Comptes Rendus

Comptes Rendus

Mathématique

Piotr Graczyk and Patrice Sawyer

Sharp Estimates of Radial Dunkl and Heat Kernels in the Complex Case

Volume 359, issue 4 (2021), p. 427-437

<https://doi.org/10.5802/crmath.188>

© Académie des sciences, Paris and the authors, 2021.

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2021, 359, n4, p. 427-437

https://doi.org/10.5802/crmath.188

Harmonic analysis, Representation theory /Analyse harmonique, Théorie des représentations

Sharp Estimates of Radial Dunkl and Heat Kernels in the Complex Case A _n

Piotr Graczyk^aand Patrice Sawyer^∗,b

aLAREMA, UFR Sciences, Université d’Angers, 2 bd Lavoisier, 49045 Angers cedex 01, France

bDepartment of Mathematics and Computer Science, Laurentian University, Sudbury, ON Canada P3E 2C6

E-mails:[email protected] (P. Graczyk), [email protected] (P. Sawyer)

Abstract.In this article, we consider the radial Dunkl geometric casek=1 corresponding to flat Riemannian symmetric spaces in the complex case and we prove exact estimates for the positive valued Dunkl kernel and for the radial heat kernel.

Résumé.Dans cet article, nous considérons le cas géométrique radial de Dunklk=1 correspondant aux espaces symétriques riemanniens plats dans le cas complexe et nous prouvons des estimations exactes pour le noyau de Dunkl à valeur positive et pour le noyau de chaleur radial.

2020 Mathematics Subject Classification.33C67, 43A90, 53C35.

Funding.The authors thank Laurentian University, Sudbury and the Défimaths programme of the Région des Pays de la Loire for their financial support.

Manuscript received 22nd December 2020, revised 10th February 2021, accepted 11th February 2021.

1. Introduction and notation

Finding good estimates of Dunkl heat kernels is a challenging and important subject, developed recently in [1]. Dunkl analysis has been developed intensely in recent years (refer to [7]). Opti- mal Dunkl heat kernel estimates have important consequences in harmonic analysis (e.g. Hardy spaces, maximal inequalities, see for instance [1]), potential theory (e.g. Green function esti- mates, Martin boundary description, see [2]) and spectral theory (see [3]). Recall that the Dunkl heat kernel is the probability transition function of the Dunkl stochastic process (refer to [5]

and [7]). Consequently, Dunkl heat kernel estimates have also stochastic applications for the Dunkl process. Establishing estimates of the heat kernels is equivalent to estimating the Dunkl kernel as demonstrated by equation (3) below.

∗Corresponding author.

In this paper we prove exact estimates in theW-radial Dunkl geometric case of multiplicity k=1, corresponding to Cartan motion groups and flat Riemannian symmetric spaces with the ambient complex groupG, the Weyl groupWand the root systemAn.

We study for the first time the non-centered heat kernel, denotedp^W_t (X,Y), on Riemannian symmetric spaces and we provide its sharp estimates. Exact estimates were obtained in [2] in the centered caseY =0 for all Riemannian symmetric spaces.

We provide exact estimates for the spherical functionsψλ(X) in the two variablesX,λwhenλ is real and, consequently, for the heat kernelp^W_t (X,Y) in the three variablest,X,Y.

We recall here some basic terminology and facts about symmetric spaces associated to Cartan motion groups.

LetG be a semisimple Lie group and let g₌k_⊕p be the Cartan decomposition ofG. We recall the definition of the Cartan motion group and the flat symmetric space associated with the semisimple Lie groupG with maximal compact subgroup K. The Cartan motion group is the semi-direct productG₀=Knp where the multiplication is defined by (k₁,X₁)·(k₂,X₂)

=(k₁k₂,Ad(k₁)(X₂)+X₁). The associated flat symmetric space is thenM=p_'G₀/K(the action of G0onpis given by (k,X)·Y =Ad(k)(Y)+X).

