An introduction to Dunkl theory and its analytic aspects

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Jean-Philippe Anker

To cite this version:

Jean-Philippe Anker. An introduction to Dunkl theory and its analytic aspects. G. Filipuk,

Y. Haraoka, S. Michalik. Analytic, Algebraic and Geometric Aspects of Differential Equations,

Birkhäuser, pp.3-58, 2017, Trends in Mathematics, 978-3-319-52841-0. �10.1007/978-3-319-52842-

7_1�. �hal-01402334�

JEAN–PHILIPPE ANKER

To appear in

Analytic, Algebraic and Geometric Aspects of Differential Equations (Będlewo, Poland, September 2015 ),

G. Filipuk, Y. Haraoka & S. Michalik (eds.), Trends Math., Birkhäuser

Contents

1. Introduction 2

2. Spherical Fourier analysis in rank 1 3

2.1. Cosine transform 3

2.2. Hankel transform on Euclidean spaces 3

2.3. Spherical Fourier analysis on real spheres 5

2.4. Spherical Fourier analysis on real hyperbolic spaces 7 2.5. Spherical Fourier analysis on homogeneous trees 12

2.6. Comments, references and further results 15

2.7. Epilogue 17

3. Rational Dunkl theory 17

3.1. Dunkl operators 18

3.2. Dunkl kernel 19

3.3. Dunkl transform 20

3.4. Heat kernel 21

3.5. Intertwining operator and (dual) Abel transform 22 3.6. Generalized translations, convolution and product formula 23

3.7. Comments, references and further results 25

4. Trigonometric Dunkl theory 27

4.1. Cherednik operators 27

4.2. Hypergeometric functions 28

4.3. Cherednik transform 30

4.4. Rational limit 30

4.5. Intertwining operator and (dual) Abel transform 31 4.6. Generalized translations, convolution and product formula 32

4.7. Comments, references and further results 33

Appendix A. Root systems 34

References 38

2010 Mathematics Subject Classification. Primary 33C67; Secondary 05E05, 20F55, 22E30, 33C80, 33D67, 39A70, 42B10, 43A32, 43A90.

Key words and phrases. Dunkl theory, special functions associated with root systems, spherical Fourier analysis.

Work partially supported by the regional project MADACA (Marches Aléatoires et processus de Dunkl – Approches Combinatoires et Algébriques, www.fdpoisson.fr/madaca). We thank the organizers of the school and conference AAGADE 2015 (Analytic, Algebraic and Geometric Aspects of Differential Equations, Mathematical Research and Conference Center, Będlewo, Poland, September 2015) for their invitation and the warm welcome at Pałac Będlewo. We also thank Béchir Amri, Nizar Demni, Léonard Gallardo, Chaabane Rejeb, Margit Rösler, Simon Ruijsenaars, Patrice Sawyer and Michael Voit for helpful comments and discussions.

1

1. Introduction

Dunkl theory is a far reaching generalization of Fourier analysis and special function theory related to root systems. During the sixties and seventies, it became gradually clear that radial Fourier analysis on rank one symmetric spaces was closely connected with certain classes of special functions in one variable :

‚ Bessel functions in connection with radial Fourier analysis on Euclidean spaces,

‚ Jacobi polynomials in connection with radial Fourier analysis on spheres,

‚ Jacobi functions (i.e. the Gauss hypergeometric function

F

) in connection with radial Fourier analysis on hyperbolic spaces.

See [51] for a survey. During the eighties, several attempts were made, mainly by the Dutch school (Koornwinder, Heckman, Opdam), to extend these results in higher rank (i.e. in several variables), until the discovery of Dunkl operators in the rational case and Cherednik operators in the trigonometric case. Together with q–special functions introduced by Macdonald, this has led to a beautiful theory, developed by several authors which encompasses in a unified way harmonic analysis on all Riemannian symmetric spaces and spherical functions thereon :

‚ generalized Bessel functions on flat symmetric spaces, and their asymmetric version, known as the Dunkl kernel,

‚ Heckman–Opdam hypergeometric functions on positively or negatively curved sym- metric spaces, and their asymmetric version, due to Opdam,

‚ Macdonald polynomials on affine buildings.

