Vector-Valued Polynomials and a Matrix Weight Function with B₂-Action

Vector-Valued Polynomials and a Matrix Weight Function with B₂-Action

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Дата:2013
Автор: Dunkl, C.F.
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Опубліковано: Інститут математики НАН України 2013
Назва видання:Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:Vector-Valued Polynomials and a Matrix Weight Function with B₂-Action / C.F. Dunkl // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 8 назв. — англ.

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spelling irk-123456789-1492142019-02-20T01:28:28Z Vector-Valued Polynomials and a Matrix Weight Function with B₂-Action Dunkl, C.F. 2013 Article Vector-Valued Polynomials and a Matrix Weight Function with B₂-Action / C.F. Dunkl // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 8 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33C52; 42C05; 33C05 DOI: http://dx.doi.org/10.3842/SIGMA.2013.007 http://dspace.nbuv.gov.ua/handle/123456789/149214 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
format Article
author Dunkl, C.F.
spellingShingle Dunkl, C.F.
Vector-Valued Polynomials and a Matrix Weight Function with B₂-Action
Symmetry, Integrability and Geometry: Methods and Applications
author_facet Dunkl, C.F.
author_sort Dunkl, C.F.
title Vector-Valued Polynomials and a Matrix Weight Function with B₂-Action
title_short Vector-Valued Polynomials and a Matrix Weight Function with B₂-Action
title_full Vector-Valued Polynomials and a Matrix Weight Function with B₂-Action
title_fullStr Vector-Valued Polynomials and a Matrix Weight Function with B₂-Action
title_full_unstemmed Vector-Valued Polynomials and a Matrix Weight Function with B₂-Action
title_sort vector-valued polynomials and a matrix weight function with b₂-action
publisher Інститут математики НАН України
publishDate 2013
url http://dspace.nbuv.gov.ua/handle/123456789/149214
citation_txt Vector-Valued Polynomials and a Matrix Weight Function with B₂-Action / C.F. Dunkl // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 8 назв. — англ.
series Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv AT dunklcf vectorvaluedpolynomialsandamatrixweightfunctionwithb2action
first_indexed 2025-07-12T21:05:55Z
last_indexed 2025-07-12T21:05:55Z
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fulltext Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 007, 23 pages Vector-Valued Polynomials and a Matrix Weight Function with B2-Action Charles F. DUNKL Department of Mathematics, University of Virginia, PO Box 400137, Charlottesville VA 22904-4137, USA E-mail: cfd5z@virginia.edu URL: http://people.virginia.edu/~cfd5z/home.html Received October 16, 2012, in final form January 23, 2013; Published online January 30, 2013 http://dx.doi.org/10.3842/SIGMA.2013.007 Abstract. The structure of orthogonal polynomials on R2 with the weight function |x21 − x22|2k0 |x1x2|2k1e−(x2 1+x2 2)/2 is based on the Dunkl operators of type B2. This refers to the full symmetry group of the square, generated by reflections in the lines x1 = 0 and x1− x2 = 0. The weight function is integrable if k0, k1, k0 +k1 > − 1 2 . Dunkl operators can be defined for polynomials taking values in a module of the associated reflection group, that is, a vector space on which the group has an irreducible representation. The unique 2-dimensional representation of the group B2 is used here. The specific operators for this group and an analysis of the inner products on the harmonic vector-valued polynomials are presented in this paper. An orthogonal basis for the harmonic polynomials is constructed, and is used to define an exponential-type kernel. In contrast to the ordinary scalar case the inner product structure is positive only when (k0, k1) satisfy − 1 2 < k0 ± k1 < 1 2 . For vector polynomials (fi) 2 i=1, (gi) 2 i=1 the inner product has the form ∫∫ R2 f(x)K(x)g(x)T e−(x2 1+x2 2)/2dx1dx2 where the matrix function K(x) has to satisfy various transformation and boundary conditions. The matrix K is expressed in terms of hypergeometric functions. Key words: matrix Gaussian weight function; harmonic polynomials 2010 Mathematics Subject Classification: 33C52; 42C05; 33C05 1 Introduction The algebra of operators on polynomials generated by multiplication and the Dunkl operators associated with some reflection group is called the rational Cherednik algebra. It is parametrized by a multiplicity function which is defined on the set of roots of the group and is invariant under the group action. For scalar-valued polynomials there exists a Gaussian-type weight function which demonstrates the positivity of a certain bilinear form on polynomials, for positive values (and a small interval of negative values) of the multiplicity function. The algebra can also be represented on polynomials with values in an irreducible module of the group. In this case the problem of finding a Gaussian-type weight function and the multiplicity-function values for which it is positive and integrable becomes much more complicated. Here we initiate the study of this problem on the smallest two-parameter, two-dimensional example, namely, the group of type B2 (the full symmetry group of the square). Griffeth [7] defined and studied analogues of nonsymmetric Jack polynomials for arbitrary irreducible representations of the complex reflection groups in the family G(r, 1, n). This paper introduced many useful methods for dealing with vector-valued polynomials. In the present paper we consider B2, which is the member G(2, 1, 2) of the family, but we use harmonic poly- nomials, rather than Griffeth’s Jack polynomials because the former play a crucial part in the analysis of the Gaussian weight. There is a detailed study of the unitary representations of the rational Cherednik algebra for the symmetric and dihedral groups in Etingof and Stoica [6]. mailto:cfd5z@virginia.edu