We write down the generalized Bessel function associated with the root system of type D by means of multivariate hypergeometric series. Our hint comes from the particular case of the Brownian motion in the Weyl chamber of type D

Demni, N.

Symmetry Integrability and Geometry Methods and Applications

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 4 (2008), 075, 7 pagesGeneralized Bessel function of Type D?Nizar DEMNISFB 701, Fakulta¨t fu¨r Mathematik, Universita¨t Bielefeld, DeutschlandE-mail: demni@math.uni-bielefeld.deReceived July 01, 2008, in final form October 24, 2008; Published online November 04, 2008Original article is available at http://www.emis.de/journals/SIGMA/2008/075/Abstract. We write down the generalized Bessel function associated with the root system oftype D by means of multivariate hypergeometric series. Our hint comes from the particularcase of the Brownian motion in the Weyl chamber of type D.Key words: radial Dunkl processes; Brownian motions in Weyl chambers; generalized Besselfunction; multivariate hypergeometric series2000 Mathematics Subject Classification: 33C20; 33C52; 60J60; 60J651 Root systems and related processesWe refer the reader to [11] for facts on root systems. Let (V, 〈·〉) be an Euclidean space of finitedimension m ≥ 1. A reduced root system R is a finite set of non zero vectors in V such that1) R ∩ Rα = {α,−α} for all α ∈ R,2) σα(R) = R,where σα is the reflection with respect to the hyperplane Hα orthogonal to ασα(x) = x− 2 〈α, x〉〈α, α〉α, x ∈ V.A simple system S is a basis of span(R) which induces a total ordering in R. A root α is positiveif it is a positive linear combination of elements of S. The set of positive roots is called a positivesubsystem and is denoted by R+. The (finite) reflection group W is the group generated by allthe reflections σα for α ∈ R. Given a root system R with positive and simple systems R+, S,define the positive Weyl chamber C by:C := {x ∈ V, 〈α, x〉 > 0 ∀α ∈ R+} = {x ∈ V, 〈α, x〉 > 0 ∀α ∈ S}and C, ∂C its closure and boundary respectively. One of the most important properties isthat the convex cone C is a fundamental domain, that is, each λ ∈ V is conjugate to one andonly one µ ∈ C. Processes related to root systems have been of great interest during the lastdecade and notably the so-called Dunkl process [17]. This V -valued process was deeply studiedin a sequence of papers by Gallardo and Yor [5, 6, 7, 8] and Chybiryakov [3] and its projectionon the Weyl chamber gives rise to a diffusion known as the W -invariant or radial Dunkl process.The generator of the latter process acts on C2c (C) asLku(x) =12∆u(x) +∑α∈R+k(α)〈α,∇u(x)〉〈α, x〉?This paper is a contribution to the Special Issue on Dunkl Operators and Related Topics. The full collectionis available at http://www.emis.de/journals/SIGMA/Dunkl operators.html2 N. Demniwith 〈α,∇u(x)〉 = 0 whenever 〈α, x〉 = 0, where k is a positive function defined on the orbitsof the W -action on V and is constant on each orbit. Such a function is called a multiplicityfunction. The semi group density (with respect to the Lebesgue measure on V ) of the radialDunkl process is written as [17]pkt (x, y) =1cktγ+m/2e−(|x|2+|y|2)/2tDWk(x√t,y√t)ωk(y)2 (1)for x, y ∈ C, whereγ =∑α∈R+k(α), ωk(y) =∏α∈R+〈α, y〉k(α).The kernel DWk is defined byDWk (x, y) =∑w∈WDk(x,wy)where Dk is the non symmetric Dunkl kernel [18], and is up to the constant factor 1/|W | theso-called generalized Bessel function since it reduces to the (normalized) Bessel function in therank-one case B1 [18]. It was identified in [1] with a multivariate hypergeometric series for bothroot systems of types