Beside Fourier analysis and special functions, this theory has also deep and fruitful interactions with

‚ algebra (double affine Hecke algebras),

‚ mathematical physics (Calogero-Moser-Sutherland models, quantum many body prob- lems),

‚ probability theory (Feller processes with jumps).

There are already several surveys about Dunkl theory available in the literature :

‚ [67] (see also [27]) about rational Dunkl theory (state of the art in 2002),

‚ [64] about trigonometric Dunkl theory (state of the art in 1998),

‚ [29] about integrable systems related to Dunkl theory,

‚ [54] and [18] about q –Dunkl theory and affine Hecke algebras,

‚ [40] about probabilistic aspects of Dunkl theory (state of the art in 2006).

These lectures are intended to give an overview of some analytic aspects of Dunkl theory.

The topics are indicated in red in Figure 1, where we have tried to summarize relations between several theories of special functions, which were alluded to above, and where arrows mean limits.

Let us describe the content of our notes. In Section 2, we consider several geometric settings (Euclidean spaces, spheres, hyperbolic spaces, homogeneous trees, . . . ) where radial Fourier analysis is available and can be applied successfully, for instance to study evolutions equations (heat equation, wave equation, Schrödinger equation, . . . ). Section 3 is devoted to the rational Dunkl theory and Section 4 to the trigonometric Dunkl theory.

In both cases, we first review the basics and next address some important analytic issues.

We conclude with an appendix about root systems and with a comprehensive bibliogra-

phy. For lack of time and competence, we haven’t touched upon other aforementioned

aspects of Dunkl theory, for which we refer to the bibliography.

affine buildings Bessel functions

Bessel functions

circular

compact

DAHA

double affine

Euclidean Dunkl theory

Dunkl theory

G{K generalized

Hecke algebras

higher rank

hyperbolic

hyperbolic spaces

Jacobi functions Jacobi polynomials

Macdonald

Macdonald

non compact

p

p–adic q–polynomials

Rⁿ

rank 1

rational

spheres

spherical functions spherical functions spherical functions

spherical functions homogeneous trees

trigonometric

U{K

Figure 1. Relation between various special function theories 2. Spherical Fourier analysis in rank 1

2.1. Cosine transform. Let us start with an elementary example. Within the frame- work of even functions on the real line R, the Fourier transform is given by

f p p λ q “ ż

R

dx f p x q cos λx and the inverse Fourier transform by

f p x q “

ż

R

dλ f p p λ q cos λx .

The cosine functions ϕ

p x q “ cos λx (λ P C) occurring in these expressions can be characterized in various ways. Let us mention

‚ Power series expansion :

ϕ

p x q “ ÿ

p´1q^ℓ

p2ℓq!

p λx q

@ λ, x P C .

‚ Differential equation : the functions ϕ “ ϕ

are the smooth eigenfunctions of p

q

, which are even and normalized by ϕ p 0 q“ 1 .

‚ Functional equation : the functions ϕ “ ϕ

are the nonzero continuous functions on R which satisfy

ϕpx`yq `ϕpx´yq

2

“ ϕ p x q ϕ p y q @ x, y P R.

2.2. Hankel transform on Euclidean spaces. The Fourier transform on R

and its inverse are given by

f p p ξ q “ ż

Rⁿ

dx f p x q e

(1) and

f p x q “ p 2π q

ż

Rⁿ

dλ f p p ξ q e

(2)

Notice that the Fourier transform of a radial function f “ f p r q on R

is again a radial function f p “ f p p λ q . In this case, (1) and (2) become

f p p λ q “

n 2

Γpⁿ2q

ż

0

dr r

f p r q j

2

p iλr q (3)

and

f p r q “

πⁿ² Γpⁿ2q

ż

0

dλ λ

f p p λ q j

2

p iλr q . (4)

Instead of the exponential function or the cosine function, (3) and (4) involve now the modified Bessel function j

2

, which can be characterized again in various ways :

‚ Relation with classical special functions and power series expansion . For every z P C, j

2

p z q “ Γ p

q `

2

˘

J

p iz q

“ ÿ

Γpⁿ2q ℓ! Γpⁿ2`ℓq

`

2

˘

“

F

p

;

q “ e

F

p

; n ´ 1; 2z q , where J

denotes the classical Bessel function of the first kind and

p

F

p a

, . . . , a

; b

, . . . , b

;z q “ ÿ

pa1qℓ...papqℓ

pb1qℓ...pbqqℓ

z^ℓ ℓ!

the generalized hypergeometric function.

