The Dunkl kernel and intertwining operator for dihedral groups
aa r X i v : . [ m a t h . C A ] O c t THE DUNKL KERNEL AND INTERTWINING OPERATOR FORDIHEDRAL GROUPS
HENDRIK DE BIE AND PAN LIAN
Abstract.
Dunkl operators associated with finite reflection groups generate acommutative algebra of differential-difference operators. There exists a uniquelinear operator called intertwining operator which intertwines between thisalgebra and the algebra of standard differential operators. There also exists ageneralization of the Fourier transform in this context called Dunkl transform.In this paper, we determine an integral expression for the Dunkl kernel,which is the integral kernel of the Dunkl transform, for all dihedral groups.We also determine an integral expression for the intertwining operator in thecase of dihedral groups, based on observations valid for all reflection groups. Asa special case, we recover the result of [Xu, Intertwining operators associatedto dihedral groups. Constr. Approx. 2019]. Crucial in our approach is asystematic use of the link between both integral kernels and the simplex in asuitable high dimensional space.
Contents
1. Introduction 12. Preliminaries 32.1. Basics of Dunkl theory 32.2. Humbert function of several variables 53. New formulas for the intertwining operator and its inverse 54. The case of dihedral groups 114.1. The generalized Bessel function 114.2. The Dunkl kernel 154.3. The intertwining operator 204.4. New proof of Xu’s result 245. Conclusions 26Acknowledgements 26References 26List of notations 281.
Introduction
Dunkl operators are differential-difference operators that generalize the standardpartial derivatives. They are constructed using a finite reflection group and aparameter function on the orbits of this group on its root system. They were
Mathematics Subject Classification.
Key words and phrases.
Dunkl operators, Intertwining operator, Dunkl transform, dihedralgroups. initially introduced by Charles Dunkl in [18], where he showed that, surprisingly,these operators still commute. Dunkl operators have found a variety of applicationsin mathematics and mathematical physics. They motivate the study of double affineHecke algebras and Cherednik algebras. They are used in the study of probabilisticprocesses and are crucial for the integration of quantum many body problems ofCalogero-Moser-Sutherland type. They have also made a lasting impact in thestudy of orthogonal polynomials and special functions in one and several variables.Apart from the seminal book [26], several excellent reviews are currently availableon this topic. We refer the reader to e.g. [22, 33] and to [3] for a most recent stateof the art.The theory of Dunkl operators is further developed using two key ingredients.The first is the
Dunkl transform , which is a generalization of the Fourier trans-form which now maps coordinate multiplication to the action of Dunkl operatorsand vice versa. It was introduced in [23] and further studied in [9], where theauthor showed the boundedness and analyticity of the associated integral kernelcalled Dunkl kernel, and obtained the Plancherel theorem. Various special casesof Paley-Wiener type theorems for the Dunkl transform were obtained in [10] andmultiplier theorems were investigated in e.g. [8, 27]. The Dunkl transform wasfurther generalized in [5], based on observations made in [4].The second important operator is the intertwining operator V . This operator isa linear and homogeneous isomorphism on the space of polynomials that maps thestandard partial derivatives to the Dunkl operators. It is abstractly proven thatthe intertwining operator can be represented as an integral operator [34]. This isimportant if one wants to extend the intertwining property to larger function spaces.Also, for certain parameter values it behaves singularly, which were determined in[25].From an abstract point of view, several satisfactory results are known about boththe Dunkl transform and the intertwining operator, as mentioned above. However,it has remained a major problem for thirty years to develop explicit formulas ofthe Dunkl kernel and the intertwining operator for concrete choices of the finitereflection group. We describe and list the known results. In the one dimensionalcase, both the Dunkl kernel and the intertwining operator are explicitly known,see [24, 33]. For the reflection groups of type A n the action of the intertwiningoperator on polynomials is determined in [19] and given as a highly complicatedintegral for n = 2 in [18]. An integral expression for the intertwining operatorfor the group B was obtained in [17]. For dihedral groups, acting in the plane,the current state of the art is as follows. For polynomials, explicit formulas giving V ( z k ¯ z ℓ ) as complicated sums of monomials are given in [21], but no integral formulais provided. Using a Laplace transform technique and knowledge on the Poissonkernel obtained in [16], an explicit and concise expression was obtained for theDunkl transform in the Laplace domain in [7]. A series expression for the Dunklkernel was given in [12]. Recently, for a restricted class of functions and restrictionson the parameter function, an integral expression was given for the intertwiningoperator in [37] using an integral over the simplex.This lack of explicit formulas has seriously hindered the further development ofharmonic analysis for Dunkl operators. The situation is better for the generalizedBessel function, which is a symmetrized version of the Dunkl kernel using theaction of the reflection group. For the root systems A n , a complicated integral HE DUNKL KERNEL AND INTERTWINING OPERATOR FOR DIHEDRAL GROUPS 3 recurrence formula for the generalized Bessel function was obtained in [1]. Anexplicit expression is derived in [36] for the intertwining operator in the symmetrizedsetting. For the dihedral case, a series expression for the generalized Bessel functionwas given in [15] and closed formulas were subsequently obtained for some specificparameters there. An explicit expression in the Laplace domain for the generalizedBessel function was obtained in [7], using the same Laplace transform technique asfor Dunkl dihedral kernel. Recently, a Laplace type expression for the generalizedBessel function for even dihedral groups with one variable specified is given in [14].The aim of the present paper is to give a complete description in the case ofdihedral groups of the Dunkl kernel, the generalized Bessel function and the inter-twining operator. For the Dunkl kernel, we obtain an expression in terms of thesecond class of Humbert functions (see Theorem 4.15, 4.18 ) or, alternatively, as anintegral over the simplex (Theorem 4.15). This is achieved by inverting the Laplacedomain expression obtained in [7]. For the intertwining operator and its inverse, wefirst determine an integral expression which is new to our knowledge, linking it tothe classical Fourier transform and the Dunkl kernel. This is achieved in Theorems3.2 and 3.10, and the formula is valid for arbitrary reflection groups. Using theexplicit formula for the Dunkl kernel in the dihedral case, we can subsequently giveexplicit expressions for the intertwining operator in that case. As expected, ourformulas specialise to those given in [37].The paper is organized as follows. In Section 2, we briefly introduce the basicsof Dunkl theory and the second class of Humbert functions. In Section 3, we givea general integral expression for the intertwining operator and its inverse. Section4 is devoted to the dihedral case. We give the explicit formulas for the generalizedBessel function, Dunkl kernel and the intertwining operator. A new proof of Xu’sresult of [37] can be found at the end of this section. We end with conclusions anda list of notations used in Section 4.2.
Preliminaries
Basics of Dunkl theory.
Let G be a finite reflection group with a fixedpositive root system R + . A multiplicity function κ : R → C on the root system R is a G -invariant function, i.e. κ ( α ) = κ ( h · α ) for all h ∈ G . For ξ ∈ R m , the Dunkloperator T ξ on R m associated with the group G and the multiplicity function κ ( α )is defined by T ξ ( κ ) f ( x ) = ∂ ξ f ( x ) + X α ∈ R + κ ( α ) h α, ξ i f ( x ) − f ( σ α x ) h α, x i , x ∈ R m where h· , ·i is the canonical Euclidean inner product in R m and σ α x := x − h x, α i α/ || α || is a reflection. In the sequel, we write T j in place of T e j ( κ ) where e j , j = 1 , · · · , m is a vector of the standard basis of R m . The Dunkl Laplacian ∆ κ is then defined by ∆ κ = P mj =1 T j .Let P denote the polynomials on R m and P n the homogeneous polynomials ofdegree n . The Dunkl operators { T j } generate a commutative algebra of differential-difference operators on P . Each T i is homogeneous of degree −
1. For p, q ∈ P , theFischer bilinear form [ p, q ] κ := ( p ( T ) q )(0)was introduced in [26]. Here p ( T ) is the differential-difference operator obtained byreplacing x j in p by T j . The Macdonald identity [32] also has a useful generalization HENDRIK DE BIE AND PAN LIAN in the Dunkl setting as follows:[ p, q ] κ = c − κ Z R m (cid:16) e − ∆ κ / p ( x ) (cid:17) (cid:16) e − ∆ κ / q ( x ) (cid:17) e −| x | / ω κ ( x ) dx where the weight function is ω κ ( x ) = Y α ∈ R + |h α, x i| κ ( α ) and c κ is the Macdonald-Mehta-Selberg constant, i.e. c κ = Z R m e −| x | / ω κ ( x ) dx. The Dunkl kernel E κ ( x, y ) is the joint eigenfunction of all the T j , T j E κ ( x, y ) = y j E κ ( x, y ) , j = 1 , . . . , m (1)and satisfies E κ (0 , y ) = 1. When κ = 0, it reduces to the ordinary exponentialfunction e h x,y i . Choosing an orthonomal basis { ϕ ν , ν ∈ Z N + } of P with respect tothe Fischer inner product, the Dunkl kernel can be expressed as E κ ( x, y ) = X ν ∈ Z N + ϕ ν ( x ) ϕ ν ( y )(2)for all x, y ∈ R m . We further introduce the generalized Bessel function defined asthe symmetric version of the Dunkl kernel by J κ ( x, y ) = 1 | G | X g ∈ G E κ ( x, g · y ) . The Dunkl transform is defined using the joint eigenfunction E κ ( x, y ) and theweight function ω κ ( x ) by F κ f ( y ) := c − κ Z R m E κ ( − ix, y ) f ( x ) ω κ ( x ) dx ( y ∈ R m ) . When κ = 0, the Dunkl transform reduces to the ordinary Fourier transform F , i.e. F f ( y ) := 1(2 π ) m/ Z R m e − i h x,y i f ( x ) dx. The definition of Dunkl transform is motivated by the following proposition (Propo-sition 7.7.2 in [26]) which will also be used in the following section.