The spherical functions for the symmetric spaceMare then given by ψ_λ(X)=

Z

K

eλ(Ad(k)(X))d k

whereλis a complex linear functional on a_⊂p, a Cartan subalgebra of the Lie algebra ofG.

To extendλtoX ∈^Ad(K)a₌p, one usesλ(X)=λ(πa(X)) whereπais the orthogonal projection with respect to the Killing form (denoted throughout this paper by〈·,·〉). Note that in [9–11],λis replaced byiλ.

Throughout this paper, we usually assume thatGis a semisimple complex Lie group. The complex root systems are respectively A_n forn ≥1 (where p consists of then×n hermitian matrices with trace 0),Bnforn≥2 (wherep₌iso(2n+1)),Cnforn≥3 (wherep₌isp(n)) and D_nforn≥4 (wherep₌iso(2n)) for the classical cases and the exceptional root systemsE₆,E₇, E₈,F₄andG₂.

The radial heat kernel is considered with respect to the invariant measureµ(d Y)=π²(Y)d Y onM, whereπ(Y)=Q

α>0α(Y).

Note also that in the curved caseM₀=G/K, the spherical functions for the symmetric space M₀are then given by

φ_λ¡ e^X¢

= Z

K

e^{(λ−ρ)H¡eXk¢}d k

whereρ is the half-sum of the roots counted with their multiplicities andH(g) is the abelian component in the Iwasawa decomposition ofg:g=k e^H(g)n.

The authors are thankful for the helpful suggestions by the anonymous referee.

2. Estimates of spherical functions and of the heat kernel

We will be developing a sharp estimate for the spherical functionψ_λ(X). We introduce the following useful convention. We will write

f(t,X,λ)³g(t,X,λ)

in a given domain off andgif there exist constantsC₁>0 andC₂>0 independent oft,Xandλ such thatC₁f(t,X,λ)≤g(t,X,λ)≤C₂f(t,X,λ) in the domain of consideration.

We conjecture the following global estimate for the spherical function in the complex case.

Conjecture 1. On flat Riemannian symmetric spaces with complex group G we have, ψ_λ(X)³ e^〈λ,X〉

Y

α>0

(1+α(λ)α(X)), λ∈a⁺,X∈a⁺. Remark 2. Recall that, denotingδ(X)=Q

α>0sinh²α(X), we have φ_λ¡

e^X¢

= π(X)

δ^1/2(X)ψ_λ(X) . (1)

Since

δ^1/2(X)³ e^ρ(X)π(X) Y

α>0

(1+α(X)) in the complex case, Conjecture 1 therefore becomes

φ_λ¡ e^X¢

³e^(λ−ρ)(X) Y

α>0

1+α(X)

1+α(λ)α(X) (2)

in the curved complex case.

Let us compare the estimate (2) we conjecture forφ_λwith the one obtained in [12, Theorem 3.4], cf. also [17, Remark 3.1]. The estimates in [12] apply in all the generality of hypergeometric functions of Heckman and Opdam. The authors show that there exist constantsC₁(λ)> 0, C₂(λ)>0 such that

C₁(λ)e^(λ−ρ)(X) Y

α>0, α(λ)=0

(1+α(X))≤φλ¡ e^X¢

≤C₂(λ)e^(λ−ρ)(X) Y

α>0, α(λ)=0

(1+α(X)) .

Given (1), corresponding estimates clearly also hold in the flat case forψ_λ(X). The interest of our result, in the caseA_n, lies in the fact that our estimate is universal in bothλandX.

The results of [12, 17] and our estimates in theA_ncase strongly suggest that the Conjecture 1 is true for any complex root system.

Note that asymptotics ofψ_λ(t X) whenλandXare singular andt→ ∞were proven in [8] for all classical complex root systems and the systemsF₄andG₂.