‚ Differential equations . The function ϕ

p r q “ j

2

p iλr q is the unique smooth solution to the differential equation

`

Br

˘

ϕ

`

`

Br

˘ ϕ

` λ

ϕ

“ 0 , which is normalized by ϕ

p 0 q“ 1. Equivalently, the function

x ÞÝÑ ϕ

p| x |q “ j

2

p iλ | x |q (5)

is the unique smooth radial normalized eigenfunction of the Euclidean Laplacian

∆

“ ÿ

j“1

`

Bxj

˘

“ `

Br

˘

`

`

Br

˘ `

∆

corresponding to the eigenvalue ´ λ

.

Remark 2.1. The function (5) is a matrix coefficient of a continuous unitary represen- tation of the Euclidean motion group R

¸ O p n q .

The function (5) is a spherical average of plane waves. Specifically, ϕ

p| x |q “

ż

Opnq

dk e

“

n 2q 2πⁿ²

ż

S^n´1

dv e

,

where u is any unit vector in R

. Hence the integral representation ϕ

p r q “

n 2q

?πΓp^n´2¹q

ż

dθ p sin θ q

e

“

n 2q

?πΓp^n´2¹q

r

ż

0

ds p r

´ s

q

cos λs .

(6)

0 e

x

R

K–orbit

Figure 2. Real sphere S

2.3. Spherical Fourier analysis on real spheres. Real spheres

S

“ t x “ p x

, x

, . . . , x

q P R

| | x |

“ x

` . . . ` x

“ 1 u

of dimension n ­ 2 are the simplest examples of Riemannian symmetric spaces of com- pact type. They are simply connected Riemannian manifolds, with constant positive sectional curvature. The Riemannian structure on S

is induced by the Euclidean metric in R

, restricted to the tangent bundle of S

, and the Laplacian on S

is given by

∆f “ ∆ r f r ˇ ˇ

Sⁿ

, where ∆ r “ ÿ

`

Bxj

˘

denotes the Euclidean Laplacian in R

and f r p x q “ f `

|x|

˘ the homogeneous extension of f to R

r t 0 u . In spherical coordinates

$ ’

’ ’

’ ’

’ &

’ ’

’ ’

’ ’

%

x

“ cos θ

, x

“ sin θ

cos θ

,

...

x

“ sin θ

sin θ

. . . sin θ

cos θ

, x

“ sin θ

sin θ

. . . sin θ

sin θ

,

the Riemannian metric, the Riemannian volume and the Laplacian read respectively ds

“ ÿ

j“1

p sin θ

q

. . . p sin θ

q

p dθ

q

, d vol “ p sin θ

q

. . . p sin θ

q dθ

. . . dθ

and

∆ “ ÿ

1

psinθ1q²...psinθj´1q²

`

Bθ_j

˘

` p n ´ j qp cot θ

q

( .

Let G “ O p n ` 1 q be the isometry group of S

and let K « O p n q be the stabilizer of e

“ p 1, 0, . . . , 0 q . Then S

can be realized as the homogeneous space G { K. As usual, we identify right–K–invariant functions on G with functions on S

, and bi–K–invariant functions on G with radial functions on S

i.e. functions on S

which depend only on x

“ cos θ

. For such functions,

ż

Sⁿ

d vol f “ 2

n 2

Γpⁿ2q

ż

dθ

p sin θ

q

f p cos θ

q and

∆f “

` p n ´ 1 qp cot θ

q

.

