Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 010, 22 pages
The Clifford Deformation of the Hermite Semigroup
Hendrik DE BIE †, Bent ØRSTED ‡, Petr SOMBERG § and Vladimir SOUCˇEK §
† Department of Mathematical Analysis, Ghent University, Galglaan 2, 9000 Gent, Belgium
E-mail: Hendrik.DeBie@UGent.be
‡ Department of Mathematical Sciences, University of Aarhus,
Building 530, Ny Munkegade, DK 8000, Aarhus C, Denmark
E-mail: orsted@imf.au.dk
§ Mathematical Institute of Charles University, Sokolovska´ 83, 186 75 Praha, Czech Republic
E-mail: somberg@karlin.mff.cuni.cz, soucek@karlin.mff.cuni.cz
Received September 21, 2012, in final form January 29, 2013; Published online February 05, 2013
http://dx.doi.org/10.3842/SIGMA.2013.010
Abstract. This paper is a continuation of the paper [De Bie H., Ørsted B., Somberg P.,
Soucˇek V., Trans. Amer. Math. Soc. 364 (2012), 3875–3902], investigating a natural radial
deformation of the Fourier transform in the setting of Clifford analysis. At the same time, it
gives extensions of many results obtained in [Ben Sa¨ıd S., Kobayashi T., Ørsted B., Compos.
Math. 148 (2012), 1265–1336]. We establish the analogues of Bochner’s formula and the
Heisenberg uncertainty relation in the framework of the (holomorphic) Hermite semigroup,
and also give a detailed analytic treatment of the series expansion of the associated integral
transform.
Key words: Dunkl operators; Clifford analysis; generalized Fourier transform; Laguerre
polynomials; Kelvin transform; holomorphic semigroup
2010 Mathematics Subject Classification: 33C52; 30G35; 43A32
1 Introduction
It is well-known that the classical Dirac operator and its Fourier symbol generate via Clifford
multiplication a natural Lie superalgebra osp(1|2) contained in the Clifford–Weyl algebra. More
surprisingly, this carries over to a natural family of deformations of the Dirac operator, see [7].
Moreover, it is possible to define a Fourier transform naturally associated to the deformed family.
The novelty of the present article is that we let group theory be the guiding principle in
defining operators and transformations, in the next step followed by a study of explicit (analytic)
properties for naturally arising eigenfunctions and kernel functions. Thus the main aim is
to find the kernel function for the Fourier transform connected with our deformation, and
also to study its associated holomorphic semigroup regarded as a particular descendant of the
Gelfand–Gindikin program analyzing representations of reductive Lie groups, see, e.g., [22] and
the discussion in [2].
Let us now recall the basic setup and results from [7] and also discuss further aspects of our
construction. The deformation family of Dunkl–Dirac operators
D = r1−
a
2Dκ + br
−a2−1x+ cr−
a
2−1xE, a, b, c ∈ R,
together with the radial deformation of the coordinate function
xa = r
a
2−1x, r =
√
√
√
√
m∑
i=1
x2i ,
2 H. De Bie, B. Ørsted, P. Somberg and V. Soucˇek
forms a realization of osp(1|2) in the Clifford–Weyl algebra. HereDκ =
m∑
i=1
eiTi with Ti the Dunkl
operators, x =
m∑
i=1
eixi and E =
m∑
i=1
xi∂xi . The ei are generators of the Clifford algebra Clm. See
also the next section for more details.
We will show in Proposition 3.2 that this realization builds a Howe dual pair with G˜. Here
the group G˜ is the double cover (contained in the Pin group) of the finite reflection group G
used in the construction of the Dunkl operators.
The Fourier transform is then defined by
FD = e
ipi2
(
1
2+
µ−1
a(1+c)
)
e
−ipi
2a(1+c)2
(D2−(1+c)2x2a),
where L = D2 − (1 + c)2x2a is the generalized Hamiltonian and µ the Dunkl dimension. The
main aim of the present paper is to find an integral expression for this Fourier transform,
FD(f)(y) =
∫
Rm
K(x, y)f(x)h(rx)dx
with h(rx)dx the measure associated to D and K(x, y) the integral kernel to be determined. Note
that this ties in with recent work on generalized Fourier transforms in different contexts, e.g.,
analysis on minimal representations of reductive groups (see [19, 20, 21]) or integral transforms
in Clifford analysis (see [6, 8]).
The deformation of the classical Hamiltonian for the harmonic oscillator is visualized in the
following figure:
∆κ − |x|2
∆− |x|2
Dunkl deformation
OO
Clifford deformation
{{
a-deformation
""
D2 + (1 + c)2|x|a |x|2−a∆− |x|a
The Dunkl deformation is by now quite standard and described for example in [11]. The a-
deformation is the subject of the paper [2] and is a scalar radial deformation of the harmonic
oscillator. Our Clifford deformation is also a radial deformation but richer in the sense that
Clifford algebra- (or spinor)-valued functions are involved.
In this paper we will thus find a series representation of the kernel function for our new
Fourier transform FD, and also study the holomorphic semigroup with generator L. The main
results are Theorem 6.1 on the operator properties of the semigroup, Theorem 7.2 on the Fourier
transform intertwining the Dirac operator and the Clifford multiplication, Proposition 7.2 on
the Bochner identities, and Proposition 7.3 on the Heisenberg uncertainty relation. Finally in
Theorem 7.3 we give the analogue of what is sometimes called the “Master formula” in the
context of Dunkl operators (see, e.g., [26, Lemma 4.5(1)] or [4]).
The paper is organized as follows. In Section 2 we repeat basic notions on Clifford algebras
and Dunkl operators needed in the rest of the paper. In Section 3 we construct intertwining
operators to reduce our radially deformed Dirac operator to its simplest form. Subsequently,
The Clifford Deformation of the Hermite Semigroup 3
in Section 4 we discuss the representation theoretic content of our deformation and solve the
spectral problem of the associated Hamiltonian. In Section 5, we obtain the reproducing kernels
for spaces of spherical monogenics, which allows us to construct the kernel of the holomorphic
semigroup in Section 6. Section 7 contains the results on the (deformed) Fourier transform.
Further properties are collected in Section 8. Finally, we summarize some results on special
functions used in the paper in Appendix A and give a list of notations in Appendix B.
2 Preliminaries
In this section we collect some basic results on Clifford algebras and Dunkl operators.
2.1 Clifford algebras
Let V be a vector space of dimension m with a given negative definite quadratic form and
let Clm be the corresponding Clifford algebra. If {ei} is an orthonormal basis of V, then Clm is
generated by ei, i = 1, . . . ,m, with the relations
eiej + eiej = 0, i 6= j, e
2
i = −1.
The algebra Clm has dimension 2m as a vector space over R. It can be decomposed as Clm =
⊕mk=0Cl
k
m with Cl
k
m the space of k-vectors defined by
Clkm := span{ei1 · · · eik , i1 < · · · < ik}.
The projection on the space of k-vectors is denoted by [·]k.
The operator .¯ is the main anti-involution on the Clifford algebra Clm defined by
ab = ba, ei = −ei, i = 1, . . . ,m.
Similarly we have the automorphism  given by
(ab) = (a)(b), (ei) = −ei, i = 1, . . . ,m.
In the sequel, we will always consider functions f taking values in Clm, unless explicitly
mentioned. Such functions can be decomposed as
f(x) = f0(x) +
m∑
i=1
eifi(x) +
∑
i<j
eiejfij(x) + · · ·+ e1 · · · emf1...m(x)
with f0, fi, fij , . . . , f1...m all real-valued functions.
Several important groups can be embedded in the Clifford algebra. Note that the space of
1-vectors in Clm is canonically isomorphic to V ∼= Rm. Hence we can define
Pin(m) =
{
s1s2 · · · sn |n ∈ N, si ∈ Cl1m such that s
2
i = −1
}
,
i.e., the Pin group is the group of products of unit vectors in Clm. This group is a double cover
of the orthogonal group O(m) with covering map p : Pin(m) → O(m), which we will describe
explicitly in the next section.