Consider the relationship between the Dunkl kernel E_k(X,Y) and the Dunkl heat kernel p_t(X,Y), as given in [13, Lemma 4.5]

pt(X,Y)= 1

2^γ+d/2c_kt^−d2−γe^−|X|

2−|Y|2

4t E_k

µ X,Y

2t

¶

, (3)

whereγis the number of positive roots and the constantc_kis the Macdonald–Mehta–Selberg integral. The formula (3) remains true for theW-invariant kernelsp^W_t andE^W. In the geometric casesk = ¹₂, 1 and 2, by [4], theW-invariant formula (3) translates in a similar relationship between the spherical functionψ_λand the heat kernelp^W_t (X,Y):

p^W_t (X,Y)= 1

2^γ+d/2c_kt^−d2−γe^−|X|

2−|Y|2

4t ψX

µY 2t

¶

. (4)

A simple direct proof of (4) fork=1 is given in [8, Remark 2.9].

Equation (4) and Conjecture 1 bring us to an equivalent conjecture for the heat kernel p^W_t (X,Y).

Conjecture 3. We have

p^W_t (X,Y)³t^−d2 e^−|X−Y|

2 4t

Y

α>0

(t+α(X)α(Y)).

Consider also the relationship between the heat kernelp^W_t (X,Y) and the heat kernelpe^W_t (X,Y) in the curved case. We have

pe^W_t (X,Y)=e^−|ρ|2t π(X)π(Y)

δ^1/2(X)δ^1/2(Y)p^W_t (X,Y) . (5) This relation follows directly from the fact that the respective radial Laplacians and radial mea- sures areπ⁻¹La◦πandπ(X)d X in the flat case andδ^−1/2(La− |ρ|²)◦δ^1/2andδ(X)d X in the curved case (L_astands for the Euclidean Laplacian ona).

In the curved complex case, Conjecture 3 becomes pe^W_t (X,Y)³e^−|ρ|2tt^−d2e^−ρ(X+Y) Y

α>0

(1+α(X)) (1+α(Y)) (t+α(X)α(Y)) e^−|X−Y|

2

4t .

Remark 4. In [6], sharp estimates ofW-invariant Poisson and Newton kernels in the complex Dunkl case were obtained, by exploiting the method of construction of theseW-invariant kernels by alternating sums. When a root systemΣacts inR^d, the sharp estimates of [6] have the common form

K^W(X,Y)³ K^Rd(X,Y) Y

α>0

¡|X−Y|²+α(X)α(Y)¢, X,Y ∈a⁺, (6) whereK^W(X,Y) is theW-invariant kernel in Dunkl setting andK^Rd(X,Y) is the classical kernel onR^d. Let us observe a common pattern in the appearance of the classical kernelsK^Rd and of products of rootsα(X)α(Y) in formulas (6) and of the Fourier kernel e^〈λ,X〉 and the classical Gaussian heat kernel and of products α(λ)α(X) in the estimates given in Conjecture 1 and Conjecture 3.

2.1. Proof of Conjecture 1 in some cases We start with a practical result.

Proposition 5. Letα1, . . . ,αr be the simple roots and let A_αi be such that〈X,A_αi〉 =αi(X)for X∈a. Suppose X∈a⁺and w∈W\ {i d}. Then we have

Y−w Y =

r

X

i=1

2a^w_i (Y)

|αi|² A_αi (7)

where a^w_i is a linear combination of simple roots with non-negative integer coefficients for each i .

Proof. Refer to [6, Proposition 3.5].

Remark 6. Note thata^w_i (Y)/|αi|²is bounded byCmax_k|αk(Y)|whereCis a constant depending only onw∈Wand, ultimately, onW.

Corollary 7. Let Y ∈a⁺and w∈W . Consider the decomposition(7)of Y−w Y . If a_k^w(Y)6=0then αkappears in a^w

k, i.e. a^w

k =Pr

i=1n_iαi with n_k>0.

Proof. Refer to [6, Corollary 3.10].

