Dunkl-Type Operators with Projection Terms Associated to Orthogonal Subsystems in Root System
In this paper, we introduce a new differential-difference operator Tξ (ξ∈RN) by using projections associated to orthogonal subsystems in root systems. Similarly to Dunkl theory, we show that these operators commute and we construct an intertwining operator between Tξ and the directional derivative ∂...
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irk-123456789-1493562019-02-22T01:23:07Z Dunkl-Type Operators with Projection Terms Associated to Orthogonal Subsystems in Root System Bouzeffour, F. In this paper, we introduce a new differential-difference operator Tξ (ξ∈RN) by using projections associated to orthogonal subsystems in root systems. Similarly to Dunkl theory, we show that these operators commute and we construct an intertwining operator between Tξ and the directional derivative ∂ξ. In the case of one variable, we prove that the Kummer functions are eigenfunctions of this operator. 2013 Article Dunkl-Type Operators with Projection Terms Associated to Orthogonal Subsystems in Root System / F. Bouzeffour // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 15 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33C15; 33D52; 35A22 DOI: http://dx.doi.org/10.3842/SIGMA.2013.064 http://dspace.nbuv.gov.ua/handle/123456789/149356 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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In this paper, we introduce a new differential-difference operator Tξ (ξ∈RN) by using projections associated to orthogonal subsystems in root systems. Similarly to Dunkl theory, we show that these operators commute and we construct an intertwining operator between Tξ and the directional derivative ∂ξ. In the case of one variable, we prove that the Kummer functions are eigenfunctions of this operator. |
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Bouzeffour, F. Dunkl-Type Operators with Projection Terms Associated to Orthogonal Subsystems in Root System Symmetry, Integrability and Geometry: Methods and Applications |
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Bouzeffour, F. |
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Bouzeffour, F. |
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Dunkl-Type Operators with Projection Terms Associated to Orthogonal Subsystems in Root System |
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Dunkl-Type Operators with Projection Terms Associated to Orthogonal Subsystems in Root System |
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Dunkl-Type Operators with Projection Terms Associated to Orthogonal Subsystems in Root System |
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Dunkl-Type Operators with Projection Terms Associated to Orthogonal Subsystems in Root System |
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Dunkl-Type Operators with Projection Terms Associated to Orthogonal Subsystems in Root System |
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dunkl-type operators with projection terms associated to orthogonal subsystems in root system |
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Інститут математики НАН України |
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2013 |
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Dunkl-Type Operators with Projection Terms Associated to Orthogonal Subsystems in Root System / F. Bouzeffour // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 15 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT bouzeffourf dunkltypeoperatorswithprojectiontermsassociatedtoorthogonalsubsystemsinrootsystem |
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2025-07-12T21:55:32Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 064, 14 pages
Dunkl-Type Operators with Projection Terms
Associated to Orthogonal Subsystems
in Root System
Fethi BOUZEFFOUR
Department of Mathematics, King Saudi University, College of Sciences,
P.O. Box 2455 Riyadh 11451, Saudi Arabia
E-mail: fbouzaffour@ksu.edu.sa
Received April 24, 2013, in final form October 16, 2013; Published online October 23, 2013
http://dx.doi.org/10.3842/SIGMA.2013.064
Abstract. In this paper, we introduce a new differential-difference operator Tξ (ξ ∈ RN ) by
using projections associated to orthogonal subsystems in root systems. Similarly to Dunkl
theory, we show that these operators commute and we construct an intertwining operator
between Tξ and the directional derivative ∂ξ. In the case of one variable, we prove that the
Kummer functions are eigenfunctions of this operator.
Key words: special functions; differential-difference operators; integral transforms
2010 Mathematics Subject Classification: 33C15; 33D52; 35A22
1 Introduction
In a series of papers [3, 4, 5, 6], C.F. Dunkl builds up the framework for a theory of differential-
difference operators and special functions related to root systems. Beside them, there are
now various further Dunkl-type operators, in particular the trigonometric Dunkl operators of
Heckman [7, 8], Opdam [14], Cherednik [2], and the important q-analogues of Macdonald and
Cherednik [13], see also [1, 11].
The main objective of this paper is to present a new class of differential-difference opera-
tors Tξ, ξ ∈ RN with the help of orthogonal projections related to orthogonal subsystems in
root systems. In other words, our operators follow from Dunkl operator after replacing the
usual reflections that exist in the definition of the operator with their corresponding ortho-
gonal projections. Several problems related to the Dunkl theory arise in the setting of our
operators, in particular, commutativity of {Tξ, ξ ∈ RN} and the existence of the intertwining
operators.