Similarly we define
Spin(m) =
{
s1s2 · · · s2n |n ∈ N, si ∈ Cl1m such that s
2
i = −1
}
,
i.e., the Spin group is the group of even products of unit vectors in Clm. This group is a double
cover of SO(m). For more information about Clifford algebras and analysis, we refer the reader
to [9, 16].
4 H. De Bie, B. Ørsted, P. Somberg and V. Soucˇek
2.2 Dunkl operators
Denote by 〈·, ·〉 the standard Euclidean scalar product in Rm and by |x| = 〈x, x〉1/2 the associated
norm. For α ∈ Rm \ {0}, the reflection rα in the hyperplane orthogonal to α is given by
rα(x) = x− 2
〈α, x〉
|α|2
α, x ∈ Rm.
A root system is a finite subset R ⊂ Rm of non-zero vectors such that, for every α ∈ R, the
associated reflection rα preserves R. We will assume that R is reduced, i.e. R ∩ Rα = {±α}
for all α ∈ R. Each root system can be written as a disjoint union R = R+ ∪ (−R+), where R+
and −R+ are separated by a hyperplane through the origin. R+ is called a positive subsystem
of the root system R. The subgroup G ⊂ O(m) generated by the reflections {rα|α ∈ R} is called
the finite reflection group associated with R. We will also assume that R is normalized such
that 〈α, α〉 = 2 for all α ∈ R. For more information on finite reflection groups we refer the
reader to [18].
If we identify α with a 1-vector in Clm (and hence α/
√
2 with an element in Pin(m)), we can
rewrite the reflection rα as
rα(x) =
1
2
αxα
with x =
m∑
i=1
eixi. Generalizing this map gives us the covering map p from Pin(m) to O(m) as
p(s)(x) = (s)xs−1, s ∈ Pin(m).
In particular, we obtain a double cover of the reflection group G as G˜ = p−1(G) (see also the
discussion in [1]).
A multiplicity function κ on the root system R is a G-invariant function κ : R → C, i.e.
κ(α) = κ(hα) for all h ∈ G. We will denote κ(α) by κα. We will always assume that the
multiplicity function is real and satisfies κ ≥ 0. This assumption is, e.g., necessary to obtain
the subsequent formula (2.1), which is crucial for the sequel.
Fixing a positive subsystem R+ of the root system R and a multiplicity function κ, we
introduce the Dunkl operators Ti associated to R+ and κ by (see [10, 13])
Tif(x) = ∂xif(x) +
∑
α∈R+
κααi
f(x)− f(rα(x))
〈α, x〉
, f ∈ C1(Rm).
An important property of the Dunkl operators is that they commute, i.e. TiTj = TjTi.
The Dunkl Laplacian is given by ∆κ =
m∑
i=1
T 2i , or more explicitly by
∆κf(x) = ∆f(x) + 2
∑
α∈R+
κα
(
〈∇f(x), α〉
〈α, x〉
−
f(x)− f(rα(x))
〈α, x〉2
)
with ∆ the classical Laplacian and ∇ the gradient operator. We also define the constant
µ =
1
2
∆κ|x|
2 = m+ 2
∑
α∈R+
κα,
called the Dunkl-dimension.
The Clifford Deformation of the Hermite Semigroup 5
It is possible to construct an intertwining operator Vκ connecting the classical derivatives ∂xj
with the Dunkl operators Tj such that TjVκ = Vκ∂xj (see, e.g., [12]). Note that explicit formulae
for Vκ are only known in a few special cases.
The weight function related to the root system R and the multiplicity function κ is given
by wκ(x) =
∏
α∈R+
|〈α, x〉|2κα . For suitably chosen functions f and g one then has the following
property of integration by parts (see [11])
∫
Rm
(Tif)gwκ(x)dx = −
∫
Rm
f (Tig)wκ(x)dx. (2.1)
For more information about Dunkl operators we refer the reader to [13, 25].
The starting point in the subsequent analysis is the Dunkl–Dirac operator, given by
Dκ =
m∑
i=1
eiTi.
Together with the vector variable x =
m∑
i=1
eixi this Dunkl–Dirac operator generates a copy
of osp(1|2), see [23] or the subsequent Theorem 3.1. In particular, we have
D2κ = −∆κ and x
2 = −|x|2 = −r2 = −
m∑
i=1
x2i .
3 Intertwining operators
Let, for a, b ∈ R, P and Q be two operators defined by
Pf(x) = rbf
((a
2
) 1
a
xr
2
a−1
)
, Qf(x) = r−
ab
2 f
((
2
a
) 1
2
xr
a
2−1
)
.
These two operators act as generalized Kelvin transformations. Indeed, one can easily compute
their composition
QP = PQ =
(
2
a
) b
2
.
We will show that these operators allow to reduce the Dirac operator D to a simpler form.
We have the following proposition, where E =
m∑
i=1
xi∂xi denotes the Euler operator. Recall
also from the introduction that xa = r
a
2−1x.
Proposition 3.1. One has the following intertwining relations
(a
2
) b−1
2
Q
(
Dκ + br
−2x+ cr−2xE
)
P = r1−
a
2Dk + βr
−a2−1x+ γr−
a
2−1xE,
(a
2
) b+1
2
QxP = xa
with β = 2b+ bc, γ = 2a(1 + c)− 1.
6 H. De Bie, B. Ørsted, P. Somberg and V. Soucˇek
Proof. In [7, Proposition 3], we already proved that
(a
2
) b−1
2
Q (Dκ)P = r
1−a2Dk + br
−a2−1x+
(
2
a
− 1
)
r−
a
2−1xE = xa.
Similarly we obtain
(a
2
) b−1
2
Q
(
r−2x
)
P = r−
a
2−1x
and
(a
2
) b−1
2
Q
(
r−2xE
)
P = br−
a
2−1x+
(
2
a
)
r−
a
2−1xE.
This completes the proof of the proposition. 
So we are reduced to the study of the operator
D = Dκ + br−2x+ cr−2xE,
where b, c ∈ R, c 6= −1. Here, the term br−2x can also be removed. Indeed, we have
r−α
(
Dκ + br
−2x+ cr−2xE
)
rα = Dκ + cr
−2xE,
when α = −b/(1 + c).
As a result of the previous discussion, we see that it is sufficient to study the function theory
for the operator
D = Dκ + cr−2xE,
where we have put a = 2, b = 0. Furthermore, we will restrict ourselves to the case c > −1 for
reasons that will become clear in Proposition 3.3. Similarly, we no longer need to consider xa
but can restrict ourselves to x. Now we repeat the basic facts concerning this operator we need
in the sequel. All the results are taken from [7], putting a = 2, b = 0.
Theorem 3.1. The operators D and x generate a Lie superalgebra, isomorphic to osp(1|2), with
the following relations
{x,D} = −2(1 + c)
(
E +
δ
2
)
,
[
E +
δ
2
,D
]
= −D,
[
x2,D
]
= 2(1 + c)x,
[
E +
δ
2
, x
]
= x,
[
D2, x
]
= −2(1 + c)D,
[
E +
δ
2
,D2
]
= −2D2,
[
D2, x2
]
= 4(1 + c)2
(
E +
δ
2
)
,
[
E +
δ
2
, x2
]
= 2x2,
where δ = 1 + µ−11+c .
Note that the square of D is a complicated operator, given by
D2 = −∆κ − (cµ) r−1∂r −
(
c2 + 2c
)
∂2r + cr
−2
∑
i
xiTi − cr
−2
∑
i<j
eiej(xiTj − xjTi).
If κ = 0, the formula for D2 simplifies a bit as now
∑
i xiTi = r∂r = E.
The Clifford Deformation of the Hermite Semigroup 7
Remark 3.1. The operator D = Dκ+cr−2xE is also considered from a very different perspective
in [3] (in the case κ = 0), where the eigenfunctions of this operator are studied.