http://people.virginia.edu/~cfd5z/home.html http://dx.doi.org/10.3842/SIGMA.2013.007 2 C.F. Dunkl We begin with a brief discussion of vector-valued polynomials and the results which hold for any real reflection group. This includes the definition of the Dunkl operators and the basic bilinear form. The next section specializes to the group B2 and contains the construction of an orthogonal basis of harmonic homogeneous polynomials, also a brief discussion of the radical. Section 4 uses this explicit basis to construct the appropriate analogue of the exponential function. Section 5 contains the derivation of the Gaussian-type weight function; it is a 2 × 2 matrix function whose entries involve hypergeometric functions. This is much more complicated than the scalar case. The method of solution is to set up a system of differential equations, find a fundamental solution and then impose several geometric conditions, involving behavior on the mirrors (the walls of the fundamental region of the group) to construct the desired solution. 2 General results Suppose R is a root system in RN and W = W (R) is the finite Coxeter group generated by {σv : v ∈ R+} (where 〈x, y〉 := N∑ i=1 xiyi, |x| := 〈x, x〉1/2, xσv := x − 2 〈x,v〉〈v,v〉v for x, y, v ∈ RN and v 6= 0). Let κ be a multiplicity function on R (u = vw for some w ∈ W and u, v ∈ R implies κ(u) = κ(v)). Suppose τ is an irreducible representation of W on a (real) vector space V of polynomials in t ∈ RN of dimension nτ . (There is a general result for these groups that real representations suffice, see [1, Chapter 11].) Let PV be the space of polynomial functions RN → V , that is, the generic f ∈ PV can be expressed as f(x, t) where f is a polynomial in x, t and f(x, t) ∈ V for each fixed x ∈ RN . There is an action of W on PV given by wf(x, t) := f(xw, tw), w ∈W. Define Dunkl operators on PV , for 1 ≤ i ≤ N by Dif(x, t) := ∂ ∂xi f(x, t) + ∑ v∈R+ κ(v) f(x, tσv)− f(xσv, tσv) 〈x, v〉 vi. There is an equivariance relation, for u ∈ RN , w ∈W N∑ i=1 uiDiw = w N∑ i=1 (uw)iDi. (1) Ordinary (scalar) polynomials act by multiplication on PV . For 1 ≤ i, j ≤ N and f ∈ PV the basic commutation rule is Dixjf(x, t)− xjDif(x, t) = δijf(x, t) + 2 ∑ v∈R+ κ(v) vivj |v|2 f(xσv, tσv). (2) The abstract algebra generated by {xi,Di : 1 ≤ i ≤ N}∪RW with the commutation relations xixj = xjxi, DiDj = DjDi, (2) and equivariance relations like (1) is called the rational Cherednik algebra of W parametrized by κ; henceforth denoted by Aκ. Then PV is called the standard module of Aκ determined by the W -module V . We introduce symmetric bilinear W -invariant forms on PV . There is a W -invariant form 〈·, ·〉τ on V ; it is unique up to multiplication by a constant because τ is irreducible. The form is extended to PV subject to 〈xif(x, t), g(x, t)〉τ = 〈f(x, t),Dig(x, t)〉τ for f, g ∈ PV and 1 ≤ i ≤ N . To be more specific: let {ξi(t) : 1 ≤ i ≤ nτ} be a basis for V . Any f ∈ PV has a unique expression f(x, t) = ∑ i fi(x)ξi(t) where each fi(x) is a polynomial, and then 〈f, g〉τ := ∑ i 〈ξi (t) , (fi (D1, . . . ,DN ) g(x, t)) |x=0〉τ , g ∈ PV . The form satisfies 〈f, g〉τ = 〈wf,wg〉τ = 〈g, f〉τ , w ∈W . B2 Matrix Weight Function 3 This is a general result for standard modules of the rational Cherednik algebra, see [4]. The proof is based on induction on the degree and the eigenfunction decomposition of the operator N∑ i=1 xiDi. Indeed N∑ i=1 xiDif(x, t) = 〈x,∇〉f(x, t) + ∑ v∈R+ κ(v) ( f(x, tσv)− f(xσv, tσv) ) , (3) where ∇ denotes the gradient (so that 〈x,∇〉 = N∑ i=1 xi ∂ ∂xi ). Because τ is irreducible there are integers cτ (v), constant on conjugacy classes of reflections (namely, the values of the character of τ) such that∑ v∈R+ κ(v)f(x, tσv) = γ(κ; τ)f(x, t), γ(κ; τ) := ∑ v∈R+ cτ (v)κ(v), for each f ∈ PV . The Laplacian is ∆κf(x, t) := N∑ i=1 D2 i f(x, t) = ∆f(x, t) + ∑ v∈R+ κ(v) { 2 〈v,∇f(x, tσv)〉 〈x, v〉 − |v|2 f(x, tσv)− f(xσv, tσv) 〈x, v〉2 } , where ∆ and ∇ denote the ordinary Laplacian and gradient, respectively. Motivated by the Gaussian inner product for scalar polynomials (case τ = 1) which is defined by 〈f, g〉G := cκ ∫ RN f(x)g(x) ∏ v∈R+ |〈x, v〉|2κ(v)e−|x| 2/2dx, where cκ is a normalizing (Macdonald–Mehta) constant, and satisfies 〈f, g〉τ = 〈 e−∆κ/2f, e−∆κ/2g 〉 G , we define a bilinear Gaussian form on PV by 〈f, g〉G := 〈 e∆κ/2f, e∆κ/2g 〉 τ . Note e−∆κ/2 := ∞∑ n=0 (−1/2)n n! ∆n κ is defined for all polynomials since ∆κ is nilpotent. From the relations ∆κ(xif(x, t)) = xi∆κf(x, t) + 2Dif(x, t), e−∆κ/2(xif(x, t)) = (xi −Di)e−∆κ/2f(x, t), we find that 〈(xi − Di)f, g〉G = 〈f,Dig〉G, for 1 ≤ i ≤ N and f, g ∈ PV . Thus the multiplier operator xi is self-adjoint for this form (since xi = Di + D∗i ). This suggests that the form may have an expression as an actual integral over RN , at least for some restricted set of the parameter values κ(v). As in the scalar case harmonic polynomials are involved in the analysis of the Gaussian form. The equation N∑ i=1 (xiDi +Dixi) = N + 2〈x,∇〉+ 2γ(κ; τ) 4 C.F. Dunkl shows that ∆κ|x|2mf = 2m|x|2(m−1)(2m− 2 +N + 2γ(κ; τ) + 2〈x,∇〉)f + |x|2m∆κf, for f ∈ PV . For n = 0, 1, 2, . . . let PV,n = {f ∈ PV : f(rx, t) = rnf(x, t), ∀ r ∈ R}, the polynomials homogeneous of degree n, and let HV,κ,n := {f ∈ PV,n : ∆κf = 0}, the harmonic homogeneous polynomials. As a consequence of the previous formula, for m = 1, 2, 3, . . . and f ∈ HV,κ,n one obtains ∆k κ ( |x|2mf(x, t) ) = 4k(−m)k(1−m−N/2− γ(κ; τ)− n)k|x|2m−2kf(x, t). (4) (The Pochhammer symbol (a)k is defined by (a)0 = 1, (a)k+1 = (a)k(a+k) or (a)k := k∏ i=1 (a+i−1). In particular ( n k ) = (−1)k (−n)k k! and (−n)k = 0 for n = 0, . . . , k − 1.) Thus ∆k κ(|x|2mf(x, t)) = 0 for k > m. With the same proofs as for the scalar case [5, Theorem 5.1.15] one can show (provided γ(κ; τ) + N 2 6= 0,−1,−2, . . .): if f ∈ PV,n then πκ,nf := bn/2c∑ j=0 1 4jj!(−γ(κ; τ)− n+ 2−N/2)j |x|2j∆j κf ∈ HV,κ,n, (5) f = bn/2c∑ j=0 1 4jj!