The outline of the content of this paper is as follows. In Section 2, we collect some definitions
and results related to root systems and Dunkl operators which will be relevant for the sequel.
In Section 3, we introduce new differential-difference operators Tξ and we prove the first main
result. In Section 4, we give an explicit formula for the intertwining operator between Tξ and
the directional derivative. In Section 5, we study the one variable case. Finally, in Section 6 we
study the cases of orthogonal subsets in root systems of type AN−1 and BN .
2 Dunkl operators
Let us begin to recall some results concerning the root systems and Dunkl operators. A useful
reference for this topic is the book by Humphreys [9]. Let α ∈ RN\{0}, we denote by sα the
mailto:fbouzaffour@ksu.edu.sa
http://dx.doi.org/10.3842/SIGMA.2013.064
2 F. Bouzeffour
reflection onto the hyperplane orthogonal to α; that is,
sα(x) = x− 2
〈x, α〉
|α|2
α,
where 〈·, ·〉 denotes the Euclidean scalar product on RN , and |x| =
√
〈x, x〉.
A root system is a finite set R of nonzero vectors in RN such that for any α ∈ R one has
sα(R) = R, and R ∩ Rα = {±α}.
A positive subsystem R+ is any subset of R satisfying R = R+ ∪ {−R+}. The Weyl group
W = W (R) (or real finite reflection group) generated by the root system R ⊂ RN is the
subgroup of orthogonal group O(N) generated by {sα, α ∈ R}. A multiplicity function on R is
a complex-valued function κ : R→ C which is invariant under the Weyl group W , i.e.,
κ(α) = κ(gα), ∀α ∈ R, ∀ g ∈W.
Let ξ ∈ RN , the Dunkl operator Dξ associated with the Weyl group W (R) and the multiplicity
function κ, is the first order differential-difference operator:
(Dξf)(x) = ∂ξf(x) +
∑
α∈R+
κ(α)〈α, ξ〉f(x)− f(sαx)
〈x, α〉
. (1)
Here ∂ξ is the direction derivative corresponding to ξ and sα is the orthogonal reflection onto
the hyperplane orthogonal to α.
The Dunkl operator Dξ is a homogeneous differential-difference operator of degree −1. By
the W -invariance of the multiplicity function κ, we have
g−1 ◦ Dξ ◦ g = Dgξ, ∀ g ∈W (R), ξ ∈ RN .
The remarkable property of the Dunkl operators is that the family {Dξ, ξ ∈ RN} generates
a commutative algebra of linear operators on the C-algebra of polynomial functions.
3 Operators of Dunkl-type
Let R be a root system. A subset R′ of R is called a subsystem of R if it satisfies the following
conditions:
i) If α ∈ R′, then −α ∈ R′;
ii) If α, β ∈ R′ and α+ β ∈ R, then α+ β ∈ R′.
A subsystem R′ of a root system R in RN consisting of pairwise orthogonal roots is called
orthogonal subsystem. In this case the related Weyl group W (R′) is a subgroup of ZN2 . For
a vector α ∈ RN \ {0}, we write
τα(x) = x− 〈x, α〉
|α|2
α, x ∈ RN ,
for the orthogonal projection onto the hyperplane (Rα)⊥ = {x, 〈x, α〉 = 0}, so that the reflec-
tion sα with respect to hyperplane orthogonal to α is related to τα by
τα =
1
2
(1 + sα).
Dunkl-Type Operators with Projection Terms 3
The hyperplane (Rα)⊥ is the invariant set of τα. If 〈α, β〉 = 0, then the orthogonal projec-
tions τα and τβ commute. The conjugate of orthogonal projection onto a hyperplane is again
an orthogonal projection onto a hyperplane: suppose u ∈ O(N) and α ∈ RN\{0} then
uταu
−1 = τuα.
Let R be a root system and R′ a positive orthogonal subsystem of R. For ξ ∈ RN , we define
the differential-difference operator Tξ by
(Tξf)(x) = ∂ξf(x) +
∑
α∈R′
κ(α)〈α, ξ〉f(x)− f(ταx)
〈x, α〉
. (2)
where κ is a multiplicity function on R′. For j = 1, . . . , N denotes Tej by Tj . The operator Tξ
can be considered as a deformation of the usual directional derivatives and when κ = 0, the
operator Tξ reduces to the corresponding directional derivative. Furthermore, there is overlap
between the notations (2) and (1). In fact, the operator (2) follows from Dunkl operator after
replacing the reflections terms that exist in (1) by orthogonal projection terms.
Example 1. In the rank-one case, the root system is of type A1 and the corresponding reflection
s and orthogonal projection τ are given by
s(x) = −x, τ(x) =
1
2
(1 + s)(x) = 0.