Let us now discuss the symmetry of the generators of osp(1|2). First we define the action of
the Pin group on C∞(Rm)⊗ Clm for s ∈ Pin(m) as
ρ(s) : C∞(Rm)⊗ Clm → C∞(Rm)⊗ Clm, f ⊗ b→ f(p
(
s−1
)
x)⊗ sb.
We then have
Proposition 3.2. Let s ∈ G˜ and define sgn(s) := sgn(p(s)). Then one has
ρ(s)x = sgn(s)xρ(s), ρ(s)D = sgn(s)Dρ(s).
Proof. This follows immediately from the definition of ρ and the G-equivariance of the Dunkl
operators. 
So up to sign, the Dirac operator D is G˜-equivariant. At this point it is interesting to remark
that an algebraic analog of the Dunkl–Dirac operator D for graded affine Hecke algebras is intro-
duced in [1] with the motivation to prove a version of Vogan’s Conjecture for Dirac cohomology.
The formulation is based on a uniform geometric parametrization of spin representations of Weyl
groups. This Dirac operator is an algebraic variant of our family deformation of the differential
Dirac operator for special values of the deformation parameters. Moreover, it satisfies the same
symmetry as in Proposition 3.2, see [1, Lemma 3.4].
There is a measure naturally associated with D. Indeed, one has
Proposition 3.3. If c > −1, then for suitable differentiable functions f and g one has
∫
Rm
(Df)gh(r)wκ(x)dx =
∫
Rm
f(Dg)h(r)wκ(x)dx
with h(r) = r1−
1+µc
1+c , provided the integrals exist.
In this proposition, ·¯ is the main anti-involution on the Clifford algebra Clm.
4 Representation space for the deformation family
of the Dunkl–Dirac operator
The function space we will work with is L2κ,c(R
m) = L2(Rm, h(r)wκ(x)dx) ⊗ Clm. This space
has the following decomposition
L2κ,c(R
m) = L2
(
R+, r
µ−1
1+c dr
)
⊗ L2(Sm−1, wκ(ξ)dσ(ξ))⊗ Clm,
where on the right-hand side the topological completion of the tensor product is understood and
with dσ(ξ) the Lebesgue measure on the sphere Sm−1. The space L2(Sm−1, wκ(ξ)dσ(ξ)) ⊗ Clm
can be further decomposed into Dunkl harmonics and subsequently into Dunkl monogenics.
This leads to
L2
(
Sm−1, wκ(ξ)dσ(ξ)
)
⊗ Clm =
∞⊕
`=0
(M` ⊕ xM`)
∣
∣
Sm−1 ,
whereM` = kerDκ∩(P` ⊗ Clm) is the space of Dunkl monogenics of degree `, with P` the space
of homogeneous polynomials of degree ` (see also [5] for more details on Dunkl monogenics).
8 H. De Bie, B. Ørsted, P. Somberg and V. Soucˇek
Using this decomposition, we have obtained in [7] a basis for L2κ,c(R
m). This basis is given
by the set {φt,`,m} (t, ` ∈ N and m = 1, . . . ,dimM`), defined as
φ2t,`,m = 2
2t(1 + c)2tt!L
γ`
2 −1
t (r
2)rβ`M (m)` e
−r2/2,
φ2t+1,`,m = −2
2t+1(1 + c)2t+1t!L
γ`
2
t (r
2)xrβ`M (m)` e
−r2/2
with Lβα the Laguerre polynomials and
β` = −
c
1 + c
`, γ` =
2
1 + c
(
`+
µ− 2
2
)
+
c+ 2
1 + c
,
and where M (m)` (m = 1, . . . ,dimM`) forms an orthonormal basis of M`, i.e.
[∫
Sm−1
M (m1)` (ξ)M
(m2)
` (ξ)wκ(ξ)dσ(ξ)
]
0
= δm1m2
with [·]0 the projection on the scalar part of the Clifford algebra. The dimension ofM` is given
by
dimRM` = dimR Clm dimR P`
(
Rm−1
)
= 2m
(`+m− 2)!
`!(m− 2)!
with P`
(
Rm−1
)
the space of homogeneous polynomials of degree ` in m− 1 variables (see [9]).
Using formula (4.10) in [7] and the proof of Theorem 3 in [7], one obtains the following
formulae for the action of D and x on the generalized Laguerre functions
2Dφt,`,m= φt+1,`,m+ C(t, `)φt−1,`,m, −2(1 + c)xφt,`,m= φt+1,`,m− C(t, `)φt−1,`,m (4.1)
with
C(2t, `) = 4(1 + c)2t, C(2t+ 1, `) = 2(1 + c)2(γ` + 2t).
These formulae determine the action of osp(1|2) on L2κ,c(R
m). Recall also that the action of G˜
on L2κ,c(R
m) is given by ρ (see Section 3).
Subsequently, we can define a creation and annihilation operator in this setting by
A+ = D− (1 + c)x, A− = D + (1 + c)x (4.2)
satisfying
A+φt,`,m = φt+1,`,m, A
−φt,`,m = C(t, `)φt−1,`,m.
Now we introduce the following inner product
〈f, g〉 =
[∫
Rm
f cgh(r)wκ(x)dx
]
0
,
where h(r) is the measure associated to D (see Proposition 3.3) and f c is the complex conjugate
of f . It is easy to check that this inner product satisfies
〈Df, g〉 = 〈f,Dg〉, 〈xf, g〉 = −〈f, xg〉. (4.3)
The related norm is defined by ||f ||2 = 〈f, f〉.
The Clifford Deformation of the Hermite Semigroup 9
Theorem 4.1. We have
〈φt1,`1,m1 , φt2,`2,m2〉 = c(t1, `1)δt1t2δ`1`2δm1m2 ,
where c(t, `) is a constant depending on t and `.
The functions φt,`,m are eigenfunctions of the Hamiltonian of a generalized harmonic oscilla-
tor.
Theorem 4.2. The functions φt,`,m satisfy the following second-order PDE
(
D2 − (1 + c)2x2
)
φt,`,m = (1 + c)
2(γ` + 2t)φt,`,m.
Proof. This follows immediately from the formula (4.1). 
Theorem 4.2 combined with the definition of A+, A− in (4.2) allows us to decompose the space
L2κ,c(R
m) under the action of osp(1|2). Clearly the odd elements A+ and A− generate osp(1|2) as
they are linear combinations of D and x. Moreover, they act between two basis vectors {φt,`,m}
of L2κ,c(R
m), so it is sufficient to consider vectors in an irreducible representation of osp(1|2)
inside the functional space. This is achieved as follows – for fixed ` and m each vector φ0,`,m
generates the irreducible representation
φ0,`,m
A+ //
L
UU
φ1,`,m
L
UU
A+ //
A−
oo φ2,`,m
L
UU
A+ //
A−
oo φ3,`,m
L
UU
A+ //
A−
oo φ4,`,m
L
UU
A+ //
A−
oo . . .
A−
oo
where
L =
1
2
{A+, A−} = D2 − (1 + c)2x2
with the action given in Theorem 4.2. In fact this highest weight representation is labeled
by ` only and we will denote it pi(`). In conclusion, we obtain the decomposition of our func-
tional space L2κ,c(R
m) into a discrete direct sum of highest weight (infinite-dimensional) Harish-
Chandra modules for osp(1|2):
L2κ,c(R
m) =
∞⊕
`=0
pi(`)⊗M`.
These results should be compared with Theorem 3.19 and Section 3.6 in [2] (where one uses sl2
instead of osp(1|2)). Also notice that the claim should be understood as an assertion on the
deformation of the Howe dual pair for osp(1|2) inside the Clifford–Weyl algebra on Rm acting
on a fixed vector space L2κ,c(R
m).
In particular, we have the following result. Recall that an operator T is essentially selfadjoint
on a Hilbert space H if T is a symmetric operator with a dense domain D(T ) ⊂ H such that for
a complete orthogonal set {fn}n in H with fn ∈ D(H), there exist {µn}n solving Tfn = µnf
for all n ∈ N.