Proposition 2.1.
Let p be a polynomial on R m and v ( y ) = P mj =1 y j for y ∈ C m ,then c − κ Z R m h e − ∆ κ / p ( x ) i E κ ( x, y ) e −| x | / ω κ ( x ) dx = e v ( y ) / p ( y ) . There exists an unique linear and homogenous isomorphism on P which satisfies V κ T j V κ = V κ ∂ j , j = 1 , , . . . , m. (3)The operator V κ is called the Dunkl intertwining operator in the literature. Theexplicit representation of this operator is only known so far for some special cases,e.g. the group Z and root system A , B , see [20, 26] and [3] for a recent review.We list the rank one case, which is frequently used in this paper. HE DUNKL KERNEL AND INTERTWINING OPERATOR FOR DIHEDRAL GROUPS 5
Example 2.2.
For k >
0, the intertwining operator in the rank one case is V k ( p )( x ) = Γ( k + 1 / / k ) Z − p ( xt )(1 − t ) k − (1 + t ) k dt. (4)Using the intertwining operator, the Dunkl kernel is expressed as E κ ( x, y ) = V κ (cid:16) e h· ,y i (cid:17) ( x ) . Throughout this paper, we only consider real multiplicity functions with κ ≥ Humbert function of several variables.
There are many ways to definehypergeometric functions of several variables. In this subsection, we introduce thesecond Humbert function of m variablesΦ ( m )2 ( β , . . . , β m ; γ ; x , . . . , x m ) := X j ,...,j m ≥ ( β ) j · · · ( β m ) j m ( γ ) j + ··· + j m x j j ! · · · x j m m j m !which is one of the confluent Lauricella hypergeometric series, see [29] (Chapter 2,2.1.1.2, page 42).When γ − P mj =1 β j and each β j , j = 1 , , . . . , m is a positive number, the hyper-geometric function Φ ( m )2 admits the following integral representationΦ ( m )2 ( β , . . . , β m ; γ ; x , . . . , x m )(5) = C ( γ ) β Z T m e P mj =1 x j t j − m X j =1 t j γ − P mj =1 β j − m Y j =1 t β j − j dt . . . dt m where C ( γ ) β = Γ( γ )Γ( γ − P mj =1 β j ) Q mj =1 Γ( β j )and T m is the open unit simplex in R m given by T m = ( t , . . . , t m ) : t j > , j = 1 , . . . , m, m X j =1 t j < . See [6, 31] for more details on this integral expression.Moreover, Section 4.24, formula (5) in [28] shows that the Laplace transform inthe variable t of Φ ( m )2 is given by L ( t γ − Φ ( m )2 ( β , . . . , β m ; γ ; λ t, . . . , λ m t ))(6) = Γ( γ ) s γ (cid:18) − λ s (cid:19) − β · · · (cid:18) − λ m s (cid:19) − β m with Re γ, Re s > , Re λ j , j = 1 , · · · , m. New formulas for the intertwining operator and its inverse
In this section, we give an integral expression of V κ in terms of the Dunkl kernelwhich will be used to derive the explicit expression for the dihedral groups insubsequent sections. We first formulate the following lemma using the ordinaryFourier transform. Note that the notation e − ∆ ( y ) / p ( iy ) used in the following meansacting with the operator e − ∆ ( y ) / on the complex valued polynomial p ( iy ). HENDRIK DE BIE AND PAN LIAN
Lemma 3.1.
For any y, z ∈ R m , let p ( z ) be a polynomial. Then we have e − ∆ ( y ) / p ( − iy ) = 1(2 π ) m/ Z R m e − i h x,y i e −| x | / p ( x ) e | y | / dx,e − ∆ ( y ) / E κ ( iy, z ) = 1(2 π ) m/ Z R m e − i h x,y i e −| x | / E κ ( − x, z ) e | y | / dx, (7) where ∆ ( y ) is the usual Laplace operator ∆ = P mj =1 ∂ j acting on the variable y .Proof. Let κ = 0 in Proposition 2.1, then one has Z R m (cid:18) e − ∆ ( y ) / p ( − iy ) (cid:19) e i h x,y i e −| y | / dy = (2 π ) m/ e −| x | / p ( x ) . Acting with the ordinary Fourier transform F on both sides leads to the first identityin the present theorem, (cid:18) e − ∆ ( y ) / p ( − iy ) (cid:19) e −| y | / = 1(2 π ) m/ Z R m e − i h x,y i e −| x | / p ( x ) dx. For the second part, we first give the following estimation, (cid:12)(cid:12)(cid:12) e − ∆ ( y ) / E κ ( iy, z ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) V κ (cid:16) e − ∆ ( y ) / e i h y, ·i (cid:17) ( z ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) V κ (cid:16) e |·| / e i h y, ·i (cid:17) ( z ) (cid:12)(cid:12)(cid:12) ≤ e | z | / (8)where the last inequality is by (cid:12)(cid:12)(cid:12) e | x | / e i h y,x i (cid:12)(cid:12)(cid:12) ≤ e | x | / and the Bochner-type repre-sentation of the intertwining operator proved in [34]. Using the above estimation(8) and the dominated convergence theorem, the second equality in the theorem isobtained by replacing p with the Dunkl kernel in the first equality. (cid:3) By the Fischer inner product, we give a general formula for the intertwiningoperator, which reveals the relationship between the Dunkl kernel and the inter-twining operator. In principle, once the Dunkl kernel is known, our theorem yieldsthe integral expression for the intertwining operator.
Theorem 3.2.
Let p be a polynomial and K ( iy, z ) := e − ∆ ( y ) / E κ ( iy, z ) , then forany z ∈ R m , the intertwining operator V κ satisfies V κ ( p )( z ) = 1(2 π ) m/ Z R m K ( iy, z ) F (cid:16) p ( · ) e −|·| / (cid:17) ( y ) dy = 1(2 π ) m/ Z R m F (cid:16) e −|·| / E κ ( · , − z ) (cid:17) ( y ) F (cid:16) e −|·| / p ( · ) (cid:17) ( y ) e | y | / dy. (9) Proof.
It is well known that the exponential function e h y,z i is the reproducing kernelof the polynomials with respect to the classical Fischer inner product, denoted by[ · , · ] , corresponding to the multiplicity function κ ( α ) = 0. More precisely, with theMacdonald identity, we have p ( z ) = h p ( y ) , e h y,z i i = 1(2 π ) m/ Z R m (cid:18) e − ∆ ( y ) / p ( y ) (cid:19)(cid:18) e − ∆ ( y ) / e h y,z i (cid:19) e −| y | / dy. HE DUNKL KERNEL AND INTERTWINING OPERATOR FOR DIHEDRAL GROUPS 7
By complexification, we have p ( z ) = h p ( − iy ) , e i h y,z i i = 1(2 π ) m/ Z R m (cid:18) e − ∆ ( y ) / p ( − iy ) (cid:19)(cid:18) e − ∆ ( y ) / e i h y,z i (cid:19) e −| y | / dy. (10)Applying V κ on both sides of (10) with respect to z , it follows that V κ ( p )( z )= 1(2 π ) m/ Z R m (cid:18) e − ∆ ( y ) / p ( − iy ) (cid:19) V κ (cid:18) e − ∆ ( y ) / e i h y, ·i (cid:19) ( z ) e −| y | / dy = 1(2 π ) m/ Z R m (cid:18) e − ∆ ( y ) / p ( − iy ) (cid:19)(cid:18) e − ∆ ( y ) / E κ ( iy, z ) (cid:19) e −| y | / dy. (11)Now, putting the integral expression (7) in (11), leads to V κ ( p )( z ) = 1(2 π ) m/ Z R m K ( iy, z ) F (cid:16) e −| x | / p ( x ) (cid:17) dy. The second identity in (9) follows from the expression (7). (cid:3)
Remark . It is known that for any polynomial p ( x ), the function F (cid:16) e −| x | / p ( x ) (cid:17) is in the Schwarz space. Furthermore, by the estimation | K ( iy, z ) | = (cid:12)(cid:12)(cid:12) V κ (cid:16) e − ∆ ( y ) / e i h y, ·i (cid:17) ( z ) (cid:12)(cid:12)(cid:12) ≤ e | z | / , the integral in (9) makes sense. Remark . At the moment, it is not clear to the authors if the above expression(3.2) can be rewritten as a R¨osler type integral [34] V κ ( p )( z ) = Z R m p ( x ) dµ z ( x ) . For a polynomial which is invariant under G , i.e. p ( g · z ) = p ( z ) , for all g ∈ G ,the reproducing kernel under the ordinary Fischer inner product is given by1 | G | X g ∈ G e h x,g · y i . Therefore, the intertwining operator acting on invariant polynomials could be ob-tained as V κ ( p )( z ) = V κ p ( · ) , | G | X g ∈ G e h· ,g · y i ( z ) . Following the proof of Theorem 3.2, we obtain the expression of the intertwin-ing operator on the invariant polynomials by replacing the Dunkl kernel with thegeneralized Bessel function.