The Dunkl-type operator Tκ associated with the projection τ and the multiplicity parameters κ
(κ ∈ C) is given by
Tκf(x) = f ′(x) + κ
f(x)− f(τ(x))
x
= f ′(x) + κ
f(x)− f(0)
x
.
Example 2. Let R = {±(e1 ± e2),±e1,±e2} be a root system of type B2 in the 2-plane and
R′ = {e1±e2} be a positive orthogonal subsystem in R. The related Dunkl-type operators to R′
and to the positive parameters (κ1, κ2) are given by
T1 = ∂x + κ1
f(x, y)− f((x+ y)/2, (x+ y)/2)
x− y
+ κ2
f(x, y)− f((x− y)/2, (x− y)/2)
x+ y
,
T2 = ∂y − κ1
f(x, y)− f((x+ y)/2, (x+ y)/2)
x− y
+ κ2
f(x, y)− f((x− y)/2, (x− y)/2)
x+ y
.
We denote by ΠN the space of polynomials and by ΠN
n the subspace of homogenous polyno-
mials of degree n.
Let R′ = {α1, . . . , αn} be a positive orthogonal subsystem of a root system R. Consider the
operator ρi defined on ΠN by
(ρif)(x) =
f(x)− f(ταix)
〈x, α〉
, i = 1, . . . , n.
It follows from the equality
(ρjf)(x) = − 1
|αj |2
∫ 1
0
∂αjf
(
x− t〈x, αj〉
|αj |2
αj
)
dt
that Tξ is a homogeneous operator of degree −1 on ΠN , that is, Tξf ∈ ΠN
n−1, for f ∈ ΠN
n , and
leaves S(R) (S(R) is the Schwartz space of rapidly decreasing functions on R) invariant.
4 F. Bouzeffour
Proposition 1. The operators ρi (i = 1, . . . , n) have the following properties:
i) for i, j = 1, . . . , n, we have [ρi, ρj ] = 0;
ii) if α is an orthogonal vector to αi, then [∂α, ρi] = 0, where the commutator of two operators
A, B is defined by [A,B] := AB −BA.
The family {α1, . . . , αn} is orthogonal, then there exist scalars ξ1, . . . , ξn and a vector ξ̂ ∈ RN
orthogonal to the subspace Rα1 ⊕ · · · ⊕ Rαn such that
ξ =
n∑
i=1
ξiαi + ξ̂.
This allows us to decompose the operator Tξ (2) associated with R′ and the multiplicity para-
meters (κ1, . . . , κn) in a unique way in the form
Tξ =
n∑
i=1
ξiTαi + ∂
ξ̂
.
We now have all ingredients to state and prove the first main result of the paper.
Theorem 1. Let ξ, η ∈ RN , then [Tξ, Tη] = 0.
Proof. A straightforward computation yields
[Tξ, Tη] =
n∑
i,j=1
ξiηj [Tαi , Tαj ] + [∂
ξ̂
, ∂η̂] +
n∑
i=1
ξi[Tαi , ∂η̂]− ηi[Tαi , ∂ξ̂].
On the other hand,
[Tαi , Tαj ] = [∂αi + κi‖αi‖ρi, ∂αj + κj‖αj‖ρj ]
= [∂αi , ∂αj ] + κj‖αj‖[∂αi , ρj ]− κi‖αi‖[∂αj , ρi] + κiκj‖αi‖‖αj‖[ρi, ρj ],
and
[Tαi , ∂ξ̂] = [∂αi , ∂ξ̂] + κi‖αi‖[ρi, ∂ξ].
From Proposition 1, we get
[Tαi , Tαj ] = 0 and [Tαi , ∂ξ̂] = 0.
This proves the result. �
One important consequence of the Theorem 1, is that the operators Tα1 , . . . , Tαm generate
a commutative algebra.
4 Intertwining operator
In this section, we give an intertwining operator between Tξ and the directional derivative ∂ξ.
Consider a positive orthogonal subsystem R′ = {α1, . . . , αn} composed of n vectors in a root
system R, and κ = (κ1, . . . , κn) ∈ Cn and ξ ∈ RN . The associated Dunkl-type operator Tξ
with R′ and κ takes the form
(Tξf)(x) = ∂ξf(x) +
n∑
j=1
κj〈αj , ξ〉
f(x)− f(ταjx)
〈x, αj〉
.
Dunkl-Type Operators with Projection Terms 5
Let h : Rn × RN → RN be the function defined by
h(t, x) = x+
n∑
j=1
(tj − 1)
〈x, αj〉
|αj |2
αj ,
where t = (t1, . . . , tn) ∈ Rn and x ∈ RN .