Proposition 4.1. Let c > −1 and κ > 0. The operator L acting on L2κ,c(R
m) is essentially self-
adjoint (i.e. symmetric and its closure is a selfadjoint operator). Moreover, L has no continuous
spectrum and its discrete spectrum is given by
Spec(L) = {2(1 + c)`+ 2(1 + c)2t+ (1 + c)(µ+ c) | `, t ∈ N}.
10 H. De Bie, B. Ørsted, P. Somberg and V. Soucˇek
Using Theorem 4.2 we can now define the holomorphic semigroup for the deformed Dirac
operator by
FωD = e
ω
(
1
2+
µ−1
2(1+c)
)
e
−ω
2(1+c)2
(D2−(1+c)2x2).
Here, ω takes values in the right half-plane of C. The special boundary value ω = ipi/2 corre-
sponds to the Fourier transform. In that case, we will use the notation FD. The functions φt,`,m
are eigenfunctions of FωD satisfying
FωD(φt,`,m) = e
−ωte−
ω`
(1+c)φt,`,m. (4.4)
Note that in the special case κ = 0, c = 0 the operator FωD reduces to the classical Hermite
semigroup (see, e.g., [17]).
Remark 4.1. One can also consider more general deformations of the Dirac operator, by adding
suitable odd powers of Γ = −xDκ − E to D as follows
D = Dκ + cr−2xE +
∑`
j=0
cjr
−1
(
Γ−
µ− 1
2
)2j+1
, cj ∈ R.
This does not alter the osp(1|2) relations, as Γ− µ−12 anti-commutes with x and has the correct
homogeneity. In particular, Γ − µ−12 can be seen as the square root of the Casimir of osp(1|2),
see [15, Example 2 in Section 2.5].
In the sequel of the paper, we will always assume κ = 0 or in other words, we do not consider
the Dunkl deformation. This is to simplify the notation of the results. Most statements can be
generalized to the Dunkl case by a suitable composition with the Dunkl intertwining operator Vκ,
except the results obtained in Section 8.
Recall that for κ = 0, the Dunkl–Dirac operator Dκ reduces to the orthogonal Dirac operator
∂x =
m∑
i=1
ei∂xi and the Dunkl dimension µ to the ordinary dimension m.
5 Reproducing kernels
In this section we determine the reproducing kernels for Mk and xMk. We start with an
auxiliary Lemma, which can be thought of as a Clifford analogue of the Funk–Hecke transform.
We define the wedge product of two vectors as
x ∧ y :=
∑
j<k
ejek(xjyk − xkyj).
Lemma 5.1. Put x = rx′ and y = sy′ with x′, y′ ∈ Sm−1. Furthermore, put λ = (m− 2)/2 and
σm = 2pim/2/Γ(m/2). Then one has, with Ml ∈M`
∫
Sm−1
Cλk (〈x
′, y′〉)M`(x
′)dσ(x′) = σm
λ
λ+ k
δk,`M`(y
′),
∫
Sm−1
Cλk (〈x
′, y′〉)x′M`(x
′)dσ(x′) = σm
λ
λ+ k
δk,`+1y
′M`(y
′),
∫
Sm−1
(x′ ∧ y′)Cλ+1k−1 (〈x
′, y′〉)M`(x
′)dσ(x′) = −σm
k
2(λ+ k)
δk,`M`(y
′),
∫
Sm−1
(x′ ∧ y′)Cλ+1k−1 (〈x
′, y′〉)x′M`(x
′)dσ(x′) = σm
k + 2λ
2(λ+ k)
δk,`+1y
′M`(y
′),
where Cλk (〈x
′, y′〉) is the k-th Gegenbauer polynomial in the variable 〈x′, y′〉.
The Clifford Deformation of the Hermite Semigroup 11
Proof. The first integral is trivial: M` is a spherical harmonic of degree ` and Cλk (〈x
′, y′〉) is
the reproducing kernel for spherical harmonics of degree k (see, e.g., [13]). The second integral
immediately follows, because x′M`(x′) ∈ H`+1.
The other two integrals are a bit more complicated. We show how to obtain the last one.
First rewrite (x′ ∧ y′)x′ = y′ − 〈x′, y′〉x′. The first term then follows from the first integral. For
the second term, we use the recursive property of Gegenbauer polynomials:
wCλ+1n−1(w) =
n
2(n+ λ)
Cλ+1n (w) +
n+ 2λ
2(n+ λ)
Cλ+1n−2(w).
The result then follows by collecting everything. 
We can use this lemma to determine the reproducing kernels. This is the subject of the
following proposition.
Proposition 5.1. For k ∈ N∗ put
Pk(x
′, y′) =
k + 2λ
2λ
Cλk (〈x
′, y′〉)− (x′ ∧ y′)Cλ+1k−1 (〈x
′, y′〉),
Qk−1(x
′, y′) =
k
2λ
Cλk (〈x
′, y′〉) + (x′ ∧ y′)Cλ+1k−1 (〈x
′, y′〉)
with P0(x′, y′) = Cλ0 (0) = 1. Then
∫
Sm−1
Pk(x
′, y′)M`(x
′)dσ(x′) = σmδk,`M`(y
′),
∫
Sm−1
Pk(x
′, y′)x′M`(x
′)dσ(x′) = 0
and
∫
Sm−1
Qk−1(x
′, y′)M`(x
′)dσ(x′) = 0,
∫
Sm−1
Qk−1(x
′, y′)x′M`(x
′)dσ(x′) = σmδk,`+1y
′M`(y
′).
Proof. This follows immediately from Lemma 5.1. 
Remark 5.1. Note that, as expected, Pk(x′, y′) + Qk−1(x′, y′) = λ+kλ C
λ
k (〈x
′, y′〉), which is the
reproducing kernel for the space of spherical harmonics of degree k.
Remark 5.2. When the dimension m = 2 and hence λ = 0, the reproducing kernel is still
well-defined by using the well-known relation [27, (4.7.8)]
lim
λ→0
λ−1Cλk (w) = (2/k) cos kθ, w = cos θ, k ≥ 1.
We will also need the following lemma.
Lemma 5.2. The reproducing kernels satisfy the following properties, for all k, l ∈ N:
∫
Sm−1
Pk(y
′, x′)P`(z
′, y′)dσ(y′) = σmδk`P`(z
′, x′),
∫
Sm−1
Pk(y
′, x′)Q`(z
′, y′)dσ(y′) = 0,
∫
Sm−1
Qk(y
′, x′)Q`(z
′, y′)dσ(y′) = σmδk`Q`(z
′, x′).
Proof. This follows immediately using Lemma 7.6 and 7.10 from [8]. 
Remark 5.3. Mind the order of the variables in the previous lemma. The kernels Pk(x′, y′)
and Qk(x′, y′) are not symmetric.
12 H. De Bie, B. Ørsted, P. Somberg and V. Soucˇek
6 The series representation of the holomorphic semigroup
The aim of the present section is to investigate basic properties of the holomorphic semigroup
defined by
FωD = e
ω
(
1
2+
µ−1
2(1+c)
)
e
−ω
2(1+c)2
(D2−(1+c)2x2), Reω ≥ 0,
acting on the space L20,c(R
m). We start with the following general statement.
Theorem 6.1. Suppose c > −1. Then
1. For any t, ` ∈ N and m ∈ {1, . . . ,dimM`}, the function φt,`,m is an eigenfunction of the
operator FωD:
FωD(φt,`,m) = e
−ωte−
ω`
(1+c)φt,`,m.
2. FωD is a continuous operator on L
2
0,c(R
m) for all ω with Reω ≥ 0, in particular
||FωD(f)|| ≤ ||f ||
for all f ∈ L20,c(R
m).
3. If Reω > 0, then FωD is a Hilbert–Schmidt operator on L
2
0,c(R
m).
4. If Reω = 0, then FωD is a unitary operator on L
2
0,c(R
m).
Proof. (1) is an immediate consequence of Theorem 4.2. For (2), let f be an element in
L20,c(R
m) and expand it with respect to the (normalized) basis {φt,`,m} as
f =
∑
t,`,m
at,`,mφt,`,m.
