ ¯ ∈
δ (δ δ )(f)
(r,a) p,q (s,b) p,q (t,c)
∗ ∗
= κ2 f H(r,a,s,b,t,c;v,w)
p,q
ZBq ZBq
(cid:0) ∆(I v v)p 2q∆(I(cid:1) w w)p 2qdvdw =: I(p)
q ∗ − q ∗ −
· − −
with a certain argument H independent of p. Similar,
(δ δ ) δ (f) =: I (p)
(r,a) p,q (s,b) p,q (t,c) ′
∗ ∗
admitsasimilarintegralrepresentationwithsomeintegrandH independent
′
of p. The integrals I(p) and I (p) are well-defined and holomorphic in p
′
{ ∈
C : Rep > 2q 1 . Furthermore, we know from the group cases above that
¯ − }
I(p) = I (p) for all integers p 2q. As
′
≥
κ = O( p q2) uniformly in p C : Rep > 2q 1 for p (2.4)
p,q
| | | | { ∈ ¯ − } → ∞
(see, for example, Eq. (3.9) of [R2]), we obtain readily that
f(p) := I(p+2q 1) I (p+2q 1) = O( p 2q2),
′
− − − | |
and Theorem 2.3 ensures that I(p) = I (p) for all p > 2q 1. Thus is
′ p,q
− ∗
associative.
Multidimensional Heisenberg convolutions 8
Finally, it is clear by Eq. (2.3) that δ is the neutral element. More-
(0,0)
over, as the support supp(δ δ ) of our convolution is obviously
(r,a) p,q (s,b)
∗
independent of p ]2q 1, [, all further hypergroup axioms from [BH] or
∈ − ∞
[J] regarding the support of convolution products are obvious, as they are
valid for the group cases with integer p 2q.
≥
2.5 Remark. The convolution (2.3) obviously satisfies the following sup-
port formula: For all (r,a),(s,b) Π R,
q
∈ × ¯
supp(δ δ )
(r,a) p,q (s,b)
∗ ⊂
(t,c) Π R : t r + s , c a + b + r s
q
⊂ { ∈ × ¯ k k ≤ k k k k | | ≤ | | | | k k·k k}
with the Euclidean norm x := tr(x x).
∗
k k
We next collect some propertpies of the hypergroups (Π R, ) for
q p,q
× ¯ ∗
p ]2q 1, [. We first turn to examples of automorphisms. For this we
∈ − ∞
first recapitulate that a homeomorphism T on Π R is called a hypergroup
q
ׯ
automorphism, if for all (r,a),(s,b) Π R,
q
∈ × ¯
T(δ δ ) = δ δ ,
(r,a) ∗ (s,b) T(r,a) ∗ T(s,b)
where the left hand side means the image of the measure under T.
2.6 Lemma. For all u U and λ > 0, the mappings
q
∈
T (r,a) := (λuru ,λ2a)
u,λ ∗
are hypergroup automorphisms on (Π R, ).
q p,q
ׯ ∗
Proof. Eq. (2.3) yields
(δ δ )(f)
Tu,λ(r,a)∗p,q Tu,λ(s,b)
= κ f λ u(r2 +s2 +ru wus+(ru wus) )u ,
p,q ∗ ∗ ∗ ∗
ZBq
(cid:0) p λ2(a+b Imtr(uru wusu ) ∆(I w w)p 2qdw.
∗ ∗ q ∗ −
− · −
(cid:1)
Using tr(ut) = tr(tu), √utu = u√tu and the substitution v = u wu, we
∗ ∗ ∗
see that this expression is equal to
κ f λu r2 +s2 +rvs+(rvs) )u ,
p,q ∗ ∗
ZBq
(cid:0) pλ2(a+b Imtr(rvs) ∆(I w w)p 2qdw
q ∗ −
− · −
= T (δ δ ) (2.5)
u,λ (r,a) p,q (s,b) (cid:1)
∗
as claimed.
Multidimensional Heisenberg convolutions 9
2.7 Remark. TheBessel hypergroups onthematrixcones Π of[R2] admit
q
many more hypergroup automorphisms. In fact, a complete classification
of all automorphisms on these Bessel hypergroups is given in [V3]. Due
to the additional term Imtr(rws) in Eq. (2.3), most of these hypergroup
automorphisms on Π cannot be extended to our Heisenberg convolutions.
q
We next turn to the (left) Haar measure which is unique up to a multi-
plicative constant by [J]:
2.8 Proposition. A left Haar measure of the hypergroup (Π R, ) is
q p,q
ׯ ∗
given by
ω (f) = f(√r,a)∆(r)p qdrda
p,q −
ZΠqׯR
for a continuous function f C (Π R) with compact support and the
c q
∈ ׯ
restriction of the Lebesgue measure dr on the vector space of all Hermitian
q q matrices.
×
Moreover, this left Haar measure is also a right Haar measure.