(γ(κ; τ) +N/2 + n− 2j)j |x|2jπκ,n−2j ( ∆j κf ) . From the definition of 〈·, ·〉τ it follows that f ∈ PV,m, g ∈ PV,n and m 6= n implies 〈f, g〉τ = 0. Also if f ∈ HV,κ,m, g ∈ HV,κ,n, m 6= n then 〈 |x|2af, |x|2bg 〉 τ = 0 for any a, b = 0, 1, 2, . . .. This follows from the previous statement when 2a + m 6= 2b + n, otherwise n = m + 2a − 2b and assume m < n (by symmetry of the form), thus 〈 |x|2af, |x|2bg 〉 τ = 〈 f,∆a κ|x|2bg 〉 τ = 0 because a > b. This shows that for generic κ there is an orthogonal decomposition of PV as a sum of |x|2mHV,κ,n over m,n = 0, 1, 2, . . .. If f, g ∈ HV,κ,n then 〈 |x|2mf, |x|2mg 〉 τ = 〈 f,∆m κ |x|2mg 〉 τ = 4mm! ( N 2 + γ(κ; τ) + n ) m 〈f, g〉τ , (6) so to find an orthogonal basis for PV,m it suffices to find an orthogonal basis for each HV,κ,n. The decomposition formula (5) implies the dimensionality result for HV,κ,n (when γ(κ; τ) + N 2 /∈ −N0). It is clear that dimPV,n = dimPR,n dimV = ( N+n−1 N−1 ) dimV , and by induction (for n ≥ 1) dimHV,κ,n = dimPV,n − dimPV,n−2 = ( N + n− 2 N − 2 ) N + 2n− 2 N + n− 2 dimV. (7) We will need a lemma about integrating closed 1-forms. Consider an N -tuple f = (f1, . . . , fN ) ∈ PNV as a vector on which W can act on the right. Say f is a closed 1-form if Difj −Djfi = 0 for all i, j. Lemma 1. Suppose f is a closed 1-form and 1 ≤ j ≤ N then Dj N∑ i=1 xifi(x, t) = (〈x,∇〉+ 1 + γ(κ; τ))fj(x, t)− ∑ v∈R+ κ(v)(f(xσv, tσv)σv)j . B2 Matrix Weight Function 5 Proof. By the commutation relations (2) Dj N∑ i=1 xifi(x, t) = N∑ i=1 xiDjfi + δijfi + 2 ∑ v∈R+ κ(v) vivj |v|2 fi(xσv, tσv)  = N∑ i=1 xiDifj + fj + 2 ∑ v∈R+ κ(v) vj |v|2 N∑ i=1 fi(xσv, tσv)vi. The calculation is finished with the use of (3). � Corollary 1. Suppose f is a closed 1-form, homogeneous of degree n and∑ v∈R+ κ(v)f(xσv, tσv)σv = λκf(x, t) for some constant λκ then Dj N∑ i=1 xifi(x, t) = (n+ 1 + γ(κ; τ)− λκ)fj(x, t) for 1 ≤ j ≤ N . 3 The group B2 The rest of the paper concerns this group. The group W (B2) is the reflection group with positive root system R+ = {(1,−1), (1, 1), (1, 0), (0, 1)}. The corresponding reflections are σ+ 12 := [ 0 1 1 0 ] , σ−12 := [ 0 −1 −1 0 ] , σ1 := [ −1 0 0 1 ] , σ2 := [ 1 0 0 −1 ] . The values of κ on the conjugacy classes {σ+ 12, σ − 12} and {σ1, σ2} will be denoted by k0 and k1, respectively. We consider the unique 2-dimensional representation τ and set V := span{t1, t2}. The generic element of PV is f(x, t) = f1(x)t1 + f2(x)t2. The reflections act on this polynomial as follows σ+ 12f(x, t) = f2(x2, x1)t1 + f1(x2, x1)t2, σ−12f(x, t) = −f2(−x2,−x1)t1 − f1(−x2,−x1)t2, σ1f(x, t) = −f1(−x1, x2)t1 + f2(−x1, x2)t2, σ2f(x, t) = f1(x1,−x2)t1 − f2(x1,−x2)t2. Here is the formula for D1 (D2 is similar) D1f(x, t) = ∂ ∂x1 f(x, t) + k1 f(x1, x2,−t1, t2)− f(−x1, x2,−t1, t2) x1 + k0 f(x1, x2, t2, t1)− f(x2, x1, t2, t1) x1 − x2 + k0 f(x1, x2,−t2,−t1)− f(−x2,−x1,−t2,−t1) x1 + x2 . Since the matrices for the reflections all have trace zero we find that γ(κ; τ) = 0. We investigate the properties of 〈·, ·〉τ by constructing bases for each HV,κ,n; note dimHV,κ,n = 4 for n ≥ 1. 6 C.F. Dunkl The form on V is given by 〈ti, tj〉τ = δij . A convenient orthogonal decomposition is based on types defined by the action of (σ1, σ2): suppose f satisfies σ1f = ε1f , σ2f = ε2f , then f is of type EE, EO, OE, OO if (ε1, ε2) = (1, 1), (1,−1), (−1, 1), (−1,−1) respectively (E is for even, O is for odd). Because the reflections are self-adjoint for the form 〈·, ·〉τ it follows immediately that polynomials of different types are mutually orthogonal. By formula (7) dimHV,κ,n = 4 for n ≥ 1. The construction of the harmonic basis {pn,i : n ≥ 0, 1 ≤ i ≤ 4} is by a recurrence; the process starts at degree 1. Set p1,1 := x1t1 + x2t2, p1,2 := −x2t1 + x1t2, p1,3 := x1t1 − x2t2, p1,4 := −x2t1 − x1t2. Thus p1,1, p1,3 are of type EE and p1,2, p1,4 are of type OO. Also 〈p1,1, p1,3〉τ = 0 = 〈p1,2, p1,4〉τ because σ+ 12p1,1 = p1,1, σ+ 12p1,3 = −p1,3, σ+ 12p1,2 = −p1,2, σ+ 12p1,4 = p1,4. The same decomposi- tions work in HV,κ,n for each odd n. By direct computation we obtain 〈p1,1, p1,1〉τ = 2(1− 2k0 − 2k1), 〈p1,2, p1,2〉τ = 2(1 + 2k0 + 2k1), 〈p1,3, p1,3〉τ = 2(1 + 2k0 − 2k1), 〈p1,4, p1,4〉τ = 2(1− 2k0 + 2k1). These values show that a necessary condition for the existence of a positive-definite inner product on PV is −1 2 < ±k0 ± k1 < 1 2 . Note this is significantly different from the analogous condition for scalar polynomials (that is, τ = 1), namely k0, k1, k0 + k1 > −1 2 . The formulae become more readable by the use of k+ := k0 + k1, k− := k1 − k0. The recurrence starts at degree 0 by setting p0,1 = p0,3 = t1 and p0,2 = −p0,4 = t2 (in the exceptional case n = 0 there are only two linearly independent polynomials). Definition 1. The polynomials pn,i ∈ PV,n for n ≥ 1 and 1 ≤ i ≤ 4 are given by[ p2m+1,1 p2m+1,3 p2m+1,2 p2m+1,4 ] := [ x1 x2 −x2 x1 ] [ p2m,1 p2m,3 p2m,2 p2m,4 ] , m ≥ 0, and [ p2m,1 p2m,2 ] := 1 2m− 1 [ x1 x2 −x2 x1 ] [ (2m− 1 + 2k−)p2m−1,3 (2m− 1− 2k−)p2m−1,4 ] ,[ p2m,3 p2m,4 ] := 1 2m− 1 [ x1 x2 −x2 x1 ] [ (2m− 1 + 2k+)p2m−1,1 (2m− 1− 2k+)p2m−1,2 ] , for m ≥ 1. The construction consists of two disjoint sequences[ p1,1 p1,2 ] 1±2k+→ [ p2,3 p2,4 ] → [ p3,3 p3,4 ] 3±2k−→ [ p4,1 p4,2 ] → [ p5,1 p5,2 ] 5±2k+→ ,[ p1,3 p1,4 ] 1±2k−→ [ p2,1 p2,2 ] → [ p3,1 p3,2 ] 3±2k+→ [ p4,3 p4,4 ] → [ p5,3 p5,4 ] 5±2k−→ . The typical steps are [p2m+1,∗] = X[p2m,∗] and [p2m+2,∗] = XC2m+1[p2m+1,◦] (∗ and ◦ denote a 2-vector with labels 1, 2 or 3, 4 as in the diagram) where X = [ x1 x2 −x2 x1 ] , C2m+1 = [2m+1+2λ 2m+1 0 0 2m+1−2λ 2m+1 ] , where λ = k+ or k− (indicated by the labels on the arrows). Each polynomial is nonzero, regardless of the parameter values. This follows from evaluation at x = (1, i). Let z := t1 + it2. B2 Matrix Weight Function 7 Proposition 1. If n = 4m+ 1 or 4m with m ≥ 1 then [pn,1 (1, i) , pn,2 (1, i) , pn,3 (1, i) , pn,4 (1, i)] = 2n−1 [z,−iz, z,−iz] ; if n = 4m+ 2 or 4m+ 3 with m ≥ 0 then [pn,1 (1, i) , pn,2 (1, i) , pn,3 (1, i) , pn,4 (1, i)] = 2n−1 [z,−iz, z,−iz] . Proof. The formula is clearly valid for n = 1. The typical step in the