As a prelude to the next Proposition, we give here a description of the Abel transformA(f) of a functionf onain the case of a flat symmetric space (in the case of the curved symmetric space refer for example to [15]). The dual of the Abel transformA^∗(f) can be defined as follows

A^∗(f)(X)= Z

K

f¡

πa(Ad(k)(X))¢ d k

where, as before,πais the orthogonal projection with respect to the Killing form. We can then define (in analogy with the curved case)

Z a

f(X)A(g)(X)d X= Z

a

A^∗(f)(X)g(X)w(X)d X wherew(X)=Q

α>0|α(X)|^mαandm_αis the multiplicity ofα. From [9, Chap IV, Theorem 10.2], we can deduce that the support of the dual Abel transform isC(X), the convex hull ofW·X. The setC(X) has non-empty interior as long as not all roots are 0 onX. From there, we conclude that we can write

A^∗(f)(X)= Z

C(X)

f(Y)K(X,Y)d Y and thatK(X,Y)d Yis a probability measure supported byC(X).

Proposition 8. Letδ>0. Supposeαi(λ)αj(X)≤δfor all i , j . Thenψ_λ(X)³e^λ(X)(the constants involved only depend onδ).

Proof. Notice that

e^{wmi nλ(X)}≤ψλ(X)= Z

C(X)

e^λ(Y)K(X,Y)d Y ≤e^λ(X) (8) wherewmi n is the element of the Weyl group giving the minimum value ofwλ(X). Now, using Proposition 5 and Remark 6 withY =λ, we see that for anyw∈W

e^λ(X)≥e^wλ(X)=e^{〈wλ−λ,X〉}e^〈λ,X〉=

r

Y

i=1

e⁻²

awi (λ)

|αi|² αi(X)

e^〈λ,X〉

≥

r

Y

i=1

e^−2C(^maxkαk(λ))αi(X)e^〈λ,X〉≥

r

Y

i=1

e^−2Cδe^〈λ,X〉. Remark 9. This case and this method apply for any radial Dunkl case; it suffices to replace K(X,Y)d Yby the so-called Rösler measureµX(d Y) in the integral in (8), see [14].

Proposition 10. A spherical functionψλ(X)on M is given by the formula ψ_λ(X)= π(ρ)

2^γπ(λ)π(X) X

w∈W

²(w)e^〈wλ,X〉, (9) whereρ= ¹₂P

α∈Σ⁺m_αα=P

α∈Σ⁺αand γ= |Σ⁺|is the number of positive roots (refer to [9, Chap. IV, Proposition 4.8 and Chap. II, Theorem 5.35]).

Proposition 11. Supposeα(λ)α(X)≥(log|W|)/2for allα>0. Then ψλ(X)³ e^λ(X)

π(λ)π(X). We are assuming here that|αi| ≥1for each i .

Proof. Supposew∈W is not the identity. In that case, a^w_i (λ) is not equal to 0 for somei. By Proposition 5 withy=λand Corollary 7,λ(X)−wλ(X)≥2a^w_i (λ)αi(X)/|αi|²≥2αi(λ)αi(X)≥ log|W|. Each terme^〈wλ,X〉 in the alternating sum (9) corresponding tow 6=id is bounded by e^−log|W|e^λ(X)=e^λ(X)/|W|. Hence, since only half the terms in the sum are negative,

|W|e^λ(X)≥ X

w∈W

²(w)e^〈wλ,X〉≥e^λ(X)−|W|

2 e^λ(X)/|W| =1

2e^λ(X).

3. The conjecture in the case of the root system

We will prove the conjecture in the case of the root system of typeA.

Theorem 12. In the case of the root system of type Anin the complex case, we have ψ_λ¡

e^X¢

³ e^〈λ,X〉 Y

i<j

¡1+¡

λi−λj¢ ¡

x_i−x_j¢¢, λ,X∈a⁺ (10) where X=[x₁, . . . ,x_n+1]with x_i≥x_i+1andλ=[λ1, . . . ,λn+1]withλi≥λi+1.

Corollary 13.