We define
χκ(f)(x) =
1
Γ(κ)
∫
[0,1]n
f(h(t, x))w(t) dt, (3)
where w(t) =
n∏
j=1
(1− tj)κj−1 and Γ(κ) =
n∏
j=1
Γ(κj).
Theorem 2. Let f ∈ C∞(RN ), then we have
Tξ ◦ χκf(x) = χκ ◦ ∂ξf(x).
Proof. For j = 1, . . . , n, we denote by θj the orthogonal projection in Rn with respect to the
hyperplane (Rej)⊥ orthogonal to the vector ej of the canonical basis (e1, . . . , en) of Rn. The
orthogonal projection θj acts on Rn as
θj(t) = (t1, . . . , tj−1, 0, tj+1, . . . , tn).
The system R is orthogonal, then for j = 1, . . . , n, we have
h(t, ταjx) = ταjx+
n∑
k=1
(tk − 1)
〈ταjx, αk〉
|αk|2
αk
= x− 〈x, αj〉
|αj |2
αj +
n∑
k=1,k 6=j
(tk − 1)
〈x, αk〉
|αk|2
αk = h(θjt, x).
Let f ∈ C∞(RN ) and ξ ∈ RN . The mapping x→ h(t, x) is linear on RN , then we can write
∂ξ(f(h(t, x))) = ∂h(t,ξ)f(h(t, x)) = ∂ξf(h(t, x)) +
n∑
j=1
(tj − 1)
〈ξ, αj〉
|αj |2
∂αjf(h(t, x)).
Hence,
∂ξχκ(f)(x) =
1
Γ(κ)
∫
[0,1]n
∂ξ(f(h(t, x)))w(t) dt =
1
Γ(κ)
∫
[0,1]n
∂ξf(h(t, x))w(t) dt
+
1
Γ(κ)
n∑
j=1
〈ξ, αj〉
|αj |2
∫
[0,1]n
(tj − 1)∂αjf(h(t, x))w(t) dt.
Since we can write
∂tjf(h(t, x)) =
〈x, αk〉
|αk|2
∂αjf(h(t, x))
and ∫ 1
0
(1− tj)κj∂tjf(h(t, x)) dt = −f(h(θj(t), x)) + κj
∫ 1
0
(1− tj)κj−1f(h(t, x)) dt,
6 F. Bouzeffour
we are lead to∫
[0,1]n
∂αjf(h(t, x))(tj − 1)w(t) dt =
|αj |2
〈x, αj〉
∫
[0,1]n
∂tjf(h(t, x))(tj − 1)w(t) dt
= κj
|αj |2
〈x, αj〉
∫
[0,1]n
(f(h(θj(t), x))− f(h(t, x)))w(t) dt
= −κjΓ(κ)
|αj |2
〈x, αj〉
(
χκ(f)(x)− χκ(f)(ταjx)
)
.
This, combined with the last expression of ∂ξ(χκf)(x), yields
∂ξχκ(f)(x) = χκ(∂ξf)(x)−
n∑
j=1
κj〈ξ, αj〉
χκ(f)(x)− χκ(f)(τjx)
〈x, αj〉
.
Therefore,
Tξ(χκf)(x) = χκ(∂ξf)(x). �
5 The one variable case
The specialization of this theory to the one variable case has its own interest, because everything
can be done there in a much more explicit way and new results for special functions in one
variable can be obtained. In this setting there is only one Dunkl-type operator Tκ associated up
to scaling and it equals to
Tκf(x) = f ′(x) + κ
f(x)− f(0)
x
. (4)
This operator leaves the space of polynomials invariant and acts on the monomials as
Tκ1 = 0, Tκx
n = (n+ κ)xn−1, n = 1, 2, . . . .
Its square is given by
T 2
κf(x) = f ′′(x) +
2κ
x
f ′(x) + κ(κ− 1)
f(x)− f(0)
x2
− κ(κ+ 1)
x
f ′(0).
Consider the confluent hypergeometric function (see [15, § 7.1])
M(a, b; z) =
∞∑
n=0
(a)n
(b)n
zn
n!
,
where (a)n is the Pochhammer symbol defined by
(a)n =
Γ(a+ n)
Γ(a)
.
This is a solution of the confluent hypergeometric differential equation
zy′′(z) + (b− z)y′(z) = ay(z).