inductive proof is[ pn+1,1 pn+1,2 ] = [ 1 i −i 1 ] [ c1pn,` −ic2pn,` ] = 2 [ pn,` −ipn,` ] , because c1 + c2 = 2. If n is odd then ` = 3, otherwise ` = 1. The same argument works for pn+1,3, pn+1,4. � The types of p2m+1,1, p2m+1,2, p2m+1,3, p2m+1,4 are EE, OO, EE, OO respectively and the types of p2m,1, p2m,2, p2m,3, p2m,4 are OE, EO, OE, EO respectively. Proposition 2. For m ≥ 0 σ+ 12  p2m+1,1 p2m+1,2 p2m+1,3 p2m+1,4  =  p2m+1,1 −p2m+1,2 −p2m+1,3 p2m+1,4  , σ+ 12  p2m,1 p2m,2 p2m,3 p2m,4  =  p2m,2 p2m,1 −p2m,4 −p2m,3  . Proof. Using induction one needs to show that the validity of the statements for 2m−1 implies the validity for 2m, and the validity for 2m implies the validity for 2m + 1, for each m ≥ 1. Suppose the statements hold for some 2m. The types of p2m+1,i are easy to verify (1 ≤ i ≤ 4). Next σ+ 12p2m+1,1 = σ+ 12(x1p2m,1 + x2p2m,2) = x2σ + 12p2m,1 + x1σ + 12p2m,2 = x2p2m,2 + x1p2m,1 = p2m+1,1, σ+ 12p2m+1,2 = σ+ 12(x1p2m,2 − x2p2m,1) = x2σ + 12p2m,2 − x1σ + 12p2m,1 = −p2m+1,2, and by similar calculations σ+ 12p2m+1,3 = −p2m+1,3 and σ+ 12p2m+1,4 = −p2m+1,4. Now suppose the statements hold for some 2m− 1. As before the types of p2m,i are easy to verify. Consider σ+ 12p2m,1 = σ+ 12 ( 2m− 1 + 2k− 2m− 1 x1p2m−1,3 + 2m− 1− 2k− 2m− 1 x2p2m−1,4 ) = −2m− 1 + 2k− 2m− 1 x2p2m−1,3 + 2m− 1− 2k− 2m− 1 x1p2m−1,4 = p2m,2, σ+ 12p2m,3 = σ+ 12 ( 2m− 1 + 2k+ 2m− 1 x1p2m−1,1 + 2m− 1− 2k+ 2m− 1 x2p2m−1,2 ) = 2m− 1 + 2k+ 2m− 1 x2p2m−1,1 − 2m− 1− 2k+ 2m− 1 x1p2m−1,2 = −p2m,4. Since (σ+ 12)2 = 1 this finishes the inductive proof. � Corollary 2. The polynomials {p2m+1,i : 1 ≤ i ≤ 4} are mutually orthogonal. Proof. These polynomials are eigenfunctions of the (self-adjoint) reflections σ1, σ+ 12 with diffe- rent pairs of eigenvalues. � 8 C.F. Dunkl We will use Corollary 1 to evaluate Dipn,j (with i = 1, 2 and 1 ≤ j ≤ 4). We intend to show that the expressions appearing in Definition 1 can be interpreted as closed 1-forms f which are eigenfunctions of f 7−→ ∑ v∈R+ κ(v)f(xσv, tσv)σv. In the present context this is the operator ρ(f1(x, t), f2(x, t)) := k1((−σ1 + σ2)f1, (σ1 − σ2)f2) + k0((σ+ 12 − σ − 12)f2, (σ + 12 − σ − 12)f1). It is easy to check that (σ1−σ2)f(x, t) = 0 = (σ+ 12−σ − 12)f(x, t) whenever f ∈ PV,2m+1 for m ≥ 0. For the even degree polynomials f(x, t) one finds (σ+ 12 − σ − 12)f = 2σ+ 12f , (σ1 − σ2)f = −2f if f is of type OE and (σ1 − σ2)f = 2f if f is of type EO. Theorem 1. For m ≥ 1 D1p2m,1 = 2m 2m− 1 + 2k− 2m− 1 p2m−1,3, D2p2m,1 = 2m 2m− 1− 2k− 2m− 1 p2m−1,4, D1p2m,2 = 2m 2m− 1− 2k− 2m− 1 p2m−1,4, D2p2m,2 = −2m 2m− 1 + 2k− 2m− 1 p2m−1,3, D1p2m,3 = 2m 2m− 1 + 2k+ 2m− 1 p2m−1,1, D2p2m,3 = 2m 2m− 1− 2k+ 2m− 1 p2m−1,2, D1p2m,4 = 2m 2m− 1− 2k+ 2m− 1 p2m−1,2, D2p2m,4 = −2m 2m− 1 + 2k+ 2m− 1 p2m−1,1, and for m ≥ 1 (in vector notation) [D1,D2]p2m+1,1 = (2m+ 1− 2k+)[p2m,1, p2m,2], [D1,D2]p2m+1,2 = (2m+ 1 + 2k+)[p2m,2,−p2m,1], [D1,D2]p2m+1,3 = (2m+ 1− 2k−)[p2m,3, p2m,4], [D1,D2]p2m+1,4 = (2m+ 1 + 2k−)[p2m,4,−p2m,3]. Proof. Use induction as above. In each case write pn,i in the form x1f1+x2f2 with D2f1 = D1f1 and apply Corollary 1. Suppose the statements are true for some 2m− 1 (direct verification for m = 1). Then p2m,1 = x1 ( 2m− 1 + 2k− 2m− 1 p2m−1,3 ) + x2 ( 2m− 1− 2k− 2m− 1 p2m−1,4 ) and D2 ( 2m− 1 + 2k− 2m− 1 p2m−1,3 ) = (2m− 1 + 2k−) (2m− 1− 2k−) 2m− 1 p2m−2,4 = D1 ( 2m− 1− 2k− 2m− 1 p2m−1,4 ) by the inductive hypothesis. Since ρp2m−1,i = 0 for 1 ≤ i ≤ 4 Corollary 1 shows that D1p2m,1 = 2m (2m−1+2k− 2m−1 p2m−1,3 ) and D2p2m,1 has the stated value. Also p2m,1 = x1 ( 2m− 1− 2k+ 2m− 1 p2m−1,2 ) + x2 ( −2m− 1 + 2k+ 2m− 1 p2m−1,1 ) , and D2 ( 2m− 1− 2k+ 2m− 1 p2m−1,2 ) = −(2m− 1 + 2k−) (2m− 1− 2k+) 2m− 1 p2m−2,1 = D1 ( −2m− 1 + 2k+ 2m− 1 p2m−1,2 ) , B2 Matrix Weight Function 9 and so D2p2m,4 = 2m ( −2m−1+2k+ 2m−1 p2m−1,1 ) . The other statements for p2m,i have similar proofs. Now assume the statements are true for some 2m. The desired relations follow from Corollary 1 and the following ρ(p2m,1, p2m,2) = 2k0(σ+ 12p2m,2, σ + 12p2m,1) + 2k1(p2m,1, p2m,2) = (2k0 + 2k1)(p2m,1, p2m,2), λκ = 2k+; ρ(p2m,2,−p2m,1) = 2k0(−σ+ 12p2m,1, σ + 12p2m,2)− 2k1(p2m,2,−p2m,1) = (−2k0 − 2k1)(p2m,2,−p2m,1), λκ = −2k+; ρ(p2m,3, p2m,4) = 2k0(σ+ 12p2m,4, σ + 12p2m,3) + 2k1(p2m,3, p2m,4) = (−2k0 + 2k1)(p2m,3, p2m,4), λκ = 2k−; ρ(p2m,4,−p2m,3) = 2k0(−σ+ 12p2m,3, σ + 12p2m,4)− 2k1(p2m,4,−p2m,3) = (2k0 − 2k1)(p2m,4,−p2m,3), λκ = −2k−. The closed 1-form condition is shown by using the inductive hypothesis. A typical calculation is p2m+1,2 = x1p2m,2 − x2p2m,1, D2p2m,2 = −2m 2m− 1 + 2k− 2m− 1 p2m−1,3 = −D1p2m,1. This completes the inductive proof. � Corollary 3. Suppose n ≥ 1 and 1 ≤ i ≤ 4 then ∆κpn,i = 0. Proof. By the above formulae D2 1pn,i = −D2 2pn,i in each case. As a typical calculation D2 1p2m,1 = D1 ( 2m 2m− 1 + 2k− 2m− 1 p2m−1,3 ) = 2m 2m− 1 (2m− 1 + 2k−)(2m+ 1− 2k−)p2m−2,3, D2 2p2m,1 = D2 ( 2m 2m− 1− 2k− 2m− 1 p2m−1,4 ) = 2m 2m− 1 (2m+ 1− 2k−)(2m+ 1 + 2k−)(−p2m−2,3). It suffices to check the even degree cases because ∆κf = 0 implies ∆κD1f = 0. � Because p2m,1 and p2m,3 are both type OE one can use an appropriate self-adjoint operator to prove orthogonality. Indeed let U12 := σ+ 12(x2D1 − x1D2), which is self-adjoint for the form 〈·, ·〉τ . Proposition 3. Suppose m ≥ 1 and 1 ≤ i ≤ 4 then U12p2m,i = 2mεip2m,i where ε1 = ε4 = −1 and ε2 = ε3 = 1. Proof. This is a simple verification. For example (x2D1 − x1D2)p2m,4 = 2m 2m− 1 (x2(2m− 1− 2k+)p2m−1,3 + x1(2m− 1 + 2k+)p2m−1,4) = 2mp2m,3, and σ+ 12p2m,3 = −p2m,4. � Corollary 4. For m ≥ 1 the polynomials {p2m,i : 1 ≤ i ≤ 4} are mutually orthogonal. Proof. These polynomials are eigenfunctions of the self-adjoint operators σ1 and U12 with different pairs of eigenvalues. � 10 C.F. Dunkl Proposition 4. For any k0, k1 and n ≥ 1 the polynomials {pn,i : 1 ≤ i ≤ 4} form a basis for HV,κ,n. Proof. The fact that pn,i 6= 0 for all n, i (Proposition 1) and the eigenvector properties from Propositions 2 and 3 imply linear independence for each set {pn,i : 1 ≤ i ≤ 4}. � We introduce the notation ν(f) := 〈f, f〉τ (without implying that the form is positive). To evaluate ν(pn,i) we will use induction and the following simple fact: suppose f = x1f1 + x2f2 and Dif = αfi for i = 1, 2 and some α ∈ Q[k0, k1] then ν(f) = 〈x1f1, f〉τ + 〈x2f2, f〉τ = 〈f1,D1f〉τ + 〈f2,D2f〉τ = α(ν(f1) + ν(f2)). By use of the various formulae we obtain ν(p2m,1) = ν(p2m,2) = 2m (( 2m−1+2k− 2m−1 )2 ν(p2m−1,3) + ( 2m−1−2k− 2m−1 )2 ν(p2m−1,4) ) , ν(p2m,3) = ν(p2m,4) = 2m (( 2m−1+2k+ 2m−1 )2 ν(p2m−1,1) + ( 2m−1−2k+ 2m−1 )2 ν(p2m−1,2) ) , and ν(p2m+1,1) = 2(2m+ 1− 2k+)ν(p2m,1), ν(p2m+1,2) = 2(2m+ 1 + 2k+)ν(p2m,1), ν(p2m+1,3) = 2(2m+ 1− 2k−)ν(p2m,3), ν(p2m+1,4) = 2(2m+ 1 + 2k−)ν(p2m,3). From these relations it follows that ν(p2m+2,1) = 8(m+ 1) (2m+ 1− 2k−)(2m+ 1 + 2k−) 2m+ 1 ν(p2m,3), ν(p2m+2,3) = 8(m+ 1) (2m+ 1− 2k+)(2m+ 1 + 2k+) 2m+ 1 ν(p2m,1). It is now a case-by-case verification for an inductive proof. Define a Pochhammer-type function (referred to as “Π” in the sequel) for notational convenience Π (a, b,m, ε1ε2ε3ε4) := ( 1 4 + a 2 ) m+ε1 ( 1 4 − a 2 ) m+ε2 ( 3 4 + b 2 ) m+ε3 ( 3 4 − b 2 ) m+ε4( 1 4 ) m+ε1 ( 1 4 ) m+ε2 ( 3 4 ) m+ε3 ( 3 4 ) m+ε4 , where m ≥ 0 and ε1ε2ε3ε4 is a list with εi = 0 or 1. To avoid repetitions of the factor 2nn! let ν ′(pn,i) := ν(pn,i)/(2 nn!). Then ν ′(p4m,1) = ν ′(p4m,2) = Π(k+, k−,m, 0000), ν ′(p4m,3) = ν ′(p4m,4) = Π(k−, k+,m, 0000), ν ′(p4m+2,1) = ν ′(p4m+2,2) = Π(k−, k+,m, 1100), ν ′(p4m+2,3) = ν ′(p4m+2,4) = Π(k+, k−,m, 1100), (8) and ν ′(p4m+1,1) = Π(k+, k−,m, 0100), ν ′(p4m+1,2) = Π(k+, k−,m, 1000), ν ′(p4m+1,3) = Π(k−, k+,m, 0100), ν ′(p4m+1,4) = Π(k−, k+,m, 1000), (9) B2 Matrix Weight Function 11 ν ′(p4m+3,1) = Π(k−, k+,m, 1101), ν ′(p4m+3,2) = Π(k−, k+,m, 1110), ν ′(p4m+3,3) = Π(k+, k−,m, 1101), ν ′(p4m+3,4) = Π(k+, k−,m, 1110). (10) These formulae, together with the harmonic decomposition of polynomials, prove that 〈·, ·〉τ is positive-definite for −1 2 < k+, k− < 1 2 (equivalently −1 2 < ±k0 ± k1 < 1 2) (this is a special case of [6, Proposition 4.4]). These norm results are used to analyze the representation of the rational Cherednik algeb- ra Aκ on PV for arbitrary parameter values. For fixed k0, k1 the radical in PV is the subspace radV (k0, k1) := {p : 〈p, q〉τ = 0 ∀ q ∈ PV }. The radical is an Aκ-module and the representation is called unitary if the form 〈·, ·〉τ is positive- definite on PV /radV (k0, k1). By Proposition 4 {|x|2mpn,i} (with m,n ∈ N0, 1 ≤ i ≤ 4, except 1 ≤ i ≤ 2 when n = 0) is a basis for PV for any parameter values (see formula (5)). Proposition 5. For fixed k0, k1 the set {|x|2mpn,i : ν(pn,i) = 0} is a basis for radV (k0, k1). Proof. Suppose p = ∑ m,n,i am,n,i|x|2mpn,i∈radV (k0, k1) then 〈p, |x|2`pk,j〉τ =a`,k,j4 ``!(k+1)`ν(pk,j) by (6), thus a`,k,j 6= 0 implies ν(pk,j) = 0. � For a given value k+ = 1 2 +m+ or k− = 1 2 +m− (or both) with m+,m− ∈ Z the polynomials with ν (pk,j) = 0 can be determined from the above norm formulae. For example let k+ = −1 2 then radV ( k0,−1 2 − k0 ) contains pn,i for (n, i) in the list (1, 2), (n, 1), (n, 2) (n ≥ 2 and n ≡ 0, 1 mod 4), (n, 3), (n, 4) (n ≥ 2 and n ≡ 2, 3 mod 4). In particular each point on the boundary of the square −1 2 < k0 ± k1 < 1 2 corresponds to a unitary representation of Aκ. 4 The reproducing kernel In the τ = 1 setting there is a (“Dunkl”) kernel E(x, y) which satisfies D(x) i E(x, y) = yiE(x, y), E(x, y) = E(y, x) and 〈E(·, y), p〉1 = p(y) for any polynomial p. We show there exists such a function in this B2-setting which is real-analytic in its arguments, provided 1 2 ± k0 ± k1 /∈ Z. This kernel takes values in V ⊗V ; for notational convenience we will use expressions of the form 2∑ i=1 2∑ j=1 fij(x, y)sitj where each fij(x, y) is a polynomial in x1, x2, y1, y2 (technically, we should write si⊗tj for the basis elements in V ⊗V ). For a polynomial f(x) = f1(x)t1 +f2(x)t2 let f(y)∗ denote f1(y)s1 + f2(y)s2. The kernel E is defined as a sum of terms like 1 ν(pn,i) pn,i(y)∗pn,i(x); by the orthogonality relations (for m,n = 0, 1, 2, . . . and 1 ≤ i, j ≤ 4) 1 ν(pn,i) pn,i(y)∗〈pn,i, pm,j〉τ = δmnδijpn,i(y)∗. By formulae (4) and (6)〈 |x|2apn,i(x), |x|2bpm,j(x) 〉 τ = δnmδabδij4 aa!(n+ 1)aν(pn,i). We will find upper bounds on {pn,i} and lower bounds on {ν(pn,i)} in order to establish con- vergence properties of E(x, y). For u ∈ R set d(u) := min m∈Z ∣∣∣∣u+ 1 2 +m ∣∣∣∣ . The condition that ν(pn,i) 6= 0 for all n ≥ 1, 1 ≤ i ≤ 4 is equivalent to d(k+)d(k−) > 0, since each factor in the numerator of Π is of the form 1 4 + 1 2(±k0 ± k1) +m or 3 4 + 1 2(±k0 ± k1) +m for m = 0, 1, 2, . . .. 12 C.F. Dunkl Definition 2. Let P0(x, y) := s1t1 + s2t2 and for n ≥ 1 let Pn(x, y) := 4∑ i=1 1 ν(pn,i) pn,i(y)∗pn,i(x). For n ≥ 0 let En(x, y) := ∑ 0≤m≤n/2 ( x2 1 + x2 2 )m ( y2 1 + y2 2 )m 4mm! (n− 2m+ 1)m Pn−2m(x, y). Proposition 6. If k0, k1 are generic or d(k+)d(k−) > 0, and n ≥ 1 then 〈En(·, y), f〉τ = f(y)∗ for each f ∈ PV,n, and D(x) i En(x, y) = yiEn−1(x, y) for i = 1, 2. Proof. By hypothesis on k0, k1 the polynomial En(x, y) exists, and is a rational function of k0, k1. The reproducing property is a consequence of the orthogonal decomposition PV,n =∑ 0≤m≤n/2 ⊕|x|2mHV,κ,n−2m. For the second part, suppose f ∈ PV,n−1, then 〈 D(x) i En(x, y), f(x) 〉 τ = 〈En(x, y), xif(x)〉τ = yif(y)∗ = yi〈En−1(x, y), f(x)〉τ . If −1 2 < ±k0 ± k1 < 1 2 the bilinear form is positive-definite which implies D(x) i En(x, y) = yiEn−1(x, y). This is an algebraic (rational) relation which holds on an open set of its arguments and is thus valid for all k0, k1 except for the poles of En. � Write En(x, y) = 2∑ i,j=1 En,ij(x, y)sitj then the equivariance relation becomes En,ij(xw, yw) = (w−1En,··(x, y)w)ij for each w ∈ W (B2). Next we show that ∞∑ n=0 En(x, y) converges to a real- entire function in x, y. For p(x, t) = p1(x)t1 + p2(x)t2 ∈ PV let β(p)(x) = p1(x)2 + p2(x)2. Lemma 2. Suppose m ≥ 0 and i = 1 or 3 then β(p2m+1,i)(x) + β(p2m+1,i+1)(x) = (x2 1 + x2 2)(β(p2m,i)(x) + β(p2m,i+1)(x)). Proof. By