Then one has, using orthogonality,
||FωD(f)||
2 =
∑
t,`,m
|at,`,m|
2e−2(Reω)te−
2(Reω)`
(1+c) ≤
∑
t,`,m
|at,`,m|
2 = ||f ||2
because Reω ≥ 0.
As for (3), we have to show that the Hilbert–Schmidt norm is finite. We compute
||FωD||
2
HS =
∑
t,k,`
||FωD(φt,k,`)||
2 =
∑
t,k,`
e−2(Reω)te−
2(Reω)k
(1+c) =
∑
t,k
e−2(Reω)te−
2(Reω)k
(1+c) dimRMk
=
∑
t,k
e−2(Reω)te−
2(Reω)k
(1+c)
(k +m− 2)!
k!(m− 2)!
2m =
∑
t
e−2(Reω)t
∑
k
e−
2(Reω)k
(1+c)
(k +m− 2)!
k!(m− 2)!
2m.
Using the ratio test, we see that these series are convergent for Reω > 0.
(4) follows immediately, because when Reω = 0 the eigenvalues all have unit norm. 
We have already observed that FωD is a Hilbert–Schmidt operator for Reω > 0 and a unitary
operator for Reω = 0. The Schwartz kernel theorem implies that FωD can be expressed by
a distribution kernel K(x, y;ω), so
(
FωDf
)
(y) =
∫
Rm
K(x, y;ω)f(x)h(rx)dx,
and K(x, y;ω)h(rx) is a tempered distribution on Rm × Rm.
The Clifford Deformation of the Hermite Semigroup 13
6.1 The case Reω > 0
Using the reproducing kernels of Section 5, we can now make a reasonable ansatz for the kernel
of the full holomorphic semigroup. We want to write this semigroup as
Fω0,c(f)(y) = σ
−1
m
∫
Rm
K(x, y;ω)f(x)h(rx)dx
with K(x, y;ω) = K0(x, y;ω) +K1(x, y;ω) and
K0 = e
− cothω2 (r
2+s2)
+∞∑
k=0
αkz
k
1+c J˜ γk
2 −1
(
iz
sinhω
)
Pk(x
′, y′),
K1 = e
− cothω2 (r
2+s2)
+∞∑
k=0
βkz
1+ k1+c J˜ γk
2
(
iz
sinhω
)
Qk(x
′, y′). (6.1)
Here r = |x|, s = |y| and z = |x||y|. We also used the notation J˜ν(t) = (t/2)−νJν(t). Now we
determine the complex constants {αk} and {βk} such that this integral transform coincides with
FωD = e
ω
(
1
2+
µ−1
2(1+c)
)
e
−ω
2(1+c)2
(D2−(1+c)2x2)
on the basis {φt,`,m}.
We calculate
σ−1m
∫
Rm
K0(x, y;ω)φ2t,`,m(x)dx = α`M
(m)
` (y
′)e−
cothω
2 s
2
s
`
1+c
(
is
2 sinhω
)−γ`/2+1
×
∫ +∞
0
rγ`/2e−
(cothω+1)
2 r
2
Jγ`/2−1
(
irs
sinhω
)
Lγ`/2−1t (r
2)dr = α`
e−2ωt2γ`/2−1
(cothω + 1)γ`/2
φ2t,`,m(y),
where we used the identity (see [2, Corollary 4.6])
2
∫ +∞
0
rα+1Jα(rβ)L
α
j (r
2)e−δr
2
dr =
(δ − 1)jβα
2αδα+j+1
Lαj
(
β2
4δ(1− δ)
)
e−
β2
4δ .
Similarly, we find
σ−1m
∫
Rm
K0(x, y;ω)φ2t+1,`,m(x)dx = 0, σ
−1
m
∫
Rm
K1(x, y;ω)φ2t,`,m(x)dx = 0,
σ−1m
∫
Rm
K1(x, y;ω)φ2t+1,`,m(x)dx = β`
e−2ωt2γ`/2
(cothω + 1)γ`/2+1
φ2t+1,`,m(y).
Hence we obtain by comparison with (4.4)
α` = e
− ω`(1+c)
(cothω + 1)γ`/2
2γ`/2−1
= 2e
ωδ
2 (2 sinhω)−γ`/2, β` =
α`
2 sinhω
.
We summarize our results in the following theorem.
Theorem 6.2. Let Reω > 0 and c > −1. Put
K(x, y;ω) = e−
cothω
2 (r
2+s2) (A(z, w) + x ∧ yB(z, w)
)
with
A(z, w) =
+∞∑
k=0
(
αk
k + 2λ
2λ
z
k
1+c J˜ γk
2 −1
(
iz
sinhω
)
+
αk−1
4 sinhω
k
λ
z
k+c
1+c J˜ γk−1
2
(
iz
sinhω
))
Cλk (w),
14 H. De Bie, B. Ørsted, P. Somberg and V. Soucˇek
B(z, w) =
+∞∑
k=1
(
−αkz
k
1+c−1J˜ γk
2 −1
(
iz
sinhω
)
+
αk−1
2 sinhω
z
k+c
1+c−1J˜ γk−1
2
(
iz
sinhω
))
Cλ+1k−1 (w)
for z = |x||y|, w = 〈x, y〉/z, α−1 = 0 and αk = 2e
ωδ
2 (2 sinhω)−γk/2.
Then these series are convergent and the integral transform defined on L20,c(R
m) by
Fω0,c(f)(y) = σ
−1
m
∫
Rm
K(x, y;ω)f(x)h(rx)dx
coincides with the operator FωD = e
ω
(
1
2+
µ−1
2(1+c)
)
e
−ω
2(1+c)2
(D2−(1+c)2x2) on the basis {φt,`,m}.
Proof. We have already shown that the integral transform coincides with the operator FωD =
e
ω
(
1
2+
µ−1
2(1+c)
)
e
−ω
2(1+c)2
(D2−(1+c)2x2) on the basis {φt,`,m}. So we only have to show that the series
are convergent. We do this for the term
+∞∑
k=0
(2 sinhω)−γk/2
k + 2λ
2λ
z
k
1+c J˜ γk
2 −1
(
iz
sinhω
)
Cλk (w)
= (2 sinhω)−δ/2
+∞∑
k=0
k + 2λ
2λ
( z
2 sinhω
) k
1+c
J˜ γk
2 −1
(
iz
sinhω
)
Cλk (w),
the other ones are treated in a similar fashion. We obtain
∣
∣
∣
∣
∣
+∞∑
k=0
k + 2λ
2λ
( z
2 sinhω
) k
1+c
J˜ γk
2 −1
(
iz
sinhω
)
Cλk (w)
∣
∣
∣
∣
∣
≤
B(λ)
2
e|Im
iz
sinhω |
+∞∑
k=0
(k + 2λ)
∣
∣
∣
z
2 sinhω
∣
∣
∣
k
1+c 1
Γ(γk/2)
k2λ−1
using formula (A.1) and (A.2). As the term Γ(γk/2) is dominant, the series clearly converges. 
6.2 The case Reω = 0
In this case, we have the following theorem.
Theorem 6.3. Let c > −1. Then for ω = iη with η 6∈ piZ, we put
K(x, y; iη) = ei
cot η
2 (r
2+s2) (A(z, w) + x ∧ yB(z, w)
)
with
A(z, w) =
+∞∑
k=0
(
αk
k + 2λ
2λ
z
k
1+c J˜ γk
2 −1
(
z
sin η
)
+
αk−1
4i sin η
k
λ
z
k+c
1+c J˜ γk−1
2
(
z
sin η
))
Cλk (w),
B(z, w) =
+∞∑
k=1
(
−αkz
k
1+c−1J˜ γk
2 −1
(
z
sin η
)
+
αk−1
2i sin η
z
k+c
1+c−1J˜ γk−1
2
(
z
sin η
))
Cλ+1k−1 (w)
for z = |x||y|, w = 〈x, y〉/z, α−1 = 0 and αk = 2e
iηδ
2 (2i sin η)−γk/2.