Corollary 1.
Let p be a G -invariant polynomial. Then for any z ∈ R m , theintertwining operator V κ satisfies V κ ( p )( z ) = 1(2 π ) m/ Z R m F (cid:16) e −|·| / J κ ( · , − z ) (cid:17) ( y ) F (cid:16) e −|·| / p ( · ) (cid:17) ( y ) e | y | / dy. HENDRIK DE BIE AND PAN LIAN
In the following, we show independently that the integral expression given in (9)actually intertwines the usual partial derivatives and the Dunkl operator, i.e. therelations (3). We use the notation [
A, B ] := AB − BA for the commutator of twooperators A, B . The following lemma from [33] will help to verify these relations.
Lemma 3.5. [33]
For j = 1 , . . . , m , we have [ y j , ∆ κ /
2] = − T j ; h y j , e − ∆ κ / i = T j e − ∆ κ / . Theorem 3.6.
For any polynomial p , the integral operator V κ ( p )( z ) = 1(2 π ) m/ Z R m K ( iy, z ) F (cid:16) p ( · ) e −|·| / (cid:17) ( y ) dy satisfies the relations V κ ∂ j = T j V κ , j = 1 , . . . , m. Proof.
By direct computation, we have V κ ( ∂ j p )( z )= 1(2 π ) m/ Z R m K ( iy, z ) F (cid:16) ( ∂ j p )( · ) e −|·| / (cid:17) ( y ) dy = 1(2 π ) m/ Z R m K ( iy, z ) F (cid:16) ∂ j ( p ( x ) e −| x | / ) + x j p ( x ) e −| x | / (cid:17) ( y ) dy = 1(2 π ) m/ Z R m K ( iy, z ) i ( y j + ∂ y j ) F (cid:16) p ( x ) e −| x | / (cid:17) ( y ) dy = 1(2 π ) m/ Z R m (cid:0) i ( y j − ∂ y j ) K ( iy, z ) (cid:1) F (cid:16) p ( x ) e −| x | / (cid:17) ( y ) dy = 1(2 π ) m/ Z R m (cid:16) e − ∆ ( y ) / ( iy j E κ ( iy, z )) (cid:17) F (cid:16) p ( x ) e −| x | / (cid:17) ( y ) dy = T j V κ ( p )( z )where we have used Lemma 3.5 in the fifth equality and the relation (1) in the laststep. (cid:3) In the following, we compute some special cases using the integral expression (9).The explicit expressions for the intertwining operator associated to some specialroot systems are reobtained.
Example 3.7.
When κ α = 0, expression (9) reduces to V κ ( p )( z ) = 1(2 π ) m/ Z R m K ( iy, z ) F (cid:16) p ( x ) e −| x | / (cid:17) ( y ) dy = 1(2 π ) m/ Z R m e i h y,z i e | z | / F (cid:16) p ( x ) e −| x | / (cid:17) ( y ) dy = e | z | / F − (cid:16) F (cid:16) p ( x ) e −| x | / (cid:17)(cid:17) ( z )= p ( z )This coincides with the result that for κ α = 0 the intertwining operator is theidentity operator. HE DUNKL KERNEL AND INTERTWINING OPERATOR FOR DIHEDRAL GROUPS 9
Example 3.8. (Rank 1 case) For the rank one case with k >
0, the Dunkl kernelis explicitly known as E k ( iy, z ) = Γ( k + 1 / / k ) Z − e ityz (1 − t ) k − (1 + t ) k dt. It follows that e − ∆ ( y ) / E k ( iy, z ) = Γ( k + 1 / / k ) Z − e ityz e t z / (1 − t ) k − (1 + t ) k dt. Now, we have V k ( p )( z ) = Γ( k + 1 / π ) / Γ(1 / k ) Z R (cid:16) e − ∆ ( y ) / p ( − iy ) (cid:17) e −| y | / × Z − e ityz e t z / (1 − t ) k − (1 + t ) k dtdy = Γ( k + 1 / π ) / Γ(1 / k ) Z − (cid:18)Z R (cid:16) e − ∆ ( y ) / p ( − iy ) (cid:17) e −| y | / e ityz dy (cid:19) × e t z / (1 − t ) k − (1 + t ) k dt = Γ( k + 1 / / k ) Z − p ( tz )(1 − t ) k − (1 + t ) k dt where we have used Proposition 2.1 again in the last equality. Example 3.9. (Root system B ) When γ = κ + κ > /
2, the generalized Besselfunction of type B (i.e. the dihedral group I ) admits the following Laplace-typeintegral representation: J κ ( iy, z ) = Z R e i h y,x i H k ( x, z ) dz where H k ( x, z ) is a positive function with explicit expression, see [2]. It followsthat e − ∆ ( y ) / J κ ( iy, z ) = Z R e i h y,x i e | x | / H k ( x, z ) dz. Now, for the root system B and p ( z ) a I -invariant polynomial, which means that p ( g · z ) = p ( z ) for g ∈ I , we have V κ ( p )( z )= 12 π Z R (cid:16) e − ∆ ( y ) / p ( − iy ) (cid:17) (cid:16) e − ∆ ( y ) / J κ ( iy, z ) (cid:17) e −| y | / dy = 12 π Z R (cid:16) e − ∆ ( y ) / p ( − iy ) (cid:17) (cid:18)Z R e − ∆ ( y ) / e i h y,x i H k ( x, z ) dz (cid:19) e −| y | / dy = 12 π Z R (cid:20)Z R (cid:16) e − ∆ ( y ) / p ( − iy ) (cid:17) (cid:16) e − ∆ ( y ) / e i h y,x i (cid:17) e −| y | / dy (cid:21) H k ( x, z ) dz = Z R p ( x ) H k ( x, z ) dz. In general, whenever a Laplace-type representation of the generalized Bessel func-tion is obtained, it is possible to obtain the intertwining operator by the methoddescribed in this example.
While the precise structure of the Dunkl intertwining operator has been unknownfor a long time, the formal inverse of this operator is easy to get, see [26]. Thisoperator can be expressed as V − κ ( p )( x ) = exp m X j =1 x j T ( y ) j p ( y ) (cid:12)(cid:12)(cid:12)(cid:12) y =0 = h e h x,y i , p ( y ) i κ and satisfies the following relation V − κ T j = ∂ j V − κ , j = 1 , , . . . , m. The integral expression of V − κ can be obtained similarly as in Theorem 3.2. Weomit the proof here. Theorem 3.10.
Let p ( z ) be a polynomial. Denote L ( iy, z ) := e − ∆ κ,y / e i h y,z i = c − κ Z R m E κ ( − ix, y ) e −h x,z i e −| x | / e | y | / ω κ ( x ) dx. Then the inverse of the intertwining operator V − κ satisfies V − κ ( p )( z ) = c − κ Z R m L ( iy, z ) F κ (cid:16) p ( · ) e −|·| / (cid:17) ( y ) ω κ ( y ) dy where F κ is the Dunkl transform. Theorem 3.11.
For any polynomial p ( x ) , the integral operator V − κ satisfies V − κ T j = ∂ j V − κ , j = 1 , . . . , m. Proof.