Definition 1 p2m+1,i = x1p2m,i + x2p2m,i+1 and p2m+1,i+1 = −x2p2m,i + x1p2m,i+1. The result follows from direct calculation using these formulae. � Lemma 3. Suppose m ≥ 1 then β(p2m,1)(x) + β(p2m,2)(x) = ( x2 1 + x2 2 )((2m− 1 + 2k− 2m− 1 )2 β(p2m−1,3)(x) + ( 2m− 1− 2k− 2m− 1 )2 β(p2m−1,4)(x) ) , β(p2m,3)(x) + β(p2m,4)(x) = ( x2 1 + x2 2 )((2m− 1 + 2k+ 2m− 1 )2 β(p2m−1,1)(x) + ( 2m− 1− 2k+ 2m− 1 )2 β(p2m−1,2)(x) ) . Proof. This follows similarly as the previous argument. � Proposition 7. Suppose m ≥ 0 and i = 1 or 3 then β(p2m,i)(x) + β(p2m,i+1)(x) ≤ 2 ( (1/2 + |k0|+ |k1|)m (1/2)m )2 ( x2 1 + x2 2 )2m , β(p2m+1,i)(x) + β(p2m+1,i+1)(x) ≤ 2 ( (1/2 + |k0|+ |k1|)m (1/2)m )2 ( x2 1 + x2 2 )2m+1 . B2 Matrix Weight Function 13 Proof. Use induction, the lemmas, and the inequality ( 2m−1±2k0±2k1 2m−1 )2 ≤ (m−1/2+|k0|+|k1| m−1/2 )2 . The beginning step is β(p1,i)(x) + β(p1,i+1)(x) = 2 ( x2 1 + x2 2 ) . � By Stirling’s formula (1/2 + |k0|+ |k1|)m (1/2)m = Γ(1/2) Γ(1/2 + |k0|+ |k1|) Γ(1/2 + |k0|+ |k1|+m) Γ(1/2 +m) ∼ Γ(1/2) Γ(1/2 + |k0|+ |k1|) m|k0|+|k1| as m → ∞. For the purpose of analyzing lower bounds on |ν ′(pn,i)| we consider an infinite product. Lemma 4. Suppose u > 0 then the infinite product ω(u; z) := ∞∏ n=0 ( 1− z2 (u+ n)2 ) converges to an entire function of z ∈ C, satisfies ω(u; z) = Γ(u)2 Γ(u+ z)Γ(u− z) , and has simple zeros at ±(u+ n), n = 0, 1, 2, . . .. Proof. The product converges to an entire function by the comparison test: ∞∑ n=1 |z|2 (u+n)2 < ∞ (this means that the partial products converge to a nonzero limit, unless one of the factors vanishes). Suppose that |z| < u then Re(u± z) ≥ u− |z| > 0 and ∞∏ n=0 ( 1− z2 (u+ n)2 ) = lim m→∞ (u+ z)m(u− z)m (u)2 m = Γ(u)2 Γ(u+ z)Γ(u− z) lim m→∞ Γ(u+ z +m)Γ(u− z +m) Γ(u+m)2 = Γ(u)2 Γ(u+ z)Γ(u− z) , by Stirling’s formula. The entire function ω(u; ·) agrees with the latter expression (in Γ) on an open set in C, hence for all z. � Corollary 5. 1. If n ≡ 0, 1 mod 4 and i = 1, 2 or n ≡ 2, 3 mod 4 and i = 3, 4 then lim n→∞ ν ′(pn,i) = ω ( 1 4 ; k+ 2 ) ω ( 3 4 ; k− 2 ) . 2. If n ≡ 0, 1 mod 4 and i = 3, 4 or n ≡ 2, 3 mod 4 and i = 1, 2 then lim n→∞ ν ′(pn,i) = ω ( 1 4 ; k− 2 ) ω ( 3 4 ; k+ 2 ) . Proof. The statements follow from formulae (8), (9) and (10). � For a ∈ R with a+ 1 2 /∈ Z let ((a)) denote the nearest integer to a. 14 C.F. Dunkl Lemma 5. For each (n1, n2) ∈ Z2 there is a constant C(n1, n2) such that ((k+)) = n1 and ((k−)) = n2 implies |ν ′(pn,i)| ≥ C(n1, n2)d(k+)d(k−). Proof. It suffices to consider Case 1 in the corollary (interchange k+ and k− to get Case 2). The limit function has zeros at k+ = ±4m+1 2 and k− = ±4m+3 2 for m = 0, 1, 2, . . .. For each n1 6= 0 and n2 6= 0 there are unique nearest zeros k+ = z1 and k− = z2 respectively; for example if n1 is odd and n1 ≥ 1 then z1 = n1 − 1 2 ; and if n2 is even and n2 ≤ −2 then z2 = n2 + 1 2 . Consider the entire function f (k+, k−) = 1 (z1 − k+)(z2 − k−) ω ( 1 4 ; k+ 2 ) ω ( 3 4 ; k− 2 ) . If n1 = 0 then replace the factor (z1 − k+) by ( 1 4 − k 2 + ) , and if n2 = 0 replace (z2 − k−) by 1. In each of these cases the quotient is an entire function with no zeros in |n1 − k+| ≤ 1 2 and |n2−k−| ≤ 1 2 . Thus there is a lower bound C1 in absolute value for all the partial products of f , valid for all k+, k− in this region. The expressions for ν ′(pn,i) (see formulae (8), (9) and (10)) involve terms in {εi} but these do not affect the convergence properties (and note 0 ≤ 4∑ i=1 εi ≤ 3 in each case). Since |z1 − k+| ≥ d(k+) and |z2 − k−| ≥ d(k−) we find that the partial products of (z1 − k+)(z2 − k−)f(k+, k−), that is, the values of ν(pn,i)/(2 nn!), are bounded below by d(k+)d(k−)C1 in absolute value. In case n1 = 0 the factor ( 1 4−k 2 + ) = d(k+)(1−d(k+)) ≥ 1 2d(k+) (and in the more trivial case n2 = 0 one has 1 > 1 2 ≥ d(k−)). � Theorem 2. For a fixed k0, k1 ∈ R satisfying d(k+)d(k−) > 0 the series E(x, y) := ∞∑ n=0 En(x, y) converges absolutely and uniformly on { (x, y) ∈ R4 : |x|2 + |y|2 ≤ R2 } for any R > 0. Proof. We have shown there is a constant C ′ > 0 such that |Pn(x, y)| ≤ C ′ |x| n|y|n 2nn! (n 2 )|k0|+|k1| , and thus the series E(x, y) = ∞∑ n=0 ∑ 0≤m≤n/2 |x|2m|y|2m 4mm!(n− 2m+ 1)m Pn−2m(x, y) = ∞∑ l=0 Pl(x, y) ∞∑ m=0 |x|2m|y|2m 4mm!(l + 1)m converges uniformly for |x|2 + |y|2 < R2. � Corollary 6. Suppose f ∈ PV then 〈E(·, y), f〉τ = f(y)∗; also D(x) i E(x, y) = yiE(x, y) for i = 1, 2. As in the scalar (τ = 1) theory the function E(x, y) can be used to define a generalized Fourier transform. 5 The Gaussian-type weight function In this section we use vector notation for PV : f(x) = (f1(x), f2(x)) for the previous f1(x)t1 + f2(x)t2. The action of W is written as (wf)(x) = f(xw)w−1. We propose to construct a 2× 2 positive-definite matrix function K(x) on R2 such that 〈f, g〉G = ∫ R2 f(x)K(x)g(x)T e−|x| 2/2dx, ∀ f, g ∈ PV , B2 Matrix Weight Function 15 and with the restriction |k0±k1| < 1 2 ; the need for this was demonstrated in the previous section. The two necessary algebraic conditions are (for all f, g ∈ PV ) 〈f, g〉G = 〈wf,wg〉G, w ∈W, (11) 〈(xi −Di)f, g〉G = 〈f,Dig〉G, i = 1, 2. (12) We will assume K is differentiable on Ω := {x ∈ R2 : x1x2(x2 1 − x2 2) 6= 0}. The integral formula is defined if K is integrable, but for the purpose of dealing with the singularities implicit in Di we introduce the region Ωε := {x : min(|x1|, |x2|, |x1 − x2|, |x1 + x2|) ≥ ε} for ε > 0. The fundamental region of W corresponding to R+ is C0 = {x : 0 < x2 < x1}. Condition (11) implies, for each w ∈W , f, g ∈ PV that∫ R2 f(xw)w−1K(x)wg(xw)T e−|x| 2/2dx = ∫ R2 f(x)w−1K ( xw−1 ) wg(x)T e−|x| 2/2dx = ∫ R2 f(x)K(x)g(x)T e−|x| 2/2dx. For the second step change the variable from x to xw−1 (note wT = w−1). Thus we impose the condition K(xw) = w−1K(x)w. This implies that it suffices to determine K on the fundamental region C0 and then extend to all of Ω by using this formula. Set ∂i := ∂ ∂xi . Recall for the scalar situation that the analogous weight function is hκ(x)2 where hκ(x) = ∏ v∈R+ |〈x, v〉|κ(v) and satisfies ∂ihκ(x) = ∑ v∈R+ κ(v) vi 〈x, v〉 hκ(x), i = 1, 2. Start by solving the equation ∂iL(x) = ∑ v∈R+ κ(v) vi 〈x, v〉 L(x)σv, i = 1, 2, (13) for a 2× 2 matrix function L on C0, extended by L(xw) = L(x)w (from the facts that w−1σvw = σvw and κ(vw) = κ(v) it follows that equation (13) is satisfied on all of Ω) and set K(x) := L(x)TL(x), with the result that K is positive-semidefinite and (note σTv = σv) ∂iK(x) = ∑ v∈R+ κ(v) vi 〈x, v〉 (σvK(x) +K(x)σv). (14) Then for f, g ∈ PV (and i = 1, 2) we find − ∂i ( f(x)K(x)g(x)T e−|x| 2/2 ) e|x| 2/2 = xif(x)K(x)g(x)T − (∂if(x))K(x)g(x)T − f(x)TK(x)(∂ig(x))T − ∑ v∈R+ κ(v) vi 〈x, v〉 { f(x)σvK(x)g(x)T + f(x)K(x)σvg(x)T } . 