A and B. Since the C-type root system is nothing but the dual rootsystem of B, it remains to write down this kernel for the root system of type D which is theremaining infinite family of irreducible root systems associated with Weyl groups. This wasbehind our motivation to write the present paper and our main result is stated asTheorem 1. For the D-type root system, the generalized Bessel function writes1|W |∑w∈WDk(x,wy) =m∏i=1(xiyi2)0F(1/k1)1(q +12,x22,y22)+ 0F(1/k1)1(q − 12,x22,y22),where 0F(1/k1)1 is the multivariate hypergeometric series of Jack parameter 1/k1, x := (x21, . . . ,x2m), k = k1 > 0 is the value of the multiplicity function and q := 1 + (m− 1)k1.Our strategy is easily explained as follows: according to Grabiner [9, p. 21], the Brownianmotion in the Weyl Chamber is a radial Dunkl process of multiplicity function k ≡ 1, that isk(α) = 1 for all α ∈ R. Indeed, it is shown in [19] that the radial Dunkl process, say XW , is theunique strong solution ofdYt = dBt +∑α∈R+k(α)dt〈α, Yt〉 , Y0 ∈ C,where B is a m-dimensional Brownian motion and k(α) > 0 for all α ∈ R. Hence, by Grabiner’sresult on Brownian motions started at C and killed when they first reach ∂C together with theDoob’s transform [16] applied with the harmonic function [9, p. 19]h(y) :=∏α∈R+〈α, y〉,the semi group density the Brownian motion in the Weyl chamber is given byp1t (x, y) =h(y)h(x)∑w∈Wdet(w)Nt,m(wy − x), (2)Generalized Bessel function of Type D 3where Nt,m is the m-dimensional heat kernel given byNt,m(x) =1(2pit)m/2e−|x|2/2t, |x|2 = 〈x, x〉, x ∈ V.On the one hand and for the irreducible root systems A, B, D, the sum over W in (2) was ex-pressed in [9] as determinants. On the other hand, the generalized Bessel function was expressedfor A and B-types root systems via multivariate hypergeometric series of two arguments and ofJack parameter which equals the inverse of one of the multiplicity values (see the end of [1]).When this parameter equals one, the multivariate series takes a determinantal form [10] whichagrees with Grabiner’s result. As a matter of fact, we will start from the expression obtainedin [9] for the Brownian motion in the Weyl chamber of type D then express it as a multivariateseries of a Jack parameter 1. Once we did, we will generalize the result to an arbitrary positiveJack parameter using an analytical tool known as the shift principle. Before proceeding, weshall recall some facts on multivariate hypergeometric series and investigate the cases of A andB-types root systems.2 Mutlivariate series and determinantal representationsWe refer the reader to [1, 2, 12] and references therein for facts on Jack polynomials and multi-variate hypergeometric series. Let τ be a partition of length m, that is a sequence of positiveintegers τ1 > · · · > τm. Let α > 0, then the Jack polynomial C(α)τ of Jack parameter α is theunique (up to a normalization) symmetric homogeneous eigenfunction of the operatorm∑i=1x2i ∂2i +2αm∑i=1x2ixi − xj ∂icorresponding to the eigenvalueρτ =m∑i=1τi[τi − (2/α)(i− 1)] + |τ |(m− 1), |τ | = τ1 + · · ·+ τm.The normalization we adopt here is(x1 + · · ·+ xm)n =∑τC(α)τ (x), n ≥ 0,where the sum is taken over all the partitions of weight |τ | = n and length m. For α = 2it reduces to the zonal polynomial [15] while for α = 1 it fits the Schur polynomial [14]. Letp, q ∈ N, then the multivariate hypergeometric series of two arguments is defined bypF(α)q ((ai)1≤i≤p, (bj)1≤j≤q;x, y) =∞∑n=0∑|τ |=n(a1)τ · · · (ap)τ(b1)τ · · · (bq)τC(α)τ (x)C(α)τ (y)C(α)τ (1)|τ |!,where (1) = (1, . . . , 1) and for a partition τ(a)(α)τ =m∏i=1(a− 1α(i− 1))τj=m∏i=1Γ(a− (i− 1)/α+ τj)Γ(a− (i− 1)/α)is the generalized Pochhammer symbol and Γ is the Gamma function. We assume of course thatthe above expression makes sense for all the coefficients (ai)1≤i≤p, (bj)1≤j≤q. An interesting4 N. Demnifeature of the hypergeometric series of Jack parameter α = 1 is that they are expressed throughunivariate hypergeometric functions pFq [13] as follows [10]pF(1)q ((m+ µi)1≤i≤p, (m+ φj)1≤j≤q;x, y) = pim(m−1)2(p−q−1+1/α)m∏i=1(m− 1)!