Since e −| x | / is G -invariant, we have by direct computation, T j ( p ( x ) e −| x | / ) = T j ( p ) e −| x | / + p ( x ) T j ( e −| x | / ) . (12)Now, the intertwining relations are verified as follows V − κ ( T j p )( z )= c − κ Z R m L ( iy, z ) F κ (cid:16) ( T j p )( · ) e −|·| / (cid:17) ( y ) ω κ ( y ) dy = c − κ Z R m L ( iy, z ) F κ (cid:16) T j ( p ( x ) e −| x | / ) + x j p ( x ) e −| x | / (cid:17) ( y ) ω κ ( y ) dy = c − κ Z R m L ( iy, z ) i ( y j + T y j ) F κ (cid:16) p ( x ) e −| x | / (cid:17) ( y ) ω κ ( y ) dy = c − κ Z R m (cid:2) i ( y j − T y j ) L ( iy, z ) (cid:3) F κ (cid:16) p ( x ) e −| x | / (cid:17) ( y ) ω κ ( y ) dy = c − κ Z R m h e − ∆ κ,y / (cid:16) iy j e h iy,z i (cid:17)i F κ (cid:16) p ( x ) e −| x | / (cid:17) ( y ) ω κ ( y ) dy = ∂ z j V − κ ( p )( z ) , where we have used formula (12) in the second line and Lemma 3.5 for the fifthequality. (cid:3) HE DUNKL KERNEL AND INTERTWINING OPERATOR FOR DIHEDRAL GROUPS 11 The case of dihedral groups
The dihedral group I k is the group of symmetries of the regular k − gon. We usecomplex coordinates z = x + ix and identify R with C . Each reflection σ j in I k is given by z → ze ij π/k , 0 ≤ j ≤ k − . In this section, we will consider theexplicit expressions for the Dunkl kernel and the intertwining operator associatedto the dihedral groups. These can be achieved mainly due to the reduction method,i.e. the general I k case can be reduced to the explicitly known cases Z and Z ,see Section 7.6 in [26]. In turn, the reduction determines the representations of theformulas. As we will see, the Dunkl kernel and the generalized Bessel function areexpressed as the compositions of two integrals. One is the Weyl fractional integral (the intertwining operator associated to the group Z ) and the other one correspondsto the reduction.On the other hand, the dihedral group I k is generated by the rotation z → ze i π/k and the reflection z → z . The action of the rotations and reflection can be seenfrom the formula, see Theorem 4.8 and Theorem 4.18. The positivity and boundsof the Dunkl kernel and generalized Bessel function will be seen directly from theformulas as well.4.1. The generalized Bessel function.
It was proved in [15] that the generalizedBessel function can be expressed as a symmetric beta integral of an infinite series.Let z = | z | e iφ , w = | w | e iφ and ξ u,v ( φ , φ ) = v cos( φ ) cos( φ ) + u sin( φ ) sin( φ ),then we have the following expression. Theorem 4.1. [15]
For the dihedral group I k , k ≥ and non-negative multiplicityfunction κ = ( α, β ) , the generalized Bessel function is given by J κ ( z, w ) = 12 Z − Z − (cid:18) f k,α + β ( | zw | , ξ u,v ( kφ , kφ ) , f k,α + β ( | zw | , − ξ u,v ( kφ , kφ ) , (cid:19) dν α ( u ) dν β ( v ) where f k,λ ( b, ξ, t ) is the infinite series f k,λ ( b, ξ, t ) = Γ( kλ + 1) (cid:18) b (cid:19) kλ ∞ X j =0 ( j + λ ) λ I k ( j + λ ) ( bt ) C ( λ ) j ( ξ ) with C ( λ ) j ( ξ ) the Gegenbauer polynomial and I k ( b ) the modified Bessel function ofthe first kind, i.e. I ν ( b ) = ∞ X n =0 ( b/ n + ν n !Γ( n + ν + 1) , b ∈ R as well as the symmetric beta measure dν α ( u ) = Γ( α + 1 / √ π Γ( α ) (1 − u ) α − du. The series f k,λ ( b, ξ, t ) admits a closed form in the Laplace domain as studiedin our previous paper [7]. By the inverse Laplace transform, it is realized that f k,λ ( b, ξ, t ) is in fact a Humbert Φ ( m )2 function, see [7] formula (19). Later, thiswas proved more generally in [13] using a different method. We summarize theresults and present a more compact expression using derivatives in the Laplacedomain in the following lemma. Lemma 4.2.
For k ≥ , the Laplace transform of f k,λ ( b, ξ, t ) with respect to t isgiven by L [ f k,λ ( b, ξ, · )]( s ) = Γ( kλ + 1) 2 kλ S ( S + s ) k − ( s − S ) k (( S + s ) k − b k ξ + ( s − S ) k ) λ +1 = − Γ( kλ )2 kλ dds (cid:18) S + s ) k − b k ξ + ( s − S ) k ) λ (cid:19) , (14) where S = √ s − b . Moreover, we have f k,λ ( b, ξ,
1) = Φ ( k )2 ( λ, . . . , λ ; kλ ; b , . . . , b k − )(15) = e b Φ ( k − ( λ, . . . , λ ; kλ ; b − b , . . . , b k − − b ) where b j = b cos (( q − jπ ) /k ) , j = 0 , . . . , k − in which q = arccos( ξ ) . Remark . The denominator ( S + s ) k − b k ξ + ( s − S ) k is a polynomial in s andsatisfies the factorization( S + s ) k − b k ξ + ( s − S ) k = 2 k k − Y l =0 (cid:18) s − b cos (cid:18) q − πlk (cid:19)(cid:19) . (16)Formula (15) follows from the Laplace transform formula (6) and the above factor-ization (16). Remark . The first identity in (15) looks more symmetric than the second one.However, it is not possible to get an integral expression for f k,λ ( b, ξ, t ) directlyfrom the integral representation (5) of Φ ( m )2 , due to the the validity of the conditionsthere. This is not a problem for the second identity.Combining the Humbert function expression of f k,λ ( b, ξ,
1) in (15) and the inte-gral expression (5) for the function Φ ( m )2 , we have the following integral expressionof the generalized Bessel function. From this integral expression, it is directly seenthat the generalized Bessel function J κ ( x, y ) is positive and that the complexifiedgeneralized Bessel function J κ ( ix, y ) is bounded by 1. Theorem 4.5.
For k ≥ , the generalized Bessel function associated to the dihedralgroup I k is given by J κ ( z, w ) = Γ( k ( α + β ))2Γ( α + β ) k Z − Z − Z T k − (cid:16) e P k − j =0 a + j t j + e P k − j =0 a − j t j (cid:17) × k − Y j =0 t α + β − j dt . . . dt k − dν α ( u ) dν β ( v )(17) where a + j = | zw | cos (cid:16) q u,v ( kφ ,kφ ) − jπk (cid:17) , a − j = | zw | cos (cid:16) π − q u,v ( kφ ,kφ ) − jπk (cid:17) , j =0 , . . . , k − , q u,v ( φ , φ ) = arccos( ξ u,v ( φ , φ )) and t = 1 − P k − j =1 t j .Remark . The integral over the simplex in (17) is the integral expression of f k,α + β ( | zw | , ξ u,v ( kφ , kφ ) , a ± j appear, it is a functionin the variables ξ u,v ( kφ , kφ ) and | zw | by its series expansion, see also a directverification in [13]. This further implies that J κ ( z, w ) is a function in the variables | zw | , Re( z k )Re( w k ) | zw | k and Im( z k )Im( w k ) | zw | k . HE DUNKL KERNEL AND INTERTWINING OPERATOR FOR DIHEDRAL GROUPS 13
It is known that the generalized Bessel function is the reproducing kernel of theinvariant polynomials under [ · , · ] κ . By the series expansion of J κ ( z, w ), we obtainthe following integral expressions for the reproducing kernels. Corollary 2.
Let p n ( z ) be a homogenous polynomial of degree n invariant underthe action of I k , k ≥
2, i.e. p n ( z ) = p n ( g · z ) , for any g ∈ I k . Then the reproducingkernel for p n ( z ) is given by J ( n ) κ ( z, w )= Γ( k ( α + β ))2Γ( n + 1)Γ( α + β ) k Z − Z − Z T k − k − X j =0 a + j t j n + k − X j =0 a − j t j n × k − Y j =0 t α + β − j dt . . . dt k − dν α ( u ) dν β ( v ) , and satisfies p n ( z ) = h p n ( w ) , J ( n ) κ ( z, w ) i κ . Remark . The polynomials invariant under the group I k form an algebra. Thisalgebra is generated by | z | and z k + z k (the Chevalley generators). As a linearspace, the dimension of the invariant polynomials of degree n can be determinedby Molien’s generating function, see [30].The integral expression (17) looks quite complicated, however, it reflects howthe dihedral group I k acts. Due to nature of the reduction method, it will beseen in the following theorem that the expression is a composition of two operators,corresponding to the reflection and rotations in dihedral groups. Alternatively, bythis expression, it is seen that J κ ( z, w ) is I k invariant. Theorem 4.8.