These series are convergent and the unitary integral transform defined in distributional sense
on L20,c(R
m) by
F iη0,c(f)(y) = σ
−1
m
∫
Rm
K(x, y; iη)f(x)h(rx)dx
coincides with the operator F iηD = e
iη
(
1
2+
µ−1
2(1+c)
)
e
−iη
2(1+c)2
(D2−(1+c)2x2) on the basis {φt,`,m}.
Proof. This follows by taking the limit ω → iη in Theorem 6.2. 
The Clifford Deformation of the Hermite Semigroup 15
7 The series representation of the Fourier transform
The Fourier transform is the very special case of the holomorphic semigroup, evaluated at
ω = ipi/2. In this case, the kernel K(x, y) = K(x, y; ipi/2) is given by the following theorem.
Theorem 7.1. Put K(x, y) = A(z, w) + x ∧ yB(z, w) with
A(z, w) =
+∞∑
k=0
z−
δ−2
2
(
αk
k + 2λ
2λ
J γk
2 −1
(z)− iαk−1
k
2λ
J γk−1
2
(z)
)
Cλk (w),
B(z, w) =
+∞∑
k=1
z−
δ
2
(
−αkJ γk
2 −1
(z)− iαk−1J γk−1
2
(z)
)
Cλ+1k−1 (w)
and z = |x||y|, w = 〈x, y〉/z, α−1 = 0 and αk = e
− ipik2(1+c) . These series are convergent and the
integral transform defined in distributional sense on L20,c(R
m) by
F0,c(f)(y) = σ
−1
m
∫
Rm
K(x, y)f(x)h(rx)dx
coincides with the operator FD = e
ipi2
(
1
2+
µ−1
2(1+c)
)
e
−ipi
4(1+c)2
(D2−(1+c)2x2) on the basis {φt,`,m}.
Proof. Using the well-known identity (see [27, Exercise 21, p. 371])
∫ +∞
0
rα+1Jα(rs)L
α
j
(
r2
)
e−r
2/2dr = (−1)jsαLαj (s
2)e−s
2/2
we can prove in the same way as leading to Theorem 6.2 that the integral transform F0,c coincides
with
FD = e
ipi2
(
1
2+
µ−1
2(1+c)
)
e
−ipi
4(1+c)2
(D2−(1+c)2x2)
on the basis φt,`,m. The theorem also follows as a special case of Theorem 6.2, taking the limit
ω → ipi/2. 
Remark 7.1. One can also define an analogue of the Schwartz space of rapidly decreasing
functions in this context. Let L = D2 − (1 + c)2x2 and denote by D(L) the domain of L
in L20,c(R
m). Then the Schwartz space is defined by
S0,c
(
Rm
)
=
∞⋂
k=0
D
(
Lk
)
and one can check that the Fourier transform F0,c is an isomorphism of this space.
Remark 7.2. In the limit case c = 0, we can check that the kernel reduces to
K(x, y) =
+∞∑
k=0
k + λ
λ
(−i)kz−λJk+λ(z)C
λ
k (w).
This is a well-known expansion of the classical Fourier kernel (see [29, Section 11.5]):
K(x, y) =
e−i〈x,y〉
Γ(m/2)2
m−2
2
.
16 H. De Bie, B. Ørsted, P. Somberg and V. Soucˇek
We can now summarize the main properties of the deformed Fourier transform in the following
theorem.
Theorem 7.2. The operator F0,c defines a unitary operator on L20,c(R
m) and satisfies the fol-
lowing intertwining relations on a dense subset:
F0,c ◦D = i(1 + c)x ◦ F0,c, F0,c ◦ x =
i
1 + c
D ◦ F0,c, F0,c ◦ E = − (E + δ) ◦ F0,c.
Moreover, F0,c is of finite order if and only if c is rational.
Proof. Every f in L20,c(R
m) can be expanded in terms of the orthogonal basis φt,`,m, satisfying
〈φt1,`1,m1 , φt2,`2,m2〉 = δt1t2δ`1`2δm1m2〈φt1,`1,m1 , φt1,`1,m1〉,
see Section 3. Note that the normalization can be computed explicitly (see [7, Theorem 6]). As
the eigenvalues of F0,c are given by (see (4.4))
(−i)te−i
pi`
2(1+c)
which clearly live on the unit circle, we conclude that
〈f, g〉 = 〈F0,c(f),F0,c(g)〉
and that F0,c is a unitary operator.
The intertwining relations are an immediate consequence of formula (4.1) combined with the
fact that φt,`,m is an eigenbasis of F0,c. The formula for E follows from the anti-commutator
(see Theorem 3.1)
{D, x} = −2(1 + c)
(
E +
δ
2
)
.
The statement on the finite order of the Fourier transform is an immediate consequence of
the explicit expression for the eigenvalues of the transform. 
Now we collect some properties of the kernel K(x, y).
Proposition 7.1. One has, with x, y ∈ Rm
K(λx, y) = K(x, λy), λ > 0, K(y, x) = K(x, y),
K(0, y) =
1
2γ0/2−1Γ(γ0/2)
, K(sxs, sys, ) = sK(x, y)s, s ∈ Spin(m),
where ·¯ is the anti-involution on the Clifford algebra Clm.
Proof. The first property is trivial. The second follows because
x ∧ y = −x ∧ y = y ∧ x.
The third property follows from Theorem 7.1. Finally, the 4th equation follows because z and
w are spin-invariant and
(sxs) ∧ (sys) = s
(
x ∧ y
)
s. 
We can also obtain Bochner identities for the deformed Fourier transform. They are given in
the following proposition.
The Clifford Deformation of the Hermite Semigroup 17
Proposition 7.2. Let M` ∈ M` be a spherical monogenic of degree `. Let f(x) = f(r) be
a radial function. Then the Fourier transform of f(r)M` and f(r)xM` can be computed as
follows
F0,c(f(r)M`) = e
− ipi`2(1+c)M`(y
′)
∫ +∞
0
r`f(r)z−
δ−2
2 J γk
2 −1
(z)h(r)rm−1dr,
F0,c(f(r)xM`) = −ie
− ipi`2(1+c) y′M`(y
′)
∫ +∞
0
r`+1f(r)z−
δ−2
2 J γk
2
(z)h(r)rm−1dr
with y = sy′, y′ ∈ Sm−1 and z = rs.
Proof. This follows immediately from Theorem 7.1 combined with Proposition 5.1. 
Remark 7.3. As a special case of this proposition, we reobtain the eigenfunctions of the
Fourier transform by putting f(r) = L
γ`
2 −1
t (r
2)rβ`e−r
2/2, resp. f(r) = L
γ`
2
t (r
2)rβ`e−r
2/2 (see
equation (4.4)).
Now we prove the following lemma.
Lemma 7.1. For all f ∈ L20,c(R
m) one has
||xf(x)||2 + ||x (F0,cf) (x)||
2 ≥ δ||f(x)||2.
The equality holds if and only if f is a multiple of e−r
2/2.
Proof. Using formula (4.3) and the unitarity of F0,c, one can compute that
||xf(x)||2 + ||x (F0,cf) (x)||
2 =
1
(1 + c)2
〈
(
D2 − (1 + c)2x2
)
f, f〉.
Now use the fact that the smallest eigenvalue of
1
(1 + c)2
(
D2 − (1 + c)2x2
)
is given by δ, see Theorem 4.2. This proves the inequality.
The equality holds when f is a multiple of an eigenfunction corresponding to the smallest
eigenvalue, i.e. when f is a multiple of e−r
2/2. 
This lemma allows us to obtain the Heisenberg inequality for the deformed Fourier transform
Proposition 7.3. For all f ∈ L20,c(R
m), the deformed Fourier transform satisfies
||xf(x)|| · ||x (F0,cf) (x)|| ≥
δ
2
||f(x)||2.
The equality holds if and only if f is of the form f(x) = λe−r
2/α.
Proof. Using Lemma 7.1, we can continue in the same way as in the proof of Theorem 5.28
in [2]. 