The spherical functions on S

are the smooth normalized radial eigenfunctions of the Laplacian on S

. Specifically,

# ∆ ϕ

“ ´ ℓ p ℓ ` n ´ 1 q ϕ

, ϕ

p e

q “ 1 ,

where ℓ P N. They can be expressed in terms of classical special functions, namely ϕ

p x

q “

C

n´1 2 q

ℓ

p x

q “

P

n

2´1,ⁿ₂´1q

ℓ

p x

q

or

ϕ

p cos θ

q “

F

` ´ ℓ, ℓ ` n ´ 1;

; sin

θ

˘ ,

where C

are the Gegenbauer or ultraspherical polynomials, P

the Jacobi polyno- mials and

F

the Gauss hypergeometric function.

Remark 2.2. We have emphasized the characterization of spherical functions on S

by a differential equation. Here are other characterizations :

‚ The spherical functions are the continuous bi–K–invariant functions ϕ on G which satisfy the functional equation

ż

K

dk ϕ p xky q “ ϕ p x q ϕ p y q @ x, y P G. (7)

‚ The spherical functions are the continuous bi–K–invariant functions ϕ on G such that f ÞÝÑ

ż

G

dx f p x q ϕ p x q (8)

defines a character of the (commutative) convolution algebra C

p K z G { K q .

‚ The spherical functions are the matrix coefficients ϕ p x q “ x π p x q v, v y ,

where π is a continuous unitary representation of G, which has nonzero K –fixed vec- tors and which is irreducible, and v is a K –fixed vector, which is normalized by | v |“ 1.

‚ Integral representation : ϕ

p cos θ

q “

n 2q

?πΓp^n´2¹q

ż

dθ

p sin θ

q

“

cos θ

` i p sin θ

qp cos θ

q ‰

“

n 2q

?πΓp^n´2¹q

p sin θ

q

ż

´sinθ1

ds p sin

θ

´ s

q

p cos θ

` i s q

.

(9)

The spherical Fourier expansion of radial functions on S

reads f p x q “ ÿ

ℓPN

d

x f, ϕ

y ϕ

p x q , where

d

“

and

x f, ϕ

y “

n`1 2 q 2π<sup>n`</sup>

1 2

ż

Sⁿ

dx f p x q ϕ

p x q “

n`1 2 q

?πΓpⁿ2q

ż

dθ

p sin θ

q

f p cos θ

q ϕ

p cos θ

q .

0 e

e

e

K –orbit

N–orbit A–orbit

Figure 3. Hyperboloid model of H

2.4. Spherical Fourier analysis on real hyperbolic spaces. Real hyperbolic spaces H

are the simplest examples of Riemannian symmetric spaces of noncompact type. They are simply connected Riemannian manifolds, with constant negative sectional curvature.

Let us recall the following three models of H

.

‚ Model 1 : Hyperboloid

In this model, H

consists of the hyperboloid sheed

t x “ p x

, x

, . . . , x

q P R

“ R ˆ R

| L p x, x q “ ´ 1, x

­ 1 u

defined by the Lorentz quadratic form L p x, x q “ ´ x

` x

` . . . ` x

. The Riemannian structure is given by the metric ds

“ L p dx, dx q , restricted to the tangent bundle of H

, and the Laplacian by ∆f “ L `

Bx

,

˘ r f ˇ ˇ

, where f r p x q “ f `

?

˘ denotes the homogeneous extension of f to the light cone t x P R

| L p x, x qă 0, x

ą 0 u .

‚ Model 2 : Upper half–space

In this model, H

consists of the upper half–space R

“ t y P R

| y

ą 0 u equipped with the Riemannian metric ds

“ y

| dy |

. The volume is given by d vol “ y

dy

. . . dy

and the Laplacian by

∆ “ y

ÿ

B²

By_j²

´ p n ´ 2 q y

.

e

y

R

K–orbit

N–orbit

A–orbit

Figure 4. Upper half–space model of H

0 e

K –orbit

N–orbit A–orbit

Figure 5. Ball model of H

‚ Model 3 : Ball

In this model, H

consists of the unit ball B

“ t z P R

| | z | ă 1 u . The Riemannian metric is given by ds

“ `

2

˘

| dz |

, the volume by d vol “ `

˘

dz

. . . dz

, the distance to the origin by r “ 2 artanh | z | “ log

, and the Laplacian by

∆ “ `

˘

ÿ

j“1 B²

Bz²_j

` p n ´ 2 q

ÿ

j“1

z

.