The generalized Bessel function J κ ( z, w ) associated to I k , k ≥ is given by J κ ( z, w ) = Z − Z − J ( w, | z | , s v, s u ) dν α ( u ) dν β ( v )(18) with s = cos( kφ ) , s = sin( kφ ) and J ( w, | z | , s , s ) = c κ,k Z T k − (cid:16) e h w,z P k − j =0 e − i jπ/k t j i + e h w,z P k − j =0 e i (2 j − π/k t j i (cid:17) × k − Y j =0 t α + β − j dt . . . dt k − where t = 1 − P k − j =1 t j and c κ,k = Γ( k ( α + β ))2Γ( α + β ) k .Proof. Note that the series f k,λ ( b, ξ,
1) is a function in variables b and ξ . Hence,the function J ( w, | z | , s , s ) is a function in the variables | zw | and ξ , ( kφ , kφ ).The integrand J ( w, | z | , s v, s u ) = 12 [ f k,λ ( | zw | , ξ u,v ( kφ , kφ ) , f k,λ ( | zw | , − ξ u,v ( kφ , kφ ) , is a function in the variables | zw | and ξ u,v ( kφ , kφ ). The latter one can be obtainedby replacing ξ , ( kφ , kφ ) in the former one by ξ u,v ( kφ , kφ ) = v cos( kφ ) cos( kφ ) + u sin( kφ ) sin( kφ ) . This fact and expressions (13) and (17) lead to the first identity.In the following, we simplify the function J ( w, | z | , s , s ) = 12 ( f k,λ ( | z || w | , ξ , ( kφ , kφ ) , f k,λ ( | z || w | , − ξ , ( kφ , kφ ) , . It is seen that q , ( kφ , kφ ) = arccos( ξ , ( kφ , kφ ))= arccos(cos( kφ ) cos( kφ ) + sin( kφ ) sin( kφ ))= k ( φ − φ ) . This yields that a + j = | zw | cos (cid:18) q , ( kφ , kφ ) − jπk (cid:19) = | zw | cos (cid:18) ( φ − φ ) − jπk (cid:19) ,a − j = | zw | cos (cid:18) π − jπ − q , ( kφ , kφ ) k (cid:19) = | zw | cos (cid:18) π − jπk − ( φ − φ ) (cid:19) . In other words, we have a + j = Re( wze j iπ/k ) = h w, ze − i jπ/k i ,a − j = Re( wze i (2 j − π/k ) = D w, ze i (2 j − π/k E , where h z, w i = Re( zw ) denote the usual Euclidean inner product for z, w ∈ C ∼ = R .Hence for this special case u = v = 1, the integrand of (17) becomes e P k − j =0 a + j t j + e P k − j =0 a − j t j (19) = e h w,z P k − j =0 e − i jπ/k t j i + e h w,z P k − j =0 e i (2 j − π/k t j i . The second formula is obtained. (cid:3)
At the end of this subsection, we express a ± j by Cartesian coordinates insteadof the polar coordinates expression (19). We denote ( k √ z ) j , ≤ j ≤ k − k different k -th roots of z . Theorem 4.9.
For k ≥ , the generalized Bessel function associated to the dihedralgroup I k is given by J κ ( z, w ) = Γ( k ( α + β ))2Γ( α + β ) k Z − Z − Z T k − h u,v,t ,...,t k − ( z, w ) × k − Y j =0 t α + β − j dt . . . dt k − dν α ( u ) dν β ( v ) where h u,v,t ,...,t k − ( z, w ) HE DUNKL KERNEL AND INTERTWINING OPERATOR FOR DIHEDRAL GROUPS 15 = exp k − X j =0 t j Re k r Re(˜ z k w k ) + i q | zw | k − (Re(˜ z k w k )) ! j +exp k − X j =0 t j Re k r − Re(˜ z k w k ) + i q | zw | k − (Re(˜ z k w k )) ! j here ˜ z k = v Re( z k ) + ui Im( z k ) , t = 1 − P k − j =1 t j and q | zw | k − (Re(˜ z k w k )) is thepositive root.Proof. We only need to express e P k − j =0 a + j t j + e P k − j =0 a − j t j in the integrand of (17)by Cartesian coordinates. It reduces to express a ± j by z, w instead of the angles q u,v ( kφ , kφ ). In fact, it is seen that a + j = | zw | cos (cid:18) q u,v ( kφ , kφ ) − jπk (cid:19) , ≤ j ≤ k − k -roots of Re(˜ z k w k ) + i q | zw | k − (Re(˜ z k w k )) where ˜ z k = v Re( z k ) + iu Im( z k ). Hence, we have e P k − j =0 a + j t j = exp k − X j =0 t j Re k r Re(˜ z k w k ) + i q | zw | k − (Re(˜ z k w k )) ! j . The expression for a − j is obtained similarly. (cid:3) Remark . The result will not be changed if we choose q | zw | k − (Re(˜ z k w k )) as the negative square root. When u = v = 1, the k -th roots of Re(˜ z k w k ) + i q | zw | k − (Re(˜ z k w k )) becomes k p z k w k . Hence, it reduces to the formula (19) a + j = Re (cid:16) k p z k w k (cid:17) j = h w, ze − i jπ/k i . However, it is not possible to express a j as an inner product of the form h w, g ( z ) i for general u and v . Remark . For the odd order dihedral group I k , the generalized Bessel functioncan be obtained from I k , see [15].4.2. The Dunkl kernel.
In this section, we give two kinds of expressions of theDunkl kernel based on the Laplace domain result obtained in [7].We still identify R with the complex plane C , and denote z = | z | e iφ , w = | w | e iφ . The Laplace domain result is obtained by introducing an auxiliary variablein the series expansion of the Dunkl kernel and then taking the Laplace transform.More precisely, the series is E κ ( z, w, t ) = 2 α + β Γ( k ( α + β ) + 1) | zw | k ( α + β ) ∞ X j =0 I j + k ( α + β ) ( | zw | t ) P j (cid:0) I k ; e iφ , e iφ (cid:1) , where P j ( I k ; e iφ , e iφ ) is the reproducing kernel of the Dunkl harmonics of degree j and I j is the modified Bessel function of first kind, see [5] for the series expansionof the general Dunkl kernel. It is proved that the above infinite series admits aclosed expression in the Laplace domain for the dihedral groups. Theorem 4.12. [7]
For the even dihedral group I k , the Laplace transform of theDunkl kernel E κ ( z, w, t ) with respect to t is given by L ( E κ ( z, w, t )) = Γ( k ( α + β ) + 1) Z − Z − f I k ( s, z, w ) dµ α ( u ) dµ β ( v ) where f I k ( s, z, w ) = 2 k ( α + β ) ( s − Re( zw )) ( s + S ) k − z k w k ) + ( s − S ) k (( s + S ) k − | zw | k ξ u,v ( kφ , kφ ) + ( s − S ) k ) α + β +1 with S = p s − | zw | and dµ γ ( ω ) = Γ( γ + 1 / / γ ) (1 + ω )(1 − ω ) γ − dω. Remark . The substitution of x by ix in the original formula of Theorem 12 in[7] has been made here. For x, y ∈ R , the Dunkl kernel studied in [7] is in fact E ( − ix, y ) = V κ [ e − i h· ,y i ]( x ) . However, the Dunkl kernel studied here is E ( x, y ) = V κ [ e h· ,y i ]( x ) . Remark . The integrand f I k ( s, z, w ) can be factored as follows, see Lemma 3in [7], f I k ( s, z, w ) = A ( s, | zw | , q , ( kφ , kφ )) B ( s, z, w )[ A ( s, | zw | , q u,v ( kφ , kφ ))] α + β +1 , where B ( s, z, w ) = s − Re( zw ) and A ( s, | zw | , q u,v ( kφ , kφ )) = k − Y ℓ =0 (cid:18) s − | zw | cos (cid:18) q u,v ( kφ , kφ + 2 πℓk (cid:19)(cid:19) . Similar to the generalized Bessel function, by the Laplace transform formula (6),the Dunkl kernel can be expressed using the Humbert function and integrals overthe simplex.
Theorem 4.15.
For each dihedral group I k and non-negative multiplicity function κ = ( α, β ) , the Dunkl kernel is given by E κ ( z, w ) = Z − Z − (cid:20) h α + β ( z, w, u, v ) + 2 − k Γ( k ( α + β ) + 1)Γ( k ( α + β + 1) + 1) | zw | k × ξ u − ,v − ( kφ , kφ ) h α + β +1 ( z, w, u, v ) (cid:21) dµ α ( u ) dµ β ( v ) , where h γ ( z, w, u, v ) = Φ ( k +1)2 ( γ, . . . , γ, kγ + 1; a , . . . , a k − , a k )= e a k Φ ( k )2 ( γ, . . . , γ ; kγ + 1; a − a k , . . . , a k − − a k ) with a j = | zw | cos (cid:16) q u,v ( kφ ,kφ )+2 πjk (cid:17) , j = 0 , . . . k − and a k = Re ( zw ) . This canbe equivalently expressed as E κ ( z, w ) = Γ( k ( α + β ) + 1)Γ( α + β ) k Z − Z − Z T k e P kj =0 a j t j dω α + β dµ α ( u ) dµ β ( v ) HE DUNKL KERNEL AND INTERTWINING OPERATOR FOR DIHEDRAL GROUPS 17 + 2 − k Γ( k ( α + β ) + 1)Γ( α + β + 1) k Z − Z − Z T k | zw | k ξ u − ,v − ( kφ , kφ ) × e P kj =0 a j t j dω α + β +1 dµ α ( u ) dµ β ( v ) where t k = 1 − P k − j =0 t j and dω ν = Q k − j =0 t ν − j dt . . . dt k − .Proof. We split f I k ( s, z, w ) into two parts f I k ( s, z, w ) = 1 B ( s, z, w )[ A ( s, | zw | , q u,v ( kφ , kφ ))] α + β (20) + | zw | k ξ u − ,v − ( kφ , kφ )2 k − B ( s, z, w )[ A ( s, | zw | , q u,v ( kφ , kφ ))] α + β +1 . Then the first expression follows from taking the inverse Laplace transform for eachterm using the Φ ( m )2 functions and then putting t = 1.The second formula is obtained by replacing the Humbert function by its integralexpression (5), which is similar to the integral expression (17) for the generalizedBessel function. (cid:3) Example 4.16.