×p∏i=1(Γ(µi + 1))mΓ(m+ µi)q∏j=1Γ(m+ φj)(Γ(φj + 1))mdet[pFq((µi + 1)1≤i≤p, (1 + φj)1≤j≤q;xlyr)]ml,r=1V (x)V (y)for all µi, φj > −1, where V stands for the Vandermonde polynomial and empty product areequal 1. For instance, when p = q = 0, we have0F(1)0 (x, y) = pi−m(m−1)2(−1+1/α)det(exiyj )mi,j=1V (x)V (y),and for p = 0, q = 1, we similarly get0F(1)1 (m+ φ;x, y) = pim(m−1)2αm∏i=1(m− 1)! Γ(m+ φ)(Γ(φ+ 1))mdet[0F1(1 + φ;xiyj)]mi,j=1V (x)V (y)for φ > −1.3 Brownian motions in Weyl chambers: types A and B revisitedBrownian motions in Weyl chambers were deeply studied in [9] and they are shown to be h-transforms in Doob’s sense of m independent real BMs killed when they first hit ∂C. They arethen interpreted as m independent particles constrained to stay in the Weyl chamber C. Theproofs of those properties use probabilistic arguments. Here, we show how, for both types Aand B, these properties follow from the above determinantal representation of multivariatehypergeometric functions of two arguments. Let us first recall that the A-type root system isdefined byAm−1 = {±(ei − ej), 1 ≤ i < j ≤ m},with positive and simple systems given byR+ = {ei − ej , 1 ≤ i < j ≤ m}, S = {ei − ei+1, 1 ≤ i ≤ m− 1},where (ei)1≤i≤m is the standard basis of Rm, W being the permutation group. In this case,V = Rm, the span of R is the hyperplane of Rm consisting of vectors whose coordinates sum tozero and C = {x ∈ Rm, x1 > · · · > xm}. Besides, there is only one orbit so that k(α) := k1 ≥ 0and the function h (product of the positive roots) is given by the Vandermonde polynomial.Next, it was shown in [1] that DWk is given up to the constant 1/|W | by 0F (1/k1)0 (x, y) (weuse another normalization than the one used in [1] so that the factor√2 is removed). Thus,when k1 = 1, (1) writes for R = Am−1p1t (x, y) =V (y)V (x)det(Nt(yj − xi))mi,j=1, (3)where Nt is the heat kernel, namelyNt(v) =1√2pite−v2/2t.Generalized Bessel function of Type D 5Thus, the V -transform property is easily seen. A similar result holds for R = Bm. This rootsystem has the following dataR = {±ei, ±ei ± ej , 1 ≤ i < j ≤ m}, R+ = {ei, 1 ≤ i ≤ m, ei ± ej , 1 ≤ i < j ≤ m},S = {ei − ei+1, 1 ≤ i ≤ m, em}, C = {y ∈ Rm, y1 > y2 > · · · > ym > 0}.W is generated by transpositions and sign changes (xi 7→ −xi) and there are two orbits so thatk = (k0, k1) thereby γ = mk0 +m(m − 1)k1. The generalized Bessel function1 is given by [1,p. 214]1|W |DWk (x, y) = 0F(1/k1)1(k0 + (m− 1)k1 + 12 ,x22,y22).It follows thatpk0,1t (x, y) = C(m)h(y)h(x)e−(|x|2+|y|2)/2ttm/2m∏i,j=1(xiyjt)det[0F1(k0 +12,(xiyj)24t2)]ni,j=1,where h(y) = V (y2)n∏i=1yi and C(m) is a constant depending on m. Taking k0 = 1 and usingthe identity [13]0F1(32, z)=C2√zsinh(2√z),for some constant C, one gets for the Brownian motion in the B-type Weyl chamberp1,1t (x, y) =h(y)h(x)det [Nt(yj − xi)−Nt(yj + xi)]mi,j=1 . (4)The h-transform property is then obvious.4 Generalized Bessel function of type DThe root system of type D is defined by [11, p. 42]R = {±ei ± ej , 1 ≤ i < j ≤ m}, R+ = {ei ± ej , 1 ≤ i < j ≤ m},and there is one orbit so that k(α) = k1 thereby γ = m(m − 1)k1. The Weyl chamber is givenby:C = {x ∈ Rm, x1 > x2 > · · · > |xm|} = C1 ∪ smC1,where C1 