Remark 2.3. Model 1 and Model 3 are mapped onto each other by the stereographic projection with respect to ´ e

, while Model 2 and Model 3 are mapped onto each other by the inversion with respect to the sphere S p´ e

, ?

2 q . This leads to the following formulae

$ ’

&

’ %

x

“

“

x

“

“

p j “ 1, . . . , n ´ 1 q x

“

“

# y

“

“

p j “ 1, . . . , n ´ 1 q y

“

“

# z

“

“

p j “ 1, . . . , n ´ 1 q z

“

“

.

Let G be the isometry group of H

and let K be the stabilizer of a base point in H

. Then H

can be realized as the homogeneous space G { K. In Model 1, G is made up of two among the four connected components of the Lorentz group O p 1, n q , and the stabilizer of e

is K “ O p n q . Consider the subgroup A « R in G consisting of

‚ the matrices

a

“

¨

˝ cosh r 0 sinh r

0 I 0

sinh r 0 cosh r

˛

‚ p r P R q in Model 1,

‚ the dilations a

: y ÞÝÑ e

y in Model 2,

and the subgroup N « R

consisting of horizontal translations n

: y ÞÝÑ y ` υ (υ P R

) in Model 2. Then we have

‚ the Cartan decomposition G “ KA

K, which corresponds to polar coordinates in

Model 3,

‚ the Iwasawa decomposition G “ NAK , which corresponds to Cartesian coordinates in Model 2.

We shall denote by a

and a

the A

and A components of g P G in the Cartan and Iwasawa decompositions.

Remark 2.4. In small dimensions, H

« SL p 2, R q{ SO p 2 q and H

« SL p 2, C q{ SU p 2 q . As usual, we identify right–K–invariant functions on G with functions on H

, and bi –K–invariant functions on G with radial functions on H

i.e. functions on H

which depend only on the distance r to the origin. For radial functions f “ f p r q ,

ż

Hⁿ

d vol f “ 2

n 2

Γpⁿ2q

ż

0

dr p sinh r q

f p r q and

∆f “

` p n ´ 1 qp coth r q

.

The spherical functions ϕ

are the smooth normalized radial eigenfunctions of the Lapla- cian on H

. Specifically, #

∆ ϕ

“ ´ λ

` ρ

( ϕ

, ϕ

p 0 q “ 1 ,

where ρ “

.

Remark 2.5. The spherical functions on H

can be characterized again in several other ways. Notably,

‚ Differential equation : the function ϕ

p r q is the unique smooth solution to the differ- ential equation

`

Br

˘

ϕ

` p n ´ 1 qp coth r q `

Br

˘ ϕ

` p λ

` ρ

q ϕ

“ 0 , which is normalized by ϕ

p 0 q“ 1.

‚ Relation with classical special functions : ϕ

p r q “ ϕ

2 2 ,´¹2

λ

p r q “

F

`

2

,

;

; ´ sinh

r ˘ ,

where ϕ

are the Jacobi functions and

F

the Gauss hypergeometric function.

‚ Same functional equations as (7) and (8).

‚ The spherical functions are the matrix coefficients ϕ

p x q “

n 2q

2πⁿ²

x π

p x q 1, 1 y ,

of the spherical principal series representations of G on L

p S

q .

‚ According to the Harish–Chandra formula ϕ

p x q “

ż

K

dk e

,

the function ϕ

is a spherical average of horocyclic waves. Let us make this integral representation more explicit :

ϕ

p r q “

n 2q 2πⁿ²

ż

S^n´1

dv x cosh r ´ p sinh r q v, e

y

“

n 2q

?π Γp^n´2¹q

ż

dθ p sin θ q

“

cosh r ´ p sinh r qp cos θ q ‰

“

n´1 2 Γpⁿ2q

?π Γp^n´2¹q

p sinh r q

ż

0

ds p cosh r ´ cosh s q

cos λs .

(10)

Remark 2.6. The asymptotic behavior of the spherical functions is given by the Harish–

Chandra expansion

ϕ

p r q “ c p λ q Φ

p r q ` c p´ λ q Φ

p r q , where

c p λ q “

and

Φ

p r q “ p 2 cosh r q

F

`

2

,

; 1 ´ iλ ; cosh

r ˘

“ e

ÿ

ℓ“0

Γ

p λ q e

, with Γ

” 1.