This function possesses the following Poisson integral representation (see [15, § 7.1])
M(a, b; z) =
Γ(b)
Γ(a)Γ(b− a)
∫ 1
0
ta−1(1− t)b−a−1ezt dt, <(b) > <(a) > 0. (5)
Dunkl-Type Operators with Projection Terms 7
Theorem 3. For λ ∈ C and κ > −1, the problem
Tκf(x) = iλf(x), f(0) = 1, (6)
has a unique analytic solution Mκ(iλx) given by
Mκ(iλx) = M(1, κ+ 1; iλx). (7)
Proof. Searching a solution of (6) in the form f(z) =
∞∑
n=0
anx
n. Replacing in (6), we obtain
∞∑
n=0
(n+ 1 + κ)an+1x
n = iλ
∞∑
n=0
anx
n.
Thus,
an+1 =
iλ
n+ 1 + κ
an and an =
(iλ)n
(κ+ 1)n
. �
Remark 1. Multiply the equation (6) by x and differentiating both sides, we see that a func-
tion u of class C2 on R, is a solution of the equation (6), if and only if, it is a solution of the
generalized eigenvalue problem
xu′′ + (κ+ 1)u′ = iλ(xu′ + u).
Proposition 2. The function Mκ(z) defined by
Mκ(z) =
Mκ(z)
Γ(κ+ 1)
=
∞∑
n=0
zn
Γ(κ+ 1 + n)
(8)
satisfies the following properties:
(i) Mκ(z) is analytic in κ and z;
(ii) M0(z) = ez;
(iii) for <(κ) > 0, the function Mκ(z), possesses the integral representation
Mκ(z) =
1
Γ(κ)
∫ 1
0
(1− t)κ−1ezt dt;
(iv) for <(κ) > 0, we have∣∣M(n)
κ (z)
∣∣ ≤ |z|ne<(z), n ∈ N, z ∈ C,
in particular,
|Mκ(iλx)| ≤ 1, λ, x ∈ R;
(v) for <(κ) > 0, and all x ∈ R∗,
lim
λ→+∞
Mκ(iλx) = 0.
8 F. Bouzeffour
Proof. (i) and (ii) are immediate. (iii) follows from (5). For n ∈ N, we have
M(n)
κ (z) =
zn
Γ(κ)
∫ 1
0
(1− t)κtnezt dt.
So we find
|M(n)
κ (z)| ≤ |z|
n
Γ(κ)
∫ 1
0
(1− t)κe<(z)t dt ≤ |z|ne<(z).
This proves (iv). (v) follows from (iii) and the Riemann–Lebesgue lemma. �
Definition 1. We define the Kummer transform on L1(R) by
∀λ ∈ R, Fκ(f)(λ) =
∫
R
f(x)Mκ(iλx)(x) dx.
When κ = 0, the transformation F0 reduces to the usual Fourier transform F that is given by
F(f)(λ) =
∫
R
f(x)eiλx dx.
Theorem 4. Let f be a function in L1(R) then Fκ(f) belongs to C0(R), where C0(R) is the
space of continuous functions having zero as limit at the infinity. Furthermore,
‖Fκ(f)‖∞ ≤ ‖f‖1.
Proof. It’s clear that Fκ(f) is a continuous function on R. From Proposition 2, we get for all
x ∈ R∗,
lim
λ→∞
f(x)Mκ(iλx) = 0 and |f(x)Mκ(iλx)| ≤ |f(x)|.
Since f is in L1(R), we conclude by using the dominated convergence theorem that Fκ(f) belongs
to C0(R) and
‖Fκ(f)‖∞ ≤ ‖f‖1.
We now turn to exhibit a relationship between the Kummer transform and the Fourier transform.
The crucial idea is to use the intertwining operator χκ. We denote by C∞(R) the space of
infinitely differentiable functions f on R, provided with the topology defined by the semi norms
‖f‖n,a = sup
0≤k≤n
x∈[−a,a]
∣∣f (k)(x)
∣∣, a > 0, n ∈ N.
In the rank-one case the intertwining operator (3) becomes
(χκf)(x) =
1
Γ(κ)
∫ 1
0
(1− t)κ−1f(tx) dt. (9)
This operator is a particular case of the so called Erdélyi–Kober fractional integral Iγ,δ, which
is given by (see [10])
(Iγ,δf)(x) =
1
Γ(δ)
∫ 1
0
(1− t)δ−1tγf(tx) dt, δ > 0, γ ∈ R.
Dunkl-Type Operators with Projection Terms 9
It was shown in [12, § 3], that the Erdélyi–Kober fractional integral has a left-inverse
Dγ,δIγ,δf = f, f ∈ C∞(R), (10)
where
Dγ,δ =
n∏
k=1
(
γ + k + x
d
dx
)
Iγ+δ,n−δ,
and n = dδe (dδe denotes the ceiling function the smallest integer ≥ δ).
As a consequence of Theorem 2, we deduce that the operator χκ (9) has the fundamental
intertwining property
Tκ ◦ χκ = χκ ◦
d
dx
.