is the Weyl chamber of type B and sm stands for the reflection with respect to thevector em acting by sign change on the variable xm. Now, Grabiner’s result reads in this casep1t (x, y) =V (y2)V (x2)det[Nt(yi − xj)−Nt(yi + xj)]mi,j=1 + det[Nt(yi − xj) +Nt(yi + xj)]mi,j=12=Cmtγ+m/2e−(|x|2+|y|2)/2tdet [sinh(xiyj/t)]mi,j=1 + det [cosh(xiyj/t)]mi,j=1V (x2/4t2)V (y2)V 2(y2),1There is an erroneous sign in one of the arguments in [1]. Moreover, to recover this expression in the Bm casefrom that given in [1], one should make substitutions a = k0 − 1/2, k1 = 1/α, q = 1 + (m− 1)k1.6 N. Demniwhere γ = m(m− 1). With the help of the determinantal representations and of [13]0F1(32, z)=C√zsinh(2√z), 0F1(12, z)= cosh(2√z),one getsp1t (x, y) =e−(|x|2+|y|2)/2tcktγ+m/2×[m∏i=1(xiyi2t)0F(1)1(m+12,x22t,y22t)+ 0F(1)1(m− 12,x22t,y22t)]V 2(y2).With regard to (1) and setting q := 1+(m−1)k1, it is natural to prove the claim of Theorem 1.Proof. It uses the so-called shift principle that we briefly outline [4]. Let E be a conjugacyclass of roots of R under the action W . Let kE be the value of the multiplicity function on thisclass. Then, the generalized Bessel function associated with the multiplicity function k′ definedbyk′(α) ={k(α) + 1 if α ∈ E,k(α) otherwise,is given by1|W |DWk′ (x, y) = C∑w∈WξE(w)Dk(x,wy)pE(x)pE(y), pE(x) :=∏α∈R+∩E〈α, x〉,where ξE(w) is defined by pE(wx) = ξE(w)pE(x) and C is some constant. Since we will deal withboth the B and D-types, it is convenient to add a superscript B or D to each corresponding item.Therefore, WB, WD denote the Weyl groups associated with the root systems of types B, Drespectively and kB, kD denote the corresponding multiplicity functions. Recall that [11] WB isthe semi-direct product of Sm and (Z/2Z)m (sign changes) while WD is the semi-direct productof Sm and (Z/2Z)m−1 (even sign changes). Let E := {ei, 1 ≤ i ≤ m}, then pE(x) is invariantunder permutations and even sign changes and skew-invariant under odd sign changes (notethat DWDk is not WB-invariant). It follows that ξE(w) = 1 for w ∈ WD while ξE(w) = −1 forw ∈ WB \WD. Now, recall that kD takes only one value while kB takes two values. Since theDunkl Laplacian ∆Bk of B reduces to ∆Dk when the value of the multiplicity function k0 on E iszero [18], the (non-symmetric) Dunkl kernel DDk associated with R = Dm is given by the Dunklkernel DBk associated with R = Bm when kBE = 0. In fact, this is true since Dk is the solutionof a spectral problem which is independent of W . As a result,1|WD|DWDk (x, y) =1|W |D∑w∈WDDDk (x,wy)=12|W |D∑w∈WB(1 + ξE(w))DB,k0=0k (x,wy)=|WB|2|WD|[1|WB|DWB(k0=0)k +pE(x)pE(y)C|WB| DWB(k0=1)k′],where we used the shift principle to derive the last line. Keeping in mind that1|WB|DWBk (x, y) = 0F(1/k1)1(k0 + (m− 1)k1 + 12 ,x22t,y22t)and using that |WB| = 2|WD| [11, p. 44] we are done with C = 2m. Generalized Bessel function of Type D 75 Concluding remarksAs the reader can see, DWDk is not equal to DWBk specialized with k0 = 0 and this shows thatthe generalized Bessel function is intimately related to W . Moreover the symmetrical of DWDkwith respect to sm gives DWBk in the special case k0 = 0 which reflects the fact that the Weylchamber of type D is the union of the one of type B and its symmetrical with respect to sm.References[1] Baker T.H., Forrester P.J., The Calogero–Sutherland model and generalized classical polynomials, Comm.Math. 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Generalized Bessel function of Type D

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Generalized Bessel function of Type D

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)