For the dihedral group I , we have E κ ( z, w ) = Z − Z − (cid:20) h α + β ( z, w, u, v ) + | zw | ξ u − ,v − ( φ , φ ) α + β + 1 × h α + β +1 ( z, w, u, v )) (cid:21) dµ α ( u ) dµ β ( v ) , where h α ( z, w, u, v )) = e Re ( zw ) ∞ X j =0 ( α ) j ( α + 1) j A j j !in which A = | zw | (( v −
1) cos φ cos φ + ( u −
1) sin φ sin φ ) . Direct computation shows that the integrand reduces to e | zw | cos( q u,v ( φ ,φ )) . Hence,the Dunkl kernel for the root system I is E κ ( z, w ) = Z − Z − e | zw | ( v cos φ cos φ + u sin φ sin φ ) dµ α ( u ) dµ β ( v )which coincides with the known results.In the following, we derive the second expression for the Dunkl kernel, which ismore compact. We start by rewriting the Laplace domain expression. Theorem 4.17.
For the even dihedral group I k , the Laplace transform of theDunkl kernel E κ ( z, w, t ) with respect to t is given by L ( E κ ( z, w, t ))= Γ( k ( α + β ) + 1) Z − Z − B ( s, z, w )[ A ( s, | zw | , q u,v ( kφ , kφ ))] α + β × (cid:20) (1 + u )(1 + v ) − α + β ( αu (1 + v ) + βv (1 + u )) (cid:21) dν α ( u ) dν β ( v ) , where B ( s, z, w ) = s − Re( zw ) and A ( s, | zw | , q u,v ( kφ , kφ )) = k − Y ℓ =0 (cid:18) s − | zw | cos (cid:18) q u,v ( kφ , kφ + 2 πℓk (cid:19)(cid:19) . Proof.
In (20), f I k ( s, z, w ) is split into two parts. The second part of (20) satisfies | zw | k ξ u − ,v − ( kφ , kφ ) B ( s, z, w )[ A ( s, | zw | , q u,v ( kφ , kφ ))] α + β +1 = | zw | k ξ u − ,v − ( kφ , kφ ) B ( s, z, w ) (cid:2) k (( S + s ) k − | zw | k ξ u,v ( kφ , kφ ) + ( s − S ) k ) (cid:3) α + β +1 = 2 k − α + β B ( s, z, w ) (cid:18) ( u − ddu A ( s, | zw | , q u,v ( kφ , kφ )] α + β + ( v − ddv A ( s, | zw | , q u,v ( kφ , kφ ))] α + β (cid:19) where as before ξ u,v ( kφ , kφ ) = v cos( kφ ) cos( kφ ) + u sin( kφ ) sin( kφ ) . This leads to the following expression for the Dunkl kernel in the Laplace domain1Γ( k ( α + β ) + 1) L ( E κ ( z, w, t ))(21)= Z − Z − B ( s, z, w )[ A ( s, | zw | , q u,v ( kφ , kφ ))] α + β dµ α ( u ) dµ β ( v )+ 1 α + β B ( s, z, w ) (cid:20)Z − Z − (cid:20) ddu (cid:18) A ( s, | zw | , q u,v ( kφ , kφ ))] α + β (cid:19) × ( u −
1) + ddv (cid:18) A ( s, | zw | , q u,v ( kφ , kφ ))] α + β (cid:19) ( v − (cid:21) dµ α ( u ) dµ β ( v ) (cid:21) . The second integral in (21) is further simplified using integration by parts Z − Z − ddu (cid:18) A ( s, | zw | , q u,v ( kφ , kφ ))] α + β (cid:19) ( u − dµ α ( u ) dµ β ( v )= Z − Z − A ( s, | zw | , q u,v ( kφ , kφ ))] α + β (cid:18) − ddu ( u − dµ α ( u ) dµ β ( v ) (cid:19) = − α Γ( α + 1 / / α ) Z − Z − A ( s, | zw | , q u,v ( kφ , kφ ))] α + β u (1 − u ) α − dudµ β ( v ) . The third integral is simplified similarly. Collecting all, we obtain the desiredresult. (cid:3)
By the inverse Laplace transform and then setting t = 1, Lemma 4.17 leads tothe following new expression of the Dunkl kernel. The proof is omitted here. Theorem 4.18.
For each dihedral group I k and positive multiplicity function κ ,the Dunkl kernel is given by E κ ( z, w ) = Z − Z − (cid:20) (1 + u )(1 + v ) − α + β ( αu (1 + v ) + βv (1 + u )) (cid:21) × h α + β ( z, w, u, v ) dν α ( u ) dν β ( v ) . HE DUNKL KERNEL AND INTERTWINING OPERATOR FOR DIHEDRAL GROUPS 19
For each odd dihedral group I k and positive multiplicity function κ , the Dunkl kernelis E κ ( z, w ) = Z − h α ( z, w, u, − u ) dν α ( u ) . In these formulas the expression h γ ( z, w, u, v ) = Φ ( k +1)2 ( γ, . . . , γ, kγ + 1; a , . . . , a k − , a k )= e a k Φ ( k )2 ( γ, . . . , γ ; kγ + 1; a − a k , . . . , a k − − a k ) is defined in Theorem 4.15. The integrand h γ ( z, w, u, v ) in Theorem 4.18 is positive, by its integral expressionover the simplex. In the following, we will show that the measure in the integral ispositive as well. This further implies that the Dunkl kernel satisfies E κ ( z, w ) > Lemma 4.19.
For u, v ∈ [ − , and α, β ≥ , we have (1 + u )(1 + v ) − α + β ( αu (1 + v ) + βv (1 + u )) ≥ . (22) Proof.
When u ≤ v ≤
0, the inequality (22) holds obviously. When u ≤ v ≥
0, we have − αu (1 + v ) ≥ u )(1 + v ) − βα + β v (1 + u ) ≥ (1 + u ) (cid:18) v − βα + β v (cid:19) ≥ . Therefore, the inequality (22) holds as well. The case when u ≥ v ≤ u ≥ v ≥
0, we have 1 + u ≥ u and 1 + v ≥ v . In this case, weconsider the quotient2 α + β ( αu (1 + v ) + βv (1 + u ))(1 + u )(1 + v ) = 2 αα + β u u + 2 βα + β v v ≤ (cid:3) Since the intertwining operator preserves homogeneous polynomials, we obtainan integral for the reproducing kernel of homogenous polynomials by a series ex-pansion.
Corollary 3.
Denote by P n the space of homogenous polynomial of degree n . Fordihedral group I k , the reproducing kernel of P n is given by V κ (cid:18) h· , w i n n ! (cid:19) ( z ) = Γ( k ( α + β ) + 1)Γ( α + β ) k Z − Z − Z T k k X j =0 a j t j n × (cid:20) (1 + u )(1 + v ) − α + β ( αu (1 + v ) + βv (1 + u )) (cid:21) dω α + β dν α ( u ) dν β ( v ) . Similarly, for the odd dihedral group I k , we have V κ (cid:18) h· , w i n n ! (cid:19) ( z ) = Γ( kα + 1)Γ( α ) k Z − Z T k k X j =0 a j t j n (1 − u ) dω α + β dν α ( u )where a j = | zw | cos (cid:16) q u,v ( kφ ,kφ )+2 πjk (cid:17) , j = 0 , . . . k − a k = Re ( zw ), t k = 1 − P k − j =0 t j and dω ν = Q k − j =0 t ν − j dt . . . dt k − . The intertwining operator.