We regard it as a second main result since it allows us to move from the complicated operator Tκ
defined in (4) to the simple derivative operator d
dx . �
Theorem 5. Let κ > 0, the operator χκ is a topological isomorphism from C∞(R) onto itself
and its inverse χ−1κ is given for all f ∈ C∞(R) by
χ−1κ f(x) = D0,κf(x) =
n∏
j=1
(
j + x
d
dx
)
(Iκ+1,n−κf)(x),
where n = dκe.
Proof. Let a > 0 and f ∈ C∞(R). For x ∈ [0, a], t ∈ [0, 1] and l ∈ N, we have the following
estimate
|tl(1− t)κ−1f (l)(xt)| ≤ ‖f‖l,a(1− t)κ−1 and
∫ 1
0
(1− t)κ−1 dt =
1
κ
.
By the theorem of derivation under the integral sign, we can prove that
χκf ∈ C∞(R) and ‖χκ(f)‖l,a ≤
1
Γ(κ+ 1)
‖f‖l,a.
Then χκ is a linear continuous mapping from C∞(R) onto its self. From formula (10) the
operator
D0,κ =
n∏
j=1
(
j + x
d
dx
)
◦ Iκ+1,n−κ
is a left-inverse of χκ. This shows that χκ is injective and D0,κ is surjective. So it suffices to
prove that D0,κ is injective.
Let f be a function in C∞(R) such that D0,κf = 0. Then the function g = Iκ+1,n−κf ∈
C∞(R) is a solution of the linear differential equation
n∏
j=1
(
1 + j + x
d
dx
)
y(x) = 0.
Since, the last differential equation has a unique C∞-solution, which is equal to y(x) = 0, it
follows that g = 0.
From (10) the operator Iκ+1,κ has a left-inverse, then f = 0. This shows that χκ is a bijective
operator. �
10 F. Bouzeffour
Let κ > 0, we define the dual intertwining operator tχκ on D(R) (D(R) is the space of
C∞-functions on R with compact support) by(
tχκf
)
(x) =
1
Γ(κ)
∫ +∞
|x|
(t− |x|)κ−1t−κf(sgn(x)t) dt, x ∈ R \ {0}.
Proposition 3. The operator tχκ is a topological automorphism of D(R), and satisfies the
transmutation relation:∫
R
(χκf)(x)g(x) dx =
∫
R
f(x)
(
tχκg
)
(x) dx, f ∈ C∞(R).
Proof. Let f ∈ C∞(R) and g ∈ D(R), we have∫
R
(χκf)(x)g(x) dx =
1
Γ(κ)
∫ +∞
0
∫ x
0
(x− t)κ−1f(t) dtg(x)x−κ dx
− 1
Γ(κ)
∫ ∞
0
∫ x
0
(x− t)κ−1f(−t) dtg(−x)x−κ dx.
Using Fubini’s theorem and a change of variable, we get∫
R
(χκf)(x)g(x) dx =
1
Γ(κ)
∫ +∞
0
∫ ∞
t
x−κ(x− t)κ−1g(x) dxf(t) dt
+
1
Γ(κ)
∫ 0
−∞
∫ ∞
−t
x−κ(x+ t)κ−1g(−x) dxf(t) dt.
Therefore,∫
R
(χκf)(x)g(x) dx =
1
Γ(κ)
∫
R
∫ ∞
|t|
x−κ(x− |t|)κ−1g(sign(t)x) dxf(t) dt
=
∫
R
f(t)
(
tχκg
)
(t) dt. �
Proposition 4. Let κ > 0, the Kummer transform Fκ satisfies the decomposition
Fκ(f) = F ◦ tχκ(f), f ∈ D(R).
Proof. The result follows from Proposition 3. �
6 Multivariable case
6.1 Direct product setting
In this subsection, we consider the direct product of the one-dimensional models, which means
that the Weyl group of the corresponding subsystem of root system is a subgroup of ZN2 .
We denote by τk (for each k from 1 to N) the orthogonal projection with respect to the
hyperplane orthogonal to ek, that is to say for every x = (x1, . . . , xN ) ∈ RN
τk(x) = x− 〈x, ek〉
|ek|2
ek = (x1, . . . , xk−1, 0, xk+1, . . . , xN ).
Let κ = (κ1, κ2, . . . , κN ) ∈ CN . The associated Dunkl type operators Tj for j = 1, . . . , N , are
given for x ∈ RN by
Tjf(x) = ∂jf(x) +
N∑
l=1
κl
f(x)− f(τl(x))
〈x, el〉
〈ek, el〉
Dunkl-Type Operators with Projection Terms 11
= ∂jf(x) + κj
f(x)− f(x1, . . . , xj−1, 0, xj+1, . . . , xN )
xj
.