φ_λ¡ e^X¢

³e^(λ−ρ)(X) Y

i<j

1+x_i−x_j 1+(x_i−x_j)¡

λi−λj

¢, p^W_t (X,Y)³t^−d2 e^−|X−Y|

2 4t

Y

i<j

¡t+(xi−xj) (yi−yj)¢,

pe^W_t (X,Y)³e^−|ρ|2tt^−d2e^−ρ(X+Y)Y

i<j

¡1+x_i−x_j¢ ¡

1+y_i−y_j¢

¡t+¡

x_i−x_j¢ ¡

y_i−y_j¢¢ e^−|X−Y|

2

4t .

We recall (refer to [16, Theorem 4.1]) the following iterative formula for the spherical functions of typeAin the complex case. Here the Cartan subalgebraafor the root systemA_n−1is isomor- phic toRⁿ. Forλ,X∈a⁺_⊂Rⁿ, we have

ψ_λ¡ e^X¢

=e^λ(X) if n=1 andψ_λ¡

e^X¢

=(n−1)!e^λnPnk=1x_k

à Y

i<j

¡x_i−x_j¢

!₋1

Z x_n−1 x_n · · ·

Z x1 x₂ ψ_λ0

¡e^Y¢ Y

i<j<n

(y_i−y_j)d y₁· · ·d y_n−1

(11)

whereλ0(U)=P_n−1

k=1(λk−λn)u_k.

Remark 14. Formula (11) represents the action of the root systemA_n−1onRⁿ. If we assume

n

X

k=1

x_k=0=

n

X

k=1

λk,

we have then the action of the root systemA_n−1onRⁿ⁻¹. We can also consider the action ofA_n−1 on anyR^mwithm≥n−1 by considering formula (9) and deciding on which entriesx_k, the Weyl groupW=S_nacts. These considerations do not affect the conclusion of Theorem 12.

3.1. Approximate factorization for An

Before proving the conjecture in the caseA_n, we will prove an interesting “factorization”.

Proposition 15. For n ≥ 1, consider the root system A_n on Rⁿ⁺¹. Let λ,X ∈ a⁺ _⊂ Rⁿ⁺¹, X⁰=[x1, . . . ,xn]so that X=[X⁰,xn+1]. Define

I⁽ⁿ⁾=I⁽ⁿ⁾(λ;X)= Z xn

x_n+1

Z xn−1 xn

· · · Z x₂

x3

Z x₁ x2

e^{−λ0(X0−Y)} Y

i<j<n

¡y_i−y_j¢ ¡ λi−λj

¢ 1+¡

y_i−y_j¢ ¡ λi−λj

¢d y₁d y₂· · ·d y_n.

Then the following approximate factorization holds I⁽ⁿ⁾³

n

Y

k=1

I_k⁽ⁿ⁾ (12)