The expressions of the generalized Bessel func-tion and the Dunkl kernel together with the result for general root systems, i.e.Theorem 3.2, lead to integral expressions for the intertwining operator.We consider the intertwining operator associated to invariant polynomials first.Recall that the generalized Bessel function is an integral of the following Humbertfunction, see Theorem 4.8, J ( w, | z | , s v, s u ) = 12 (cid:16) Φ ( k )2 (cid:0) λ, . . . , λ ; kλ ; a +0 , . . . , a + k − (cid:1) + Φ ( k )2 (cid:0) λ, . . . , λ ; kλ ; a − , . . . , a − k − (cid:1)(cid:17) where s = cos( kφ ), s = sin( kφ ), a + j = | zw | cos (cid:16) q u,v ( kφ ,kφ ) − jπk (cid:17) , a − j = | zw | cos (cid:16) π − q u,v ( kφ ,kφ ) − jπk (cid:17) , j = 0 , . . . , k − Theorem 4.20.
Let p ( x ) be a polynomial invariant under the action of I k , k ≥ ,i.e. p ( x ) = p ( g · x ) , for any g ∈ I k . Then, the intertwining operator V κ associatedto I k and κ = ( α, β ) is given by V κ ( p )( z ) = Z − Z − p u,v ( z ) dν α ( u ) dν β ( v ) where p u,v ( z ) = [ p ( w ) , J ( w, | z | , s v, s u )] = 12 π Z R F (cid:16) e −|·| / J ( z, · , u, v ) (cid:17) ( y ) F (cid:16) e −|·| / p ( · ) (cid:17) ( y ) e | y | / dy. This is equivalently expressed as V κ ( p )( z )= Γ( k ( α + β ))2Γ( α + β ) k Z − Z − Z T k − h p ( w ) , (cid:16) e P k − j =0 a + j t j + e P k − j =0 a − j t j (cid:17)i × k − Y j =0 t α + β − j dt . . . dt k − dν α ( u ) dν β ( v ) where a ± j is defined in Theorem 4.5. Example 4.21.
For the group I , denote x = ( x , x ) and y = ( y , y ). By setting k = 1 in (15), the generalized Bessel function associated to the group I is given by J κ ( x, y ) = 12 Z − Z − (cid:16) e vx y + ux y + e − ( vx y + ux y ) (cid:17) dν α ( u ) dν β ( v ) . By Theorem 4.20, the intertwining operator for the I -invariant polynomials is givenby V κ ( p )( x )= 12 Z − Z − h p ( x ) , e vx y + ux y + e − ( vx y + ux y ) i dν α ( u ) dν β ( v )= 12 Z − Z − ( p ( vx , ux ) + p ( − vx , − ux )) dν α ( u ) dν β ( v ) HE DUNKL KERNEL AND INTERTWINING OPERATOR FOR DIHEDRAL GROUPS 21 = Z − Z − p ( vx , ux ) dν α ( u ) dν β ( v )which coincides with the known results.We study the intertwining operator for the root system B again. Example 4.22. (Root system B ) For x = ( x , x ) , y = ( y , y ) and κ = ( α, β ), bysetting k = 2 in Theorem 4.8, the generalized Bessel function takes the followingform in Cartesian coordinates, (see also [2], [7], [15]) J κ ( x, y ) = Z − Z − ˜ I α + β − / r Z x,y ( u, v )2 ! dν α ( u ) dν β ( v )where ˜ I ν ( t ) = Γ( ν + 1) ∞ X n =0 ( t/ n n !Γ( n + ν + 1)and Z x,y ( u, v ) = ( x + x )( y + y ) + u ( x − x )( y − y ) + 4 vx x y y . In order to obtain the intertwining operator for the invariant polynomials, we onlyneed to compute p u,v ( y ) defined in Theorem 4.20, i.e. p u,v ( y ) = " p ( x ) , ˜ I α + β − / r Z x,y ( u, v )2 ! . By the Mehler-Sonine type integral expression of the Bessel function and the re-producing property of the exponential, we have p u,v ( y )= c B " p ( x ) , Z { t + t ≤ } e x at + ax ( ct + bt ) (1 − t − t ) α + β − / dt dt = c B Z { t + t ≤ } p ( at , a ( ct + bt ))(1 − t − t ) α + β − / dt dt where c B = ( α + β − / /π , a, b and c have been determined explicitly in [2] as a = (cid:18) y + y + u ( y − y )2 (cid:19) / ,b = (cid:0) ( y − y ) (1 − u ) + 4 y y (1 − v ) (cid:1) / y + y + u ( y − y ) ,c = 2 vy y y + y + u ( y − y ) . Hence, for α + β > / I -invariant polynomial p ( y ), the intertwining operatorassociated to B is given by V κ ( p )( y ) = c B Z − Z − Z { t + t ≤ } p ( at , a ( ct + bt )) × (1 − t − t ) α + β − / dt dt dν α ( u ) dν β ( v ) . It is seen that the measure given above is positive, therefore the integral transformis a positive operator as expected.
Let us now turn to the general case, which follows by combining Theorem 3.2with Theorem 4.18.
Theorem 4.23.
For polynomials p ( z ) , the intertwining operator V κ for the dihedralgroup I k is given by V κ ( p )( z ) = Z − Z − (cid:18) (1 + u )(1 + v ) − α + β ( αu (1 + v ) + βv (1 + u )) (cid:19) × P α + β ( z, u, v ) dν α ( u ) dν β ( v ) . The intertwining operator V κ for the odd dihedral group I k is given by V κ ( p )( z ) = Z − P α ( z, u, v )(1 − u ) dν α ( u ) . In these formulas, we put P γ ( z, u, v )) = [ p ( w ) , h γ ( z, w, u, v )] = 12 π Z R F (cid:16) e −|·| / h γ ( z, · , u, v ) (cid:17) ( y ) F (cid:16) e −|·| / p ( · ) (cid:17) ( y ) e | y | / dy with h γ ( z, w, u, v ) = Φ ( k +1)2 ( γ, . . . , γ, kγ + 1; a , . . . , a k − , a k ) defined in Theorem 4.15. We verify this directly for the group I . Example 4.24.
Let x = ( x , x ) and y = ( y , y ). For the rank one case, theintertwining operator is given by V κ ( p )( x ) = Z − P α ( x, u, v )(1 − u ) dν α ( u )where P γ ( x, u, v )) = [ p ( y ) , h γ ( x, y, u, v )] = α Z [ p ( y ) , e x y +(1+ ut − t ) x y ] t α − dt. Computing the Fischer inner product, the intertwining operator is expressed as V κ ( p )( x , x ) = α Z − Z p ( x , ( ut + 1 − t ) x ) t α − dt (1 − u ) dν α ( u )= α Z − Z p ( x , ( ut + 1 − t ) x ) t α − dtdµ α ( u )+ α Z − Z p ( x , ( ut + 1 − t ) x ) t α − ( − u ) dtdν α ( u ) . In the following, we only consider the polynomial x n , because in the present casethe intertwining operator has no influence on the variable x . We claim that thefollowing equality holds α Z − Z ( ut + 1 − t ) n t α − dt ( − u ) dν α ( u )(23) HE DUNKL KERNEL AND INTERTWINING OPERATOR FOR DIHEDRAL GROUPS 23 = Z − Z n ( u − ut + 1 − t ) n − t α dtdµ α ( u ) . The identity (23) can be proved by computing both the left and right hand sideexplicitly. Indeed, the left hand side is − Z − Z α ( ut + 1 − t ) n t α − dt u (1 − u ) α − du = − α n X j =0 (cid:18) nj (cid:19) Z − u j +1 (1 − u ) α − du Z t j + α − (1 − t ) n − j dt = − α n X j =0 (cid:18) nj (cid:19) Γ( j + 1)Γ( α )Γ( j + α + 1) Γ( j + α )Γ( n − j + 1)Γ( n + α + 1)= − Γ( α + 1)Γ( n + 1)Γ( n + α + 1) n X j =0 (1 + ( − j +1 )2 Γ( j )Γ( j + α )Γ( j )Γ( j + α + 1) . The right hand side is Z − Z n ( u − ut + 1 − t ) n − t α dt (1 + u )(1 − u ) α − du = − Z − Z n ( ut + 1 − t ) n − t α dt (1 − u ) α du = − n − X j =0 n (cid:18) n − j (cid:19) Z − u j (1 − u ) α du Z t j + α (1 − t ) n − − j dt = − n − X j =0 n !(1 + ( − j )2 j !( n − − j )! Γ( j +12 )Γ( α + 1)Γ( j +12 + α + 1) Γ( j + α + 1)Γ( n − j )Γ( n + 1 + α )= − Γ( α + 1)Γ( n + 1)Γ( n + α + 1) n − X j =0 (1 + ( − j )2 Γ( j +12 )Γ( j + α + 1)Γ( j + 1)Γ( j +12 + α + 1)= − Γ( α + 1)Γ( n + 1)Γ( n + α + 1) n X j =1 (1 + ( − j +1 )2 Γ( j )Γ( j + α )Γ( j )Γ( j + α + 1) . Comparing both sides, we obtain the identity (23).Now, with the identity (23), the intertwining operator for x n becomes V κ ( x n ) = α Z − Z ( ut + 1 − t ) x ) n t α − dtdµ α ( u )(24) + Z − Z n ( u − x (( ut + 1 − t ) x ) n − t α dtdµ α ( u ) . On the other hand, by direct verification or using integration by parts, we have α Z ( ut + 1 − t ) n t α − dt + Z ( u − n ( ut + 1 − t ) n − t α dt = u n . (25)Hence, combining (24) and (25), we obtain V κ ( x n ) = Z − ( x u ) n dµ α ( u ) , which is the well-known expression.4.4. New proof of Xu’s result.