These operators form a commuting system. The generalized Laplacian associated with Tj is
defined in a natural way as
∆κ =
N∑
j=1
T 2
j .
A straightforward computation yields
∆κ = ∆ + 2
N∑
j=1
κjx
−1
j ∂jf(x)−
N∑
j=1
(κ2j + κj)x
−1
j ∂jf(x1, . . . , xj−1, 0, xj+1, . . . , xN )
+
N∑
j=1
(κ2j − κj)x−2j
(
f(x)− f(x1, . . . , xj−1, 0, xj+1, . . . , xN )
)
.
This operator will play in our context a similar role to that of the Euclidean Laplacian in
the classical harmonic analysis. Obviously, the trivial choice of the multiplicity function κ = 0,
reduces our situation to the analysis related to the classical Laplacian ∆.
Let κ = (κ1, . . . , κN ) ∈ (0,∞)N . For x, λ ∈ RN , we consider the function Mκ(λ, x) which is
given as the tensor products
Mκ(λ, x) =
N∏
j=1
Mκj (iλjxj).
Theorem 6. For λ = (λ1, . . . , λN ) ∈ CN , the function Mκ(λ, x) is the unique analytic solution
of the system
Tξu(x) = i〈λ, ξ〉u(x), u(0) = 1, ∀ ξ ∈ CN .
6.2 Dunkl-type operators associated to an orthogonal subsystem
in a root system of type AN−1
Let R be a root system of type AN−1
R = {±(ei − ej), 1 ≤ i < j ≤ N}.
Define a positive orthogonal subsystem R′ = {α1, . . . , α[N/2]} of R by setting:
αi = e2i−1 − e2i, i = 1, . . . , [N/2].
We denote by τj (for each j from 1 to [N/2]) the orthogonal projection onto the hyperplane
perpendicular to αj , that is to say for every x = (x1, . . . , xN ) ∈ RN
τix = (x1, . . . , x2i−1, x2i, . . . , xN ),
where x2i−1 = x2i = 1
2(x2i−1 + x2i), i = 1, . . . , [N/2]. The vector ξ ∈ RN can be decomposed
uniquely in the form
ξ =
[N/2]∑
i=1
ξi(e2i−1 − e2i) + ξ̂,
where ξ̂ is an orthogonal vector to the linear space generated by R′ = {α1, . . . , α[N/2]}.
12 F. Bouzeffour
A straightforward computation shows that the operator Tξ (ξ ∈ RN ) associated with R′ and
the multiplicity parameters (κ1, . . . , κ[N/2]) has the following decomposition
Tξ =
[N/2]∑
i=1
ξiTαi + ∂
ξ̂
=
2[N/2]∑
i=1
(−1)i+1ξ[ i+1
2
]Ti + ∂
ξ̂
,
where
Ti = ∂i − (−1)iκ[ i+1
2
]ρ[ i+1
2
], i = 1, . . . , 2[N/2],
and
(ρif)(x) =
f(x)− f(τix)
x2i−1 − x2i
.
The intertwining operator (3) becomes
χκ(f)(x) =
1
Γ(κ)
∫
[0,1]n
f(h(t, x))w(t) dt,
where
h(t, x) = x+
[N/2]∑
i=1
ti − 1
2
(x2i−1 − x2i)(e2i−1 − e2i).
Proposition 5. Let λ = (λ1, . . . , λN ) ∈ CN , and κ = (κ1, . . . , κ[N/2]) ∈ (0,∞)[N/2]. The
following eigenvalue problem
Tξf = i〈λ, ξ〉f, f(0) = 1, ∀ ξ ∈ CN , (11)
has a unique analytic solution Mκ(λ, x) given by
Mκ(λ, x) = ei〈λ,h(0,x)〉
[N/2]∏
j=1
Mκj
(
i
2
(λ2j−1 − λ2j)(x2j−1 − x2j)
)
.
Proof. According to Theorem 2, χκ is an intertwining operator between Tξ and ∂ξ. So, the
function χκ(ei〈λ,·〉) is the unique C∞-solution of problem (11).
Since we can write
〈λ, h(t, x)〉 = 〈λ, h(0, x)〉+
[N/2]∑
j=1
tj
2
(λ2j−1 − λ2j)(x2j−1 − x2j),
we are lead to
Mκ(λ, x) =
ei〈λ,h(0,x)〉
Γ(κ)
∫
[0,1][N/2]
e
i
2
[N/2]∑
j=1
tj(λ2j−1−λ2j)(x2j−1−x2j)
w(t) dt
= ei〈λ,h(0,x)〉
[N/2]∏
j=1
1
Γ(κj)
∫
[0,1]
e
i
2
(λ2j−1−λ2j)(x2j−1−x2j)(1− tj)κj−1 dtj .