where

I₁⁽ⁿ⁾= Z x1

x2

e^{−(λ1−λn+1) (x1−y1)}d y₁ and

I_k⁽ⁿ⁾= Z x_k

x_k+1

e⁻(λk−λn+1)(^xk−yk)^k−1Y

j=1

¡x_j−y_k¢ ¡ λj−λk

¢ 1+¡

xj−y_k¢ ¡ λj−λk

¢d y_k for 1<k≤n. Proof. Sinceu/(1+u) is an increasing function, we clearly have

I⁽ⁿ⁾≤ Z x_n

xn+1

Z x_n−1 xn

· · · Z x₂

x3

Z x₁ x2

e^{−λ0(X0−Y)} Y

i<j<n

¡x_i−y_j¢ ¡ λi−λj

¢ 1+¡

xi−yj¢ ¡

λi−λj¢d y1d y2· · ·d yn. On the other hand,

I⁽ⁿ⁾≥ Z xn

(x_n+x_n+1)/2

Z x_n−1 (x_n−1+x_n)/2· · ·

Z x2 (x₂+x₃)/2

Z x1 (x₁+x₂)/2

e^−λ0(^X0−Y) Y

i<j<n

¡y_i−y_j¢ ¡ λi−λj¢ 1+¡

y_i−y_j¢ ¡ λi−λj

¢d y₁d y₂· · ·d y_n

≥ Z x_n

(xn+xn+1)/2

Z x_n−1 (xn−1+xn)/2· · ·

Z x₂ (x₂+x₃)/2

Z x₁ (x₁+x₂)/2

e^{−λ0(X0−Y)} Y

i<j<n

¡(x_i+x_i+1) /2−y_j¢ ¡ λi−λj

¢ 1+¡

(xi+xi+1) /2−yj¢ ¡

λi−λj¢d y1d y2· · ·d yn

³ Z x_n

(xn+xn+1)/2

Z x_n−1 (xn−1+xn)/2· · ·

Z x₂ (x2+x3)/2

Z x₁ (x1+x2)/2

e^−λ0(^X0−Y) Y

i<j<n

¡x_i−y_j¢ ¡ λi−λj

¢ 1+¡

x_i−y_j¢ ¡

λi−λj¢d y₁d y₂· · ·d y_n=

=

n

Y

k=1

Z x_k

(^xk+x_k+1)^/2e⁻(λk−λn+1)^(xk−yk)k−1

Y

j=1

(x_j−y_k)¡ λj−λk

¢ 1+(x_j−y_k)¡

λj−λk

¢d y_k=

n

Y

k=1

A⁽ⁿ⁾_k since

¡(x_i+x_i+1) /2−y_j¢ ¡ λi−λj

¢ 1+¡

(x_i+x_i+1) /2−y_j¢ ¡

λi−λj¢≤

¡x_i−y_j¢ ¡ λi−λj

¢ 1+¡

x_i−y_j¢ ¡ λi−λj¢ while

¡(xi+xi+1) /2−yj¢ ¡ λi−λj¢ 1+¡

(x_i+x_i+1) /2−y_j¢ ¡ λi−λj

¢≥

¡xi−yj¢ /2¡

λi−λj¢ 1+¡

x_i−y_j¢ /2¡

λi−λj

¢≥1 2

¡xi−yj¢ ¡ λi−λj¢ 1+¡

x_i−y_j¢ ¡ λi−λj

¢. Now, let

B⁽ⁿ⁾

k =

Z(^xk+x_k+1)^/2

x_k+1

e⁻(λk−λn+1)(xk−yk)^k−1Y

j=1

¡x_j−y_k¢ ¡ λj−λk

¢ 1+¡

x_j−y_k¢ ¡

λj−λk¢d y_k and note thatI_k⁽ⁿ⁾=A⁽ⁿ⁾_k +B_k⁽ⁿ⁾.

Now, using the change of variable 2w=x_k−y_k, we have B_k⁽ⁿ⁾=2

Z (^xk−x_k+1)^/2 (xk−xk+1)/4

e⁻²(λk−λn+1)^wkY⁻¹

j=1

¡xj−x_k+2w¢ ¡ λj−λk

¢ 1+¡

x_j−x_k+2w¢ ¡ λj−λk

¢d w

≤4

Z (^xk−x_k+1)^/2 (^xk−x_k+1)^/4

e⁻²(λk−λn+1)^wk−1Y

j=1

¡x_j−x_k+w¢ ¡ λj−λk

¢ 1+¡

xj−x_k+w¢ ¡ λj−λk

¢d w

≤4

Z (xk−xk+1)/2 0

e⁻(λk−λn+1)^wkY⁻¹

j=1

¡x_j−x_k+w¢ ¡ λj−λk

¢ 1+¡

x_j−x_k+w¢ ¡ λj−λk

¢d w=4A⁽ⁿ⁾_k ,

where the last equality comes from the change of variablew=x_k−y_kin the expression forA⁽ⁿ⁾_k . ThereforeI⁽ⁿ⁾

k =A⁽ⁿ⁾

k +B⁽ⁿ⁾

k ≤5A⁽ⁿ⁾

k . The result follows.