Now we can obtain the Master formula for the kernel of the Fourier transform. We use the
formula (see [14, p. 50])
∫ +∞
0
Jν(at)Jν(bt)e
−γ2t2tdt =
1
2
γ−2e
−a
2+b2
4γ2 Iν
(
ab
2γ2
)
, Re ν > −1, Re γ2 > 0, (7.1)
where Iν(z) = e−i
piν
2 Jν(iz).
We then obtain
18 H. De Bie, B. Ørsted, P. Somberg and V. Soucˇek
Theorem 7.3 (Master formula). Let s > 0. Then one has
∫
Rm
K
(
y, x; i
pi
2
)
K
(
z, y;−i
pi
2
)
e−sr
2
yh(ry)dy = σme
−ωδ2 K(z, x;ω)e−
|x|2+|z|2
2
1−coshω
sinhω
with 2s = sinhω.
Proof. First observe that K
(
y, x; ipi2
)
= K(y, x) and that K(z, y;−ipi2 ) is the complex conjugate
of K
(
z, y; ipi2
)
.
We rewrite the kernel K obtained in Theorem 7.1 in terms of the reproducing kernels Pk
and Qk, i.e. as K(x, y) = K0(x, y) +K1(x, y) with
K0(x, y) =
+∞∑
k=0
αk
(
|x||y|
)− δ−22 J γk
2 −1
(|x||y|)Pk(x
′, y′),
K1(x, y) =
+∞∑
k=0
βk
(
|x||y|
)− δ−22 J γk
2
(|x||y|)Qk(x
′, y′),
where αk = e
− ipik2(1+c) and βk = −iαk.
When passing to spherical co-ordinates, the integral simplifies, using Lemma 5.2, to
σm
+∞∑
k=0
(|x||z|)−
δ−2
2 Pk(z
′, x′)
∫ +∞
0
re−sr
2
J γk
2 −1
(r|x|)J γk
2 −1
(r|z|)dr
+ σm
+∞∑
k=0
(|x||z|)−
δ−2
2 Qk(z
′, x′)
∫ +∞
0
re−sr
2
J γk
2
(r|x|)J γk
2
(r|z|)dr.
The radial integral can be computed explicitly using (7.1). Comparing with formula (6.1) and
Theorem 6.2 leads to the statement of the theorem. 
Remark 7.4. For the Dunkl transform (see, e.g., [24, 28]) and for the Clifford–Fourier transform
(see [8]) one can compute even a more general integral of the form
∫
Rm
K
(
y, x; i
pi
2
)
K
(
z, y;−i
pi
2
)
f(ry)h(ry)dy
with f(ry) an arbitrary radial function of suitable decay. This is done by using the addition
formula for the Bessel function
u−λJλ(u) = 2
λΓ(λ)
∞∑
k=0
(k + λ)(r2|x||z|)−λJk+λ(r|x|)Jk+λ(r|z|)C
λ
k (〈x
′, z′〉)
with u = r
√
|x|2 + |z|2 − 2〈x, z〉 instead of formula (7.1). Here, we cannot do that, as the orders
of the Bessel functions do not match the order of the Gegenbauer polynomials.
Remark 7.5. Theorem 7.3 is the starting point for the study of a generalized heat equation,
see, e.g., [26, Lemma 4.5(1)] in the context of Dunkl operators.
8 Further results for the kernel
In this section we will always be working in the non-Dunkl case, i.e. we put the multiplicity
function κ = 0. Theorem 7.1 implies that the kernel of our deformed Fourier transform is
a function of the type
K(x, y) = f(z, w) + x ∧ yg(z, w)
The Clifford Deformation of the Hermite Semigroup 19
with f , g scalar functions of the variables z = |x||y| and w = 〈x, y〉/z. On the other hand, this
kernel needs to satisfy the system of PDEs
DyK(x, y) = −i(1 + c)K(x, y)x, (K(x, y)Dx) = −i(1 + c)yK(x, y),
as can be deduced from Theorem 7.2. In order to rewrite this system in terms of the vari-
ables z, w, we first observe that
∂xf(z, w) = r
−2xz∂zf(z, w) +
(
z−1y − r−2xw
)
∂wf(z, w),
Ef(z, w) = z∂zf(z, w), ∂x
(
x ∧ y
)
= (1−m)y.
Using these identities, one obtains that the kernel is determined by the following 2 PDEs:
(m− 1 + c)g + (1 + c)z∂zg +
1
z
∂wf + i(1 + c)f − i(1 + c)zwg = 0,
(1 + c)z∂zf − w∂wf − czwg − (1 + c)z
2w∂zg + z(w
2 − 1)∂wg + i(1 + c)z
2g = 0. (8.1)
Remark 8.1. Note that, contrary to the case of the classical Fourier transform and the Dunkl
transform, where the kernel is uniquely determined by the system of PDEs
Tj,xK(x, y) = iyjK(x, y), j = 1, . . . ,m
this is not the case for the kernel of the radially deformed Fourier transform. In fact, one can
observe that there exist several different types of solutions of (8.1). This is discussed in detail
in [6] for a similar system of PDEs in the context of the so-called Clifford–Fourier transform
(see [8]).
Now we show that it is sufficient to solve this system in dimension m = 2 and m = 3. Recall
that the kernel K(x, y) is given in Theorem 7.1. To know this kernel, it is hence sufficient to
know the series
Aλ =
+∞∑
k=0
αk(k + λ)J γk
2 −1
(z)Cλk (w), Dλ =
+∞∑
k=0
αk−1J γk−1
2
(z)Cλk (w),
Bλ =
+∞∑
k=0
αkJ γk
2 −1
(z)Cλk (w), Eλ =
+∞∑
k=1
αkJ γk
2 −1
(z)Cλ+1k−1 (w),
Cλ =
+∞∑
k=0
αk−1(k + λ)J γk−1
2
(z)Cλk (w), Fλ =
+∞∑
k=1
αk−1J γk−1
2
(z)Cλ+1k−1 (w),
because then one has
K =
1
2λ
z−
δ−2
2 (Aλ − iCλ) +
1
2
z−
δ−2
2 (Bλ + iDλ)− z
− δ2x ∧ y (Eλ + iFλ) .
Using the well-known property of the Gegenbauer polynomials 2λCλ+1k−1 (w) = ∂wC
λ
k (w), we
observe the following recursion relations
Aλ+1 = e
i pi2(1+c)
1
2λ
∂wAλ, Bλ+1 = e
i pi2(1+c)
1
2λ
∂wBλ, Cλ+1 = e
−i pi2(1+c)
1
2λ
∂wCλ,
Dλ+1 = e
−i pi2(1+c)
1
2λ
∂wDλ, Eλ =
1
2λ
∂wBλ, Fλ =
1
2λ
∂wDλ.
We conclude that it suffices to know Aλ, Bλ, Cλ and Dλ for λ = 0, 1/2 or m = 2, 3. At this
point, the problem of finding explicit expressions for these functions for special values of the
deformation parameter c is still open.
20 H. De Bie, B. Ørsted, P. Somberg and V. Soucˇek
A Properties of Laguerre and Gegenbauer polynomials
The generalized Laguerre polynomials L(α)k for k ∈ N are defined as
L(α)k (t) =
k∑
j=0
Γ(k + α+ 1)
j!(k − j)!Γ(j + α+ 1)
(−t)j
and satisfy the orthogonality relation (when α > −1)
∫ ∞
0
tαL(α)k (t)L
(α)
l (t) exp(−t)dt = δkl
Γ(k + α+ 1)
k!
.
The Gegenbauer polynomials C(α)k (t) are a special case of the Jacobi polynomials. For k ∈ N
and α > −1/2 they are defined as
C(α)k (t) =
bk/2c∑
j=0
(−1)j
Γ(k − j + α)
Γ(α)j!(k − 2j)!
(2t)k−2j
and satisfy the orthogonality relation
∫ 1
−1
C(α)k (t)C
(α)
l (t)
(
1− t2
)α− 12dt = δkl
pi21−2αΓ(k + 2α)
k!(k + α)(Γ(α))2
.