In this section, we reobtain the intertwiningoperator given in [37]. We start from the odd dihedral group I k with multiplicityfunction α . In this case, the Laplace transform of the Dunkl kernel is given by12 kα Γ( kα + 1) L ( E κ ( z, w, t ))(26)= Z − ( s + S ) k − z k w k ) + ( s − S ) k ( s − Re( zw ))(( s + S ) k − | zw | k ξ u, ( kφ , kφ ) + ( s − S ) k ) α +1 dµ α ( u )where ξ u,v ( kφ , kφ ) = v cos( kφ ) cos( kφ ) + u sin( kφ ) sin( kφ ) . This is obtained by the relations between the Dunkl kernel of I k and I k , see alsoTheorem 12 in [7].Denote w p = e i pπk , then w kp = e ipπ = cos( pπ ), for p = 0 , , . . . , k −
1. Putting w = w p in formula (26), the Dunkl kernel E κ ( z, w p , t ) in the Laplace domainbecomes L ( E κ ( z, e i pπk , t ))= 2 kα Γ( kα + 1) ( s + S ) k − z k w k ) + ( s − S ) k ( s − Re( zw ))(( s + S ) k − | zw | k ξ u, ( kφ , pπ ) + ( s − S ) k ) α +1 = Γ( kα + 1) 1 (cid:0) s − | z | cos( φ − pπk ) (cid:1) α +1 Q k − j =1 (cid:0) s − | z | cos( φ − pπk − jπk ) (cid:1) α where the first identity is because ξ u, ( kφ , pπ ) = cos( kφ ) cos( pπ ) + u sin( kφ ) sin( pπ ) = ( − p cos( kφ )which is independent of u and R − dµ α ( u ) = 1. The inverse Laplace transformimmediately shows that the Dunkl kernel in this special case is V κ (cid:16) e h· ,w p i (cid:17) ( z ) = E κ ( z, w p )(27) = e a Φ ( k − ( α, . . . , α ; kα + 1; a − a , . . . , a k − − a )= c α,k Z T k − e h e ipπ/k ,z P k − j =0 e − i jπ/k t j i t α k − Y j =1 t α − j dt . . . dt k − where c α,k = Γ( kα +1) α Γ( α ) k , a j = | z | cos( φ − pπk − πjk ), j = 0 , . . . , k − t =1 − P k − j =1 t j . This formula also follows from setting w = w p in Theorem 4.18.It is known that the intertwining operator preserves homogenous polynomials,i.e. V κ ( P n ) ⊂ P n where P n is the space of homogenous polynomials of degree n .Hence, formula (27) yields V κ ( h· , w p i n )( z )= c α,k Z T k − * w p , z k − X j =0 e − i jπ/k t j + n t α k − Y j =1 t α − j dt . . . dt k − . HE DUNKL KERNEL AND INTERTWINING OPERATOR FOR DIHEDRAL GROUPS 25
By a limit discussion, it further leads to the intertwining operator for functions ofthe form f ( h e i pπk , z i ) as V κ (cid:16) f (cid:16) h· , e i pπk i (cid:17)(cid:17) ( z )= c α,k Z T k − f * e i pπk , z k − X j =0 e − i jπ/k t j + t α k − Y j =1 t α − j dt . . . dt k − where t = 1 − P k − j =1 t j and c α,k = Γ( kα +1) α Γ( α ) k , which is the formula given in [37],Theorem 1.1.Based on the above proof, we understand Xu’s formula in the another way, whichis the following corollary. Corollary 4.
For polynomials p ( z ), the intertwining operator V κ for the dihedralgroup I k with k odd at the lines z = | z | e i qπk , ≤ q ≤ k − V κ ( p ( · )) (cid:16) | z | e i qπk (cid:17) = c α,k Z T k − p k − X j =0 | z | e i ( q − j ) π/k t j t α k − Y j =1 t α − j dt . . . dt k − , where t = 1 − P k − j =1 t j and c α,k = Γ( kα +1) α Γ( α ) k . Proof.
The same method used for deriving the formula (27) leads to a similarformula for the Dunkl kernel E κ (cid:0) | z | e i qπk , w (cid:1) . With this formula, we have V κ ( p ( · )) (cid:16) | z | e i qπk (cid:17) = h p ( w ) , E κ (cid:16) | z | e i qπk , w (cid:17)i = c α,k Z T k − " p ( w ) , e (cid:28) w, | z | P k − j =0 e i ( q − j ) πk tj (cid:29) t α k − Y j =1 t α − j dt . . . dt k − = c α,k Z T k − p | z | k − X j =0 e i ( q − j ) πk t j t α k − Y j =1 t α − j dt . . . dt k − . (cid:3) In particular, Corollary 4 offers a way to compute the intertwining action onpolynomial p ( z ) when V κ ( p ( · ))( z ) is radial.At the end of this section, we show that the same method leads to a partial andsimple formula for the dihedral group I k with κ = ( α, β ) without other difficulties.In this case, the Weyl fractional integral vanishes as well. Example 4.25.
Consider w p = e i pπk and z = | z | e i ( q +1 / πk , p, q = 0 , , , . . . , k − ξ u,v (( q + 1 / π, pπ ) = v cos(( q + 1 / π ) cos( pπ ) + u sin(( q + 1 / π ) sin( pπ ) = 0 . In this case, the Dunkl kernel at the line z = | z | e i ( q +1 / πk and w = w p is only givenby an integral over the simplex, V κ (cid:16) e h· ,w p i (cid:17) ( z ) = E κ ( z, w p ) = Γ( k ( α + β ) + 1)Γ( α + β ) k Z T k − e h e ipπ/k ,z P k − j =0 e − i jπ/k t j i t α k − Y j =1 t α − j dt . . . dt k − . The same discussion as in the above shows that for functions f ( h e i pπk , z i ), theintertwining operator at the line re i ( q +1 / πk is given by V κ (cid:16) f (cid:16)D · , e i pπk E(cid:17)(cid:17) (cid:16) | z | e i ( q +1 / πk (cid:17) = Γ( k ( α + β ) + 1)Γ( α + β ) k Z T k − f * e i pπk , | z | e i ( q +1 / πk k − X j =0 e − i jπ/k t j + × t α k − Y j =1 t α − j dt . . . dt k − , where t = 1 − P k − j =1 t j . Remark . The same method together with the integral expression of the gen-eralized Bessel function leads to an integral expression of the intertwining operatorfor the I k invariant polynomials. Remark . A similar approach was used to derive the intertwining operator for aspecial class of functions for symmetric groups, where again the Humbert functionsappear, see [11]. 5.
Conclusions
In this paper, an integral expression of the intertwining operator and it inverse isgiven for arbitrary reflection groups which is based on the classical Fourier transformand the Dunkl kernel. For the dihedral case, explicit expressions for the generalizedBessel function and Dunkl kernel are obtained by inverting the Laplace domainresult of our previous paper [7] using the second class of Humbert functions. Withthese explicit formulas, we obtain several integral expressions for the intertwiningoperators in the symmetric and the non-symmetric settings. The positivity and thebound of the Dunkl kernel can be observed directly from our integral expressions.
Acknowledgements
This work was supported by the Research Foundation Flanders (FWO) underGrant EOS 30889451.
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List of notations
For the reader’s convenience, we list the notations used in Section 4 below. z | z | e iφ w | w | e iφ h z, w i Re( zw ) dν α ( u ) Γ( α + 1 / √ π Γ( α ) (1 − u ) α − dudµ α ( u ) Γ( α + 1 / √ π Γ( α ) (1 + u )(1 − u ) α − duξ u,v ( φ , φ ) v cos( φ ) cos( φ ) + u sin( φ ) sin( φ ) q u,v ( φ , φ ) arccos( ξ u,v ( φ , φ )) a + j | zw | cos (cid:18) q u,v ( kφ ,kφ ) − jπk (cid:19) , 0 ≤ j ≤ k − a − j | zw | cos (cid:18) π − q u,v ( kφ ,kφ ) − jπk (cid:19) , ≤ j ≤ k − a k Re( zw ) a j | zw | cos (cid:18) q u,v ( kφ ,kφ )+2 jπk (cid:19) , ≤ j ≤ k − Department of Electronics and Information Systems, Faculty of Engineering andArchitecture, Ghent University, Krijgslaan 281, 9000 Gent, Belgium.
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