The next Proposition 16 gives an inductive way of estimatingI⁽ⁿ⁺¹⁾, knowingI⁽ⁿ⁾andI⁽ⁿ⁻¹⁾. Proposition 16. Consider the root system A_n+1onRⁿ⁺². Letλ,X ∈a⁺_⊂Rⁿ⁺². Assumeα1(X)≥ αn+1(X). Then

I⁽ⁿ⁺¹⁾(λ;X)³I⁽ⁿ⁾(λ1, . . . ,λn,λn+2;x₁, . . . ,x_n+1) (x₁−x_n+1) (λ1−λn+1) 1+(x1−xn+1) (λ1−λn+1) I⁽ⁿ⁾(λ2, . . . ,λn+1,λn+2;x₂, . . . ,x_n+2)

I⁽ⁿ⁻¹⁾(λ2, . . . ,λn,λn+2;x₂, . . . ,x_n+1). Proof. We start with an outline of the proof.

(i) I⁽ⁿ⁺¹⁾is estimated by a product ofn+1 factorsI_k⁽ⁿ⁺¹⁾(λ;X).

(ii) The product of the firstnfactorsI₁⁽ⁿ⁺¹⁾(λ;X), . . . ,I_n⁽ⁿ⁺¹⁾(λ;X) gives an estimate of the term I⁽ⁿ⁾(λ1, . . . ,λn,λn+2;X⁰) by Proposition 15.

(iii) In the last factor I_n⁽ⁿ⁺¹⁾₊₁ (λ;X), we “draw off” one term from under the integral, us- ing the additional hypothesisα1(X)≥αn+1(X). The remaining integral corresponds to I_n⁽ⁿ⁾(λ2, . . . ,λn+2;x2, . . . ,xn+2).

(iv) The last factorI_n⁽ⁿ⁾ofI⁽ⁿ⁾is estimated byI⁽ⁿ⁾/I⁽ⁿ⁻¹⁾, up to a change of variables (we re-use the idea of (ii)).

Sincex_n+2≤y_n+1≤x_n+1andx_n+1−x_n+2≤x₁−x₂, we getx₁−x_n+1≤x₁−y_n+1≤x₁−x_n+2

≤2(x1−xn+1) and we have I_n+1⁽ⁿ⁺¹⁾³

Z xn+1 x_n+2

e^{−(λn+1−λn+2)}(x_n+1−yn+1)

¡x₁−y_n+1¢

(λ1−λn+1) 1+¡

x₁−y_n+1¢

(λ1−λn+1)

n

Y

j=2

¡x_j−y_n+1¢ ¡

λj−λn+1¢ 1+¡

x_j−y_n+1¢

¡λj−λn+1

¢d y_n+1

³ (x₁−x_n+1) (λ1−λn+1) 1+(x1−xn+1) (λ1−λn+1)

Z xn+1 x_n+2

e^{−(λn+1−λn+2)}(x_n+1−yn+1)

n

Y

j=2

¡xj−yn+1¢ ¡

λj−λn+1¢ 1+¡

x_j−y_n+1¢ ¡

λj−λn+1

¢d y_n+1.

Hence, noting thatI₁⁽ⁿ⁺¹⁾(λ;X)· · ·I_n⁽ⁿ⁺¹⁾(λ;X)³I⁽ⁿ⁾(λ1, . . . ,λn,λn+2;X⁰), we have I⁽ⁿ⁺¹⁾(λ;X)³I⁽ⁿ⁾¡

λ1, . . . ,λn,λn+2;X⁰¢ (x₁−x_n+1) (λ1−λn+1) 1+(x₁−xn+1) (λ1−λn+1) Z xn+1

x_n+2

e^{−(λn+1−λn+2)}(x_n+1−yn+1)Yⁿ

j=2

¡x_j−yn+1

¢ ¡λj−λn+1

¢ 1+¡

x_j−y_n+1¢ ¡

λj−λn+1

¢d y_n+1.