Proof. We first recall that the Heisenberg groups H are unimodular with
p,q
the usual Lebesgue measure dλ as Haar measure. Therefore, by general
results on orbit hypergroups (see e.g. [J]), the image ϕ(dλ) of dλ under the
canonical projection ϕ : H Π R is a left and right Haar measure on
p,p q
→ × ¯
the hypergroup (Π R, ). Moreover, the computation in Section 3.1 of
q p,q
× ¯ ∗
[R2] shows that
ϕ(dλ)(r,a) = c ∆(r)p qdrda M+(Π R)
p,q − q
· ∈ × ¯
with a certain known constant c > 0. This proves the result for integers
p,q
p 2q.
≥
For the general case we must check that
(δ δ )(f)∆(s)p qdsdb
(r,a) p,q (√s,b) −
ZΠq ZR¯ ∗
= (δ δ )(f)∆(s)p qdsdb
(√s,b) p,q (r,a) −
ZΠq ZR¯ ∗
= f(√s,b)∆(s)p qdsdb (2.6)
−
ZΠq ZR¯
for all f C (Π R), r Π , a R and p C with Rep > 2q 1,
c q q
∈ × ¯ ∈ ∈ ¯ ∈ ¯ −
where we use Eq. (2.3) also for the convolution for complex p. Clearly, all
expressions are analytic in p for fixed f,r,a,q. Moreover, by Eq. (2.3), all
three expressions are bounded by
C f κ MRe(p q)
p,q −
k k∞ ·
with some constant C and
M :=sup ∆(s) : (s,b) Π R, supp(δ δ ) suppf =
{ ∈ q × ¯ (r,a) ∗p,q (√s,b) ∩ 6 ∅}
=sup ∆(s) : (s,b) (r, a) supp(f) .
p,q
{ ∈ − ∗ }
Multidimensional Heisenberg convolutions 10
Using the estimate (2.4) for κ and the estimate for the support of con-
p,q
volution products in Remark 2.5, we obtain that the necessary estimate in
Carlson’s theorem 2.3 holds whenever r and the support of f are con-
k k
tained in a sufficiently small neighborhood of (0,0). Therefore, (2.6) holds
in this case.
Finally, if f C (Π R) and r Π are arbitrary, then we choose
c q q
∈ × ¯ ∈
a sufficiently small scaling parameter λ such that λr and the support of
f (s,a) := f(λ 1s,λ 2a) are sufficiently small such that (2.6) holds for λr
λ − −
and f . As the scaling map T is a hypergroup automorphism, it follows
λ Iq,λ
readily that (2.6) for λr and f is equivalent to (2.6) for r and f. This
λ
completes the proof.
2.9 Remark. Eq. (2.3) implies that for p > 2q 1 and (r,a),(s,b) Π R
q
− ∈ ׯ
with positive definite matrices r,s, the convolution product δ δ
(r,a) p,q (s,b)
∗
admits a density w.r.t. the Lebesgue measure and hence by the preceding
proposition w.r.t. the Haar measure of the hypergroup (Π R, ).
q p,q
× ¯ ∗
In fact, in the case p > 2q 1 consider the linear map
−
w (r2 +s2 +rws+(rws) ,Imtr(rws))
∗
7→
from B R2q2 to Π R Rq2 1 which has a Jacobi matrix with maximal
q ⊂ ¯ ◦qׯ ⊂ −
rank q2 1. As the square root mapping on the interior Π of Π is a
− ◦q q
diffeomorphism, the claim follows immediately from the convolution (2.3).
We next turn to the subhypergroups of (Π R, ). Recapitulate for
q p,q
× ¯ ∗
this that a closed set X Π R is called a subhypergroup, if for all
q
⊂ × ¯
x,y X, we have x¯ X and x y := supp(δ δ ) X. We next
x y
∈ ∈ { } ∗ { } ∗ ⊂
determine all subhypergroups of (Π R, ). We begin with examples of
q p,q
× ¯ ∗
subhypergroups.
2.10 Proposition. Let p > 2q 1, k 1,...,q , and u U . Then
q
− ∈ { } ∈
r˜ 0
X := u u ,a : r˜ Π ,a R
k,u 0 0 ∗ ∈ k ∈¯
(cid:26)(cid:18) (cid:18) (cid:19) (cid:19) (cid:27)
is a subhypergroup of (Π R, ), and the mapping
q p,q
ׯ ∗
r˜ 0
(r˜,a) u u ,a
∗
7→ 0 0
(cid:18) (cid:18) (cid:19) (cid:19)
is a hypergroup isomorphism between the Heisenberg hpergroup (Π R, )
k p,k
ׯ ∗
and the subhypergroup (X , ).
k,u p,q
∗
Proof. The X are obviously subhypergroups by Eq. (2.3). Moreover, us-
k,Iq
ing the automorphism T of Lemma 2.6, we see that the X are subhy-
u,1 k,u
pergroups for arbitrary u U .
q
∈
In order to check that the subhypergroup X is isomorphic with the
k,u
hypergroup (Π R, ), we may assume u = I without loss of general-
k p,k q
× ¯ ∗
ity. Here we first consider the group cases with integer p 2q. Here, the
≥