One can prove that there exists a constant B(α) such that
sup
−1≤t≤1
∣
∣
∣
∣
1
α
C(α)k (t)
∣
∣
∣
∣ ≤ B(α)k
2α−1, ∀ k ∈ N, (A.1)
see [2, Lemma 4.9].
The Bessel function Jν(z) is defined using the following Taylor series
Jν(z) =
∞∑
k=0
(−1)k
k!Γ(k + ν + 1)
(z
2
)2k+ν
.
For z ∈ C and ν ≥ −1/2 one has the inequality (see, e.g., [27])
∣
∣
∣
∣
(z
2
)−ν
Jν(z)
∣
∣
∣
∣ ≤
1
Γ(ν + 1)
e| Im z|. (A.2)
B List of notations
List of notations used in this paper:
m dimension of Rm,
κ multiplicity function on root system,
µ Dunkl-dimension,
c deformation parameter of D,
ω semigroup parameter with Reω ≥ 0,
∂x ordinary Dirac operator,
Dk Dunkl Dirac operator,
The Clifford Deformation of the Hermite Semigroup 21
D radially deformed Dirac operator,
FωD exponential form of the holomorphic semigroup,
Fω0,c integral form of the holomorphic semigroup,
FD exponential form of the Fourier transform,
F0,c integral form of the Fourier transform.
We also have the following definitions:
µ = m+ 2
∑
α∈R+
κα, λ =
m− 2
2
, σm = 2pi
m/2/Γ(m/2), δ = 1 +
µ− 1
1 + c
,
β` = −
c
1 + c
`, ` ∈ N, γ` =
2
1 + c
(
`+
µ− 2
2
)
+
c+ 2
1 + c
, ` ∈ N.
Notations for variables. Let x and y be vector variables in Rm. Then we denote
z = |x||y|, w = 〈x, y〉/z.
When using spherical co-ordinates, we use x = rx′ with x′ ∈ Sm−1, hereby implicitly identifying
a vector in the Clifford algebra with a vector in Rm.
Acknowledgements
H. De Bie would like to thank E. Opdam and J. Stokman for valuable input and discussions
during his visit to the Korteweg–de Vries Institute for Mathematics in Amsterdam. This visit
was supported by a Postdoctoral Fellowship of the Research Foundation – Flanders (FWO). The
last two authors would like to acknowledge support of the research grant GA CˇR P201/12/G028.
References
[1] Barbasch D., Ciubotaru D., Trapa P.E., Dirac cohomology for graded affine Hecke algebras, Acta Math.
209 (2012), 197–227, arXiv:1006.3822.
[2] Ben Sa¨ıd S., Kobayashi T., Ørsted B., Laguerre semigroup and Dunkl operators, Compos. Math. 148 (2012),
1265–1336, arXiv:0907.3749.
[3] Cac¸a˜o I., Constales D., Krausshar R.S., On the role of arbitrary order Bessel functions in higher dimensional
Dirac type equations, Arch. Math. (Basel) 87 (2006), 468–477.
[4] Cherednik I., Markov Y., Hankel transform via double Hecke algebra, in Iwahori–Hecke algebras and their
representation theory (Martina–Franca, 1999), Lecture Notes in Math., Vol. 1804, Springer, Berlin, 2002,
1–25, math.QA/0004116.
[5] De Bie H., De Schepper N., Clifford–Gegenbauer polynomials related to the Dunkl Dirac operator, Bull.
Belg. Math. Soc. Simon Stevin 18 (2011), 193–214.
[6] De Bie H., De Schepper N., Sommen F., The class of Clifford–Fourier transforms, J. Fourier Anal. Appl.
17 (2011), 1198–1231, arXiv:1101.1793.
[7] De Bie H., Ørsted B., Somberg P., Soucˇek V., Dunkl operators and a family of realizations of osp(1|2),
Trans. Amer. Math. Soc. 364 (2012), 3875–3902, arXiv:0911.4725.
[8] De Bie H., Xu Y., On the Clifford–Fourier transform, Int. Math. Res. Not. 2011 (2011), no. 22, 5123–5163,
arXiv:1003.0689.
[9] Delanghe R., Sommen F., Soucˇek V., Clifford algebra and spinor-valued functions. A function theory for the
Dirac operator, Mathematics and its Applications, Vol. 53, Kluwer Academic Publishers Group, Dordrecht,
1992.
[10] Dunkl C.F., Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311
(1989), 167–183.
22 H. De Bie, B. Ørsted, P. Somberg and V. Soucˇek
[11] Dunkl C.F., Hankel transforms associated to finite reflection groups, in Hypergeometric Functions on Do-
mains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemp. Math., Vol. 138,
Amer. Math. Soc., Providence, RI, 1992, 123–138.
[12] Dunkl C.F., de Jeu M.F.E., Opdam E.M., Singular polynomials for finite reflection groups, Trans. Amer.
Math. Soc. 346 (1994), 237–256.
[13] Dunkl C.F., Xu Y., Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its
Applications, Vol. 81, Cambridge University Press, Cambridge, 2001.
[14] Erde´lyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, Vol. II, Mc-Graw
Hill, New York, 1953.
[15] Frappat L., Sciarrino A., Sorba P., Dictionary on Lie algebras and superalgebras, Academic Press Inc., San
Diego, CA, 2000.
[16] Gilbert J.E., Murray M.A.M., Clifford algebras and Dirac operators in harmonic analysis, Cambridge Studies
in Advanced Mathematics, Vol. 26, Cambridge University Press, Cambridge, 1991.
[17] Howe R., The oscillator semigroup, in The Mathematical Heritage of Hermann Weyl (Durham, NC, 1987),
Proc. Sympos. Pure Math., Vol. 48, Amer. Math. Soc., Providence, RI, 1988, 61–132.
[18] Humphreys J.E., Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics,
Vol. 29, Cambridge University Press, Cambridge, 1990.
[19] Kobayashi T., Mano G., Integral formulae for the minimal representation of O(p, 2), Acta Appl. Math. 86
(2005), 103–113.
[20] Kobayashi T., Mano G., The inversion formula and holomorphic extension of the minimal representation
of the conformal group, in Harmonic Analysis, Group Representations, Automorphic Forms and Invariant
Theory, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., Vol. 12, World Sci. Publ., Hackensack, NJ,
2007, 151–208, math.RT/0607007.
[21] Kobayashi T., Mano G., The Schro¨dinger model for the minimal representation of the indefinite orthogonal
group O(p, q), Mem. Amer. Math. Soc. 213 (2011), no. 1000, vi+132 pages, arXiv:0712.1769.
[22] Ol’shanski˘ı G.I., Complex Lie semigroups, Hardy spaces and the Gel’fand–Gindikin program, Dif ferential
Geom. Appl. 1 (1991), 235–246.
[23] Ørsted B., Somberg P., Soucˇek V., The Howe duality for the Dunkl version of the Dirac operator, Adv. Appl.
Clif ford Algebr. 19 (2009), 403–415.
[24] Ro¨sler M., A positive radial product formula for the Dunkl kernel, Trans. Amer. Math. Soc. 355 (2003),
2413–2438, math.CA/0210137.
[25] Ro¨sler M., Dunkl operators: theory and applications, in Orthogonal Polynomials and Special Functions
(Leuven, 2002), Lecture Notes in Math., Vol. 1817, Springer, Berlin, 2003, 93–135, math.CA/0210366.
[26] Ro¨sler M., Generalized Hermite polynomials and the heat equation for Dunkl operators, Comm. Math. Phys.
192 (1998), 519–542, q-alg/9703006.
[27] Szego¨ G., Orthogonal polynomials, American Mathematical Society, Colloquium Publications, Vol. 23,
4th ed., American Mathematical Society, Providence, R.I., 1975.
[28] Thangavelu S., Xu Y., Convolution operator and maximal function for the Dunkl transform, J. Anal. Math.
97 (2005), 25–55, math.CA/0403049.
[29] Watson G.N., A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, 1944.