Finally, Z x_n+1

xn+2

e^{−(λn+1−λn+2)}(^xn+1−yn+1)Yⁿ

j=2

¡x_j−y_n+1¢ ¡

λj−λn+1

¢ 1+¡

xj−yn+1¢ ¡

λj−λn+1¢d yn+1

=

n

Y

k=1

Z _xk+1

x_k+2

e⁻(λk+1−λn+2)(^xk+1−y_k+1)^kY⁻¹

j=1

¡x_j+1−y_k+1¢ ¡

λj+1−λk+1¢ 1+¡

x_j+1−y_k+1¢ ¡

λj+1−λk+1

¢d y_k+1

n−1Y

k=1

Z x_k+1 xk+2

e⁻(λk+1−λn+2)(xk+1−yk+1)^k−1Y

j=1

¡xj+1−y_k+1¢ ¡

λj+1−λk+1

¢ 1+¡

x_j+1−y_k+1¢ ¡

λj+1−λk+1

¢d y_k+1

=I⁽ⁿ⁾(λ2, . . . ,λn+1,λn+2;x₂, . . . ,x_n+2) I⁽ⁿ⁻¹⁾(λ2, . . . ,λn,λn+2;x₂, . . . ,x_n+1).

Remark 17. Whenn=1, the result of Proposition 16 remains valid if we setI⁽⁰⁾=1.

We now prove our main result.

Proof of Theorem 12. We use induction on the rank. In the case ofA1, we have ψλ¡

e^X¢

=e^λ2(x1+x2)(x1−x2)⁻¹ Z x₁

x2

e^(λ1−λ2)yd y

=e^λ2(x1+x2)(x₁−x₂)⁻¹e^{(λ1−λ2)x1}−e^{(λ1−λ2)x2} λ1−λ2

=e^λ1x1+λ2x21−e^{−(λ1−λ2) (x1−x2)}

(λ1−λ2) (x₁−x₂) ³e^λ1x1+λ2x2 1

1+(λ1−λ2) (x₁−x₂) since 1−e^−u³u/(1+u) foru≥0.

Assume that the result is true forAr, 1≤r≤n,n≥1. Using (11) and the induction hypothesis, we have forr=1, . . . ,n+1 andλ,Xin positive Weyl chamber inR^r+1

π(X)π(λ⁰)e^−λ(X)ψ_λ([x1, . . . ,xr,xr+1])

=r!π(λ⁰)e^−λ(X)e^λr+1

r+1P k=1

x_k Zxr xr+1

· · · Z x1

x2

ψ_λ0¡

e^Y¢ Y

i<j<r+1

¡y_i−y_j¢

d y₁· · ·d y_r

³ Z x_r

x_r+1

Z x_r−1 x_r · · ·

Z x₂ x₃

Z x₁ x₂

e^−λ0(^X0−Y) Y

i<j<r+1

¡y_i−y_j¢ ¡ λi−λj

¢ 1+¡

y_i−y_j¢ ¡ λi−λj

¢d y₁d y₂· · ·d yr

whereX⁰=diag[x₁, . . . ,x_r] andλ⁰=[λ1, . . . ,λr]. Using the notation introduced in Proposition 15, we have

π(X)π(λ⁰)e^−λ(X)ψ[λ1, ...,λr+1]([x₁, . . . ,xr+1])=r!I^(r)(λ1, . . . ,λr+1,x₁, . . . ,xr+1) . Still using the induction hypothesis, we have

π(X)π(λ⁰)e^−λ(X)ψ[λ1, ...,λr+1]([x₁, . . . ,x_r+1])=r!I^(r)(λ1, . . . ,λr+1;x₁, . . . ,x_r+1)

³ π(X)π(λ⁰) Q

i<j≤r+1

¡1+¡ λi−λj

¢ ¡x_i−x_j¢¢

(13)

forr=1, . . . ,n.

It remains to show that (13) holds forr=n+1, i.e. that

I⁽ⁿ⁺¹⁾(λ1, . . . ,λn+2;x₁, . . . ,x_n+2)³ π(X)π(λ⁰) Q

i<j≤n+2

¡1+¡ λi−λj

¢ ¡x_i−x_j¢¢.