The spherical Fourier transform (or Harish–Chandra transform) of radial functions on H

is defined by

H f p λ q “ ż

Hⁿ

dx f p x q ϕ

p x q “

n 2q 2πⁿ²

ż

0

dr p sinh r q

f p r q ϕ

p r q (11) and the inversion formula reads

f p x q “ 2

π

Γ p

q ż

0

dλ | c p λ q|

H f p λ q ϕ

p x q . (12) Remark 2.7.

‚ The Plancherel density reads

| c p λ q|

“

ź

j“0

p λ

` j

q in odd dimension, and

| c p λ q|

“

λ tanh πλ ź

“ λ

` p j `

q

‰ in even dimension. Notice the different behaviors

| c p λ q|

„

| λ |

at infinity, and

| c p λ q|

„

n´1 2 q² 2^2n´4Γpⁿ2q²

λ

at the origin.

‚ Observe that (11) and (12) are not symmetric, unlike (1) and (2), or (3) and (4).

The spherical Fourier transform (11), which is somewhat abstract, can be bypassed by considering the Abel transform, which is essentially the horocyclic Radon transform restricted to radial functions. Specifically,

A f p r q “ e

ż

N

dn f p na

q

“

n´1 2

Γp^n´2¹q

ż

|r|

ds sinh s p cosh s ´ cosh r q

f p s q .

Then the following commutative diagram holds, let say in the Schwartz space setting : S

p R q

H Õ Ô F S

p H

q ÝÑ

A S

p R q

Here S

p H

q denotes the L

radial Schwartz space on H

, which can be identified

with p cosh r q

S

p R q , F the Euclidean Fourier transform on R and each arrow is an

isomorphism. Thus the inversion of the spherical Fourier transform H boilds down to the inversion of the Abel transform A . In odd dimension,

A

g p r q “ p 2π q

`

´

˘

g p r q while, in even dimension,

A

g p r q “

2^n´

1 2 πⁿ²

ż

|r|

? ds

coshs´coshr

` ´

˘`

´

˘

g p s q . Consider finally the transform

A

g p x q “ ż

K

dk e

g p h p kx qq , which is dual to the Abel transform, i.e.,

ż

Hⁿ

dx f p x q A

g p x q “ ż

´8

dr A f p r q g p r q ,

and which is an isomorphism between C

p R q and C

p H

q . It is given explicitly by A

g p r q “

n´1 2 Γpⁿ2q

?πΓp^n´2¹q

p sinh r q

ż

0

ds p cosh r ´ cosh s q

g p s q and its inverse by

p A

q

f p r q “

?π

2^n´

1 2 Γpⁿ2q

BBr

`

sinhr B sinhr

˘

p sinh r q

f p r q ( in odd dimension and by

p A

q

f p r q “

2^n´

1 2 pⁿ2´1q!

B Br

`

sinhr B Br

˘

ż

? ds

coshr´coshs

p sinh s q

f p s q in even dimension.

Remark 2.8. Notice that the spherical function ϕ

p r q is the dual Abel transform of the cosine function cos λs.

Applications. Spherical Fourier analysis is an efficient tool for solving invariant PDEs on H

. Here are some examples of evolution equations.

‚ The heat equation #

B

u p x, t q “ ∆

u p x, t q u p x, 0 q “ f p x q

can be solved explicitly by means of the inverse Abel transform. Specifically, u p x, t q “ f ˚ h

p x q ,

where the heat kernel is given by h

p r q “

2^n`

1

2 πⁿ²

t

e

`

´

˘

e

2 4t

in odd dimension and by

h

p r q “ p 2π q

t

e

ż

|r|

? ds

coshs´coshr

` ´

˘` ´

˘

e

2 4t

in even dimension. Moreover, the following global estimate holds : h

p r q — e

e

e

2 4t

ˆ

# t

p 1 ` r q if t ­ 1 ` r,

t

p 1 ` r q

if 0 ă t ¬ 1 ` r. (13)