If we now use (7) and (8) we get
Mκ(λ, x) = ei〈λ,h(0,x)〉
[N/2]∏
j=1
Mκj
(
i
2
(λ2j−1 − λ2j)(x2j−1 − x2j)
)
. �
Dunkl-Type Operators with Projection Terms 13
6.3 Dunkl-type operators associated to orthogonal subsystem
in root system of type BN
Throughout this subsection R is a root system of type BN which is given by
R = {±ei ± ej , 1 ≤ i < j ≤ N ; ±ei 1 ≤ i ≤ N},
and R′ is a positive orthogonal subsystem R′ in the root system R given by
R′ = {α±i = e2i−1 ± e2i, 1 ≤ i ≤ [N/2]}.
Denote by τ±i (for each i from 1 to [N/2]) the orthogonal projection onto the hyperplane per-
pendicular to α±i , that is to say for every x = (x1, . . . , xN ) ∈ RN
τ±i x =
(
x1, . . . , x
±
2i−1, x
±
2i, . . . , xN
)
,
where x±2i−1 = x±2i = 1
2(x2i−1±x2i). In this case, the Dunkl type operator Tξ associated with R′
and the multiplicity parameters
(
κ±1 , . . . , κ
±
[N/2]
)
takes the form
(Tξf)(x) = ∂ξf(x) +
[N/2]∑
j=1
κ−j 〈α
−
j , ξ〉
f(x)− f(τ−j x)
〈x, α−j 〉
+ κ+j 〈α
+
j , ξ〉
f(x)− f(τ+j x)
〈x, α+
j 〉
.
In particular, for i = 1, . . . , 2[N/2] we have
Ti = ∂i − (−1)iκ−
[ i+1
2
]
ρ−
[ i+1
2
]
+ κ+
[ i+1
2
]
ρ+
[ i+1
2
]
.
where
(ρ±i f)(x) =
f(x)− f(τ±i x)
x2i−1 ± x2i
.
The operator Tξ has also the following decomposition
Tξ =
2[N/2]∑
i=1
(
ξ+
[ i+1
2
]
+ (−1)i+1ξ−
[ i+1
2
]
)
Ti + εξN∂N ,
where
ξ =
[N/2]∑
i=1
ξ+i α
+
i + ξ−i α
−
i + εξNeN , and ε =
{
1 if N is odd,
0 if N is even.
Proposition 6. Let λ = (λ1, . . . , λN ) ∈ CN and
κ = (κ+, . . . , κ+[N/2], κ
−, . . . , κ−[N/2]) ∈ (0,∞)2[N/2].
The following eigenvalue problem
Tξf = i〈λ, ξ〉f, f(0) = 1, ∀ ξ ∈ CN ,
has a unique analytic Mκ(λ, x) given by
Mκ(λ, x) = ei〈λ,h(0,x)〉
[N/2]∏
j=1
Mκ−j
(
i
2
(λ2j−1 − λ2j)(x2j−1 − x2j)
)
×Mκ+j
(
i
2
(λ2j−1 + λ2j)(x2j−1 + x2j)
)
.
14 F. Bouzeffour
Acknowledgements
This research is supported by NPST Program of King Saud University, project number 10-
MAT1293-02. I would like to thank the editor and the anonymous referees for their helpful
comments and remarks.
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http://dx.doi.org/10.1007/s11139-013-9481-3
http://dx.doi.org/10.1155/S1073792892000199
http://dx.doi.org/10.1007/BF01161629
http://dx.doi.org/10.2307/2001022
http://dx.doi.org/10.1112/S0024611502013825
http://arxiv.org/abs/math.RT/0108185
http://dx.doi.org/10.1017/CBO9780511565717
http://dx.doi.org/10.1017/CBO9780511565717
http://dx.doi.org/10.1007/BF01239517
http://dx.doi.org/10.1093/qmath/os-11.1.193
http://dx.doi.org/10.1080/00036811.2010.502117
http://arxiv.org/abs/1006.1140
http://dx.doi.org/10.1017/CBO9780511542824
http://dx.doi.org/10.1002/9781118032572
http://dx.doi.org/10.1002/9781118032572
1 Introduction
2 Dunkl operators
3 Operators of Dunkl-type
4 Intertwining operator
5 The one variable case
6 Multivariable case
6.1 Direct product setting
6.2 Dunkl-type operators associated to an orthogonal subsystem in a root system of type AN-1
6.3 Dunkl-type operators associated to orthogonal subsystem in root system of type BN
References
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