16 C.F. Dunkl Figure 1. Region of integration. Consider the second necessary condition (12) 〈(xi − Di)f, g〉G − 〈f,Dig〉G = 0, that is, the following integral must vanish∫ Ωε { ((xi −Di)f(x))K(x)g(x)T − f(x)K(x)(Dig(x ))T } e−|x| 2/2dx = − ∫ Ωε ∂i ( f(x)K(x)g(x)T e−|x| 2/2 ) dx + ∑ v∈R+ κ(v) ∫ Ωε vi 〈x, v〉 { f(xσv)σvK(x)g(x)T + f(x)K(x)σvg(xσv) T } dx. In the second part, for each v ∈ R+ change the variable to xσv, then the numerator is invariant, because σvK(xσv) = K(x)σv, and 〈xσv, v〉 = −〈x, v〉, and thus each term vanishes (note Ωε is W -invariant). So establishing the validity of the inner product formula reduces to showing lim ε→0+ ∫ Ωε ∂i ( f(x)K(x)g(x)T e− |x| 2/2 ) dx = 0 for i = 1, 2. By the polar identity it suffices to prove this for g = f . Set Q(x) = f(x)K(x)f(x)T e−|x| 2/2. By symmetry (σ+ 12D1σ + 12 = D2) it suffices to prove the formula for i = 2. Consider the part of Ωε in {x1 > 0} as the union of {x : ε < |x2| < x1 − ε} and {x : ε < x1 < |x2| − ε} (with vertices (2ε,±ε), (ε,±2ε) respectively). In the iterated integral evaluate the inner integral over x2 on the segments {(x1, x2) : ε ≤ |x2| ≤ x1 − ε} and {(x1 − ε, x2) : x1 ≤ |x2|} for a fixed x1 > 2ε; obtaining (due to the exponential decay) −(Q(x1 − ε, x1)−Q(x1, x1 − ε))− (Q(x1, ε)−Q(x1,−ε)) − (Q(x1,−x1 + ε)−Q(x1 − ε,−x1)). See Fig. 1 for a diagram of Ωε and a typical inner integral. By differentiability hij(x1, ε)− hij(x1,−ε) = C1(x1)ε+O ( ε2 ) , hij(x1 − ε, x1)− hij(x1, x1 − ε) = C2(x1)ε+O ( ε2 ) , where hij(x) = fi(x)fj(x)e−|x| 2/2 for 1 ≤ i, j ≤ 2; the factors C1, C2 depend on x1 but there is a global bound |Ci(x1)| < C0 depending only on f because of the exponential decay. Thus B2 Matrix Weight Function 17 the behavior of K(x1, ε) and K(x1, x1 − ε) is crucial in analyzing the limit as ε → 0+, for x2 = −x1, 0, x1. It suffices to consider the fundamental region 0 < x2 < x1. At the edge {x2 = 0} corresponding to σ2 K(x1,−ε) = K((x1, ε)σ2) = σ2K(x1, ε)σ2, and (Q(x1, ε)−Q(x1,−ε)) = (h11(x1, ε)− h11(x1,−ε))K(x1, ε)11 + (h22(x1, ε)− h22(x1,−ε))K(x1, ε)22 − 2(h12(x1, ε) + h12(x1,−ε))K(x1, ε)12. At the edge {x1 − x2 = 0} corresponding to σ+ 12 K(x1 − ε, x1) = K((x1, x1 − ε)σ+ 12) = σ+ 12K(x1, x1 − ε)σ+ 12. For conciseness set x(0) = (x1, x1 − ε) and x(1) = (x1 − ε, x1). Then Q ( x(0) ) −Q ( x(1) ) = 2 ( h12 ( x(0) ) − h12 ( x(1) )) K ( x(0) ) 12 + 1 2 ( h11 ( x(0) ) − h11 ( x(1) ) + h22 ( x(0) ) − h22 ( x(1) ))( K ( x(0) ) 11 +K ( x(0) ) 22 ) + 1 2 ( h11 ( x(0) ) + h11 ( x(1) ) − h22 ( x(0) ) − h22 ( x(1) ))( K ( x(0) ) 11 −K ( x(0) ) 22 ) . To get zero limits as ε → 0+ some parts rely on the uniform continuity of hij and bounds on certain entries of K, and the other parts require lim ε→0+ K(x1, ε)12 = 0, lim ε→0+ (K(x1, x1 − ε)11 −K(x1, x1 − ε)22) = 0. This imposes various conditions on K near the edges, as described above. We turn to the solution of the system (13) and rewrite ∂1L(x) = L(x) { k1 x1 [ −1 0 0 1 ] + 2k0x2 x2 1 − x2 2 [ 0 1 1 0 ]} , ∂2L(x) = L(x) { k1 x2 [ 1 0 0 −1 ] − 2k0x1 x2 1 − x2 2 [ 0 1 1 0 ]} . The reflections σ−12 and σ+ 12 were combined into one term: (1,−1) x1−x2 − (1,1) x1+x2 = (2x2,−2x1) x21−x22 . Since (x1∂1 + x2∂2)L(x) = 0 we see that L is positively homogeneous of degree 0. Because of the homogeneity the system can be transformed to an ordinary differential system by setting u = x2 x1 . Then the system is transformed to d du L(u) = L(u) { k1 u [ 1 0 0 −1 ] − 2k0 1− u2 [ 0 1 1 0 ]} . It follows from this equation that d du detL(u) = 0. Since the goal is to find a positive-definite matrix K we look for a fundamental solution for L, that is detL(x) 6= 0. If L(x) is a solution then so is ML(x) for any nonsingular constant matrix. Thus K(x) = L(x)TMTML(x) satisfies the differential equation (14) and some other condition must be imposed to obtain the desired (unique) solution for the weight function. The process starts by solving for a row of L, say (f1(u), f2(u)), that is d du f1(u) = k1 u f1(u)− 2k0 1− u2 f2(u), d du f2(u) = − 2k0 1− u2 f1(u)− k1 u f2(u). 18 C.F. Dunkl A form of solutions can be obtained by computer algebra, then a desirable solution can be verified. Set f1(u) = |u|k1 ( 1− u2 )−k0g1 ( u2 ) , f2(u) = |u|k1 ( 1− u2 )−k0ug2 ( u2 ) , then the equations become (with s := u2) (s− 1) d ds g1(s) = k0g1(s) + k0g2(s), s(s− 1) d ds g2(s) = k0g1(s) + { k0s− ( 1 2 + k1 ) (s− 1) } g2(s), and the solution regular at s = 0 is g1(s) = F ( −k0, k1 + 1 2 − k0; k1 + 1 2 ; s ) , g2(s) = − k0 k1 + 1 2 F ( 1− k0, k1 + 1 2 − k0; k1 + 3 2 ; s ) . (We use F to denote the hypergeometric function 2F1; it is the only type appearing here.) The verification uses two hypergeometric identities (arbitrary parameters a, b, c with −c /∈ N0) (s− 1) d ds F (a, b; c; s) = −aF (a, b; c; s) + a(c− b) c F (a+ 1, b; c+ 1; s), s(s− 1) d ds F (a+ 1, b; c+ 1; s) = (c− bs)F (a+ 1, b; c+ 1; s)− cF (a, b; c; s). To get the other solutions we use the symmetry of the system, replace k1 by −k1 and inter- change f1 and f2. We have a fundamental solution L(u) given by L(u)11 = |u|k1 ( 1− u2 )−k0F (−k0, 1 2 − k0 + k1; k1 + 1 2 ;u2 ) , L(u)12 = − k0 k1 + 1 2 |u|k1 ( 1− u2 )−k0uF (1− k0, 1 2 − k0 + k1; k1 + 3 2 ;u2 ) , L(u)21 = − k0 1 2 − k1 |u|−k1 ( 1− u2 )−k0uF (1− k0, 1 2 − k0 − k1; 3 2 − k1;u2 ) , L(u)22 = |u|−k1 ( 1− u2 )−k0F (−k0, 1 2 − k0 − k1; 1 2 − k1;u2 ) . Observe that lim u→0+ detL(u) = 1, thus detL(u) = 1 for all u. We can write L in the form L(x1, x2) = [ |x2|k1x−k11 0 0 |x2|−k1xk11 ] [ c11(x) x2 x1 c12(x) x2 x1 c21(x) c22(x) ] , where each cij is even in x2 and is real-analytic in 0 < |x2| < x1. In fact L(x) is thus defined on C0 ∪ C0σ2. It follows that K = (ML)TML is integrable near {x2 = 0} if |k1| < 1 2 , and lim ε→0+ K(x1, ε)12 = 0 exactly when MTM is diagonal. The standard identity [8, 15.8.1] F (a, b; c;u) = (1− u)c−a−bF (c− a, c− b; c;u) (15) shows that there is a hidden symmetry for k0( 1− u2 )−k0F (−k0, 1 2 − k0 + k1; k1 + 1 2 ;u2 ) = ( 1− u2 )k0F (k0, 1 2 + k0 + k1; k1 + 1 2 ;u2 ) , (16) B2 Matrix Weight Function 19 and similar equations for the other entries of L. Consider Q(x1,−ε)−Q(x1, ε) = K(x1, ε)11(h11(x1,−ε)− h11(x1, ε)) − 2K(x1, ε)12(h12(x1,−ε) + h12(x1, ε)) +K(x1, ε)22(h22(x1,−ε)− h22(x1, ε)). With diagonal MTM we find (note x1 > 2ε in the region, so ε x1 < 1 2) K(x1, ε)11 = O ( x−2k1 1 ε2k1 ) +O ( x2k1−2 1 ε2−2k1 ) , K(x1, ε)12 = O ( x−1−2k1 1 ε1+2k1 ) +O ( x−1+2k1 1 ε1−2k1 ) , K(x1, ε)22 = O ( x2k1 1 ε−2k1 ) +O ( x−2−2k1 1 ε2+2k1 ) . By the exponential decay we can assume that the double integral is over the box max(|x1|, |x2|) ≤ R for some R <∞. Note ∫ R 2ε ( ε x1 )α dx1 = 1 1+α ( R1−αεα − 21−αε ) . These bounds (recall −1 2 < k0, k1 < 1 2) show that lim ε→0+ ∫ R 2ε (Q(x1,−ε)−Q(x1, ε))dx1 = 0. Next we analyze the behavior of this solution in a neighborhood of t = 1, that is the ray {(x1, x1) : x1 > 0}. The following identity [8, 15.10.21] is used F (a+ d, a+ c; c+ d;u) = Γ(c+ d)Γ(−2a) Γ(c− a)Γ(d− a) F (a+ d, c+ a; 1 + 2a; 1− u) + (1− u)−2a Γ(c+ d)Γ(2a) Γ(c+ a)Γ(d+ a) F (d− a, c− a; 1− 2a; 1− u). Let η(k0, k1) := Γ ( 1 2 + k1 ) Γ(2k0) Γ ( 1 2 + k0 + k1 ) Γ(k0) = 22k0−1 √ π Γ ( 1 2 + k1 ) Γ ( 1 2 + k0 ) Γ ( 1 2 + k0 + k1 ) ; the latter equation follows from Γ(2a)/Γ(a) = 22a−1Γ(a+ 1 2)/ √ π (the duplication formula). We will need the identity η(k0, k1)η(−k0,−k1) + η(k0,−k1)η(−k0, k1) = 1 2 , proved by use of Γ ( 1 2 + a ) Γ ( 1 2 − a ) = π cosπa . Thus L(u)11 = |u|k1 { η(k0, k1) ( 1− u2 )−k0F (−k0, 1 2 − k0 + k1; 1− 2k0; 1− u2 ) + η(−k0, k1) ( 1− u2 )k0F (k0, 1 2 + k0 + k1; 1 + 2k0; 1− u2 )} , by use of identity (16), and also L(u)12 = u|u|k1 { −η(k0, k1) ( 1− u2 )−k0 2F1 ( 1− k0, 1 2 − k0 + k1; 1− 2k0; 1− u2 ) + η(−k0, k1) ( 1− u2 )k0 2F1 ( 1 + k0, 1 2 + k0 + k1; 1 + 2k0; 1− u2 )} . Transform again using (15) to obtain F ( 1− k0, 1 2 − k0 + k1; 1− 2k0; 1− u2 ) = |u|−2k1−1F ( −k0, 1 2 − k0 − k1; 1− 2k0; 1− u2 ) , 20 C.F. Dunkl and so on. All the hypergeometric functions we use are of one form and it is convenient to introduce H(a, b; s) := F ( a, a+ b+ 1 2 ; 2a+ 1; 1− s ) . By using similar transformations as for L11 and L12 we find L(u) = [ η(−k0, k1) η(k0, k1) η(−k0,−k1) −η(k0,−k1) ][( 1− u2 )k0 0 0 ( 1− u2 )−k0 ] × [ H(k0, k1;u2) H(k0,−k1;u2) H(−k0, k1;u2) −H(−k0,−k1;u2) ] [ |u|k1 0 0 |u|−k1 ] . Let Γ denote the first matrix in the above formula. By direct calculation we find that a necessary condition for lim ε→0+ (K(x1, x1 − ε)11 −K(x1, x1 − ε)22) = 0, where K(u) = (M ′Γ−1L(u))T (M ′Γ−1L(u)) is that M ′TM ′ is diagonal. The proof that∫ R 2ε (Q(x1, x1 − ε)−Q(x1 − ε, x1))dx1 → 0 is similar to the previous case; a typical term in K(u)11 −K(u)22 is( 1− u2 )2k0(H(k0, k1;u2 )2|u|2k1 −H(k0,−k1;u2 )2|u|−2k1 ) which is O ( (1− u)1+2k0 ) , tending to zero as u→ 1− for |k0| < 1 2). It remains to combine the two conditions: K(u) = (ML(u))T (ML(u)) and the previous one to find a unique solution for M : MTM = (Γ−1)TDΓ−1 and both MTM and D are positive- definite diagonal. Indeed D = c [ η(−k0,−k1)η(−k0, k1) 0 0 η(k0, k1)η(k0,−k1) ] , MTM = 2c [ η(−k0,−k1)η(k0,−k1) 0 0 η(k0, k1)η(−k0, k1) ] , for some c > 0. Thus the desired matrix weight (in the region C0, 0 < x2 < x1, u = x2/x1) is given by K(u)11 = d1L(u)2 11 + d2L(u)2 21, K(u)12 = K(u)21 = d1L(u)11L(u)12 + d2L(u)21L(u)22, K(u)22 = d1L(u)2 12 + d2L(u)2 22, where d1 = c Γ ( 1 2 − k1 )2 cosπk0Γ ( 1 2 + k0 − k1 ) Γ ( 1 2 − k0 − k1 ) , d2 = c Γ ( 1 2 + k1 )2 cosπk0Γ ( 1 2 + k0 + k1 ) Γ ( 1 2 − k0 + k1 ) . B2 Matrix Weight Function 21 Figure 2. K, k0 = 0.3, k1 = 0.1. Also detK = d1d2 = c2 ( 1− tan2 πk0 tan2 πk1 ) . The expressions for Kij can be rewritten some- what by using the transformations (16). Observe that the conditions −1 2 < ±k0 ± k1 < 1 2 are needed for d1, d2 > 0. The normalization constant is to be determined from the con- dition ∫ R2 K(x)11e −|x|2/2dx = 1. By the homogeneity of K this is equivalent to evaluat- ing ∫ π −πK(cos θ, sin θ)11dθ (or ∫ π/4 0 (K11 + K22)(cos θ, sin θ)dθ) (the integral looks difficult be- cause K11 involves squares of hypergeometric functions with argument tan2 θ). Numerical ex- periments suggest the following conjecture for the normalizing constant c = cosπk0 cosπk1 2π . We illustrate K for k0 = 0.3, k1 = 0.1 with plots of K(cos θ, sin θ) for 0 < θ < π 4 . Fig. 2 shows the values of K11, K12, K22. For behavior near x1 = x2 introduce σ = 1√ 2 [ 1 1 1 −1 ] , σσ+ 12σ = [ 1 0 0 −1 ] . Fig. 3 displays σKσ (thus (σKσ)12 = 1 2(K11 −K22); the (2, 2)-entry is rescaled by 0.1). The degenerate cases k0 = 0 and k1 = 0 (when the group aspect reduces to Z2×Z2) provide a small check on the calculations: for k0 = 0 the weight is K(x) = c [ |x2/x1|k1 0 0 |x1/x2|k1 ] , and for k1 = 0 and by use of the quadratic transformation F ( a, a+ 1 2 ; 2a+1; 1−u2 ) = ( 1+u 2 )−2a (for u near 1)(see [8, 15.4.17]) we obtain K(x) = cσ [ |x1 − x2|k0 |x1 + x2|−k0 0 0 |x1 − x2|−k0 |x1 + x2|k0 ] σ. An orthogonal basis for PV for the Gaussian inner product can be given in terms of Laguerre polynomials and the harmonic polynomials from the previous section. Recall from equations (4) 22 C.F. Dunkl Figure 3. σKσ, k0 = 0.3, k1 = 0.1. and (6) (specializing N = 2 and γ(κ; τ) = 0) that ∆k κ ( |x|2mf(x, t) ) = 4k(−m)k(−m− n)k|x|2m−2kf(x, t),〈 |x|2mf, |x|2mg 〉 τ = 4mm!(n+ 1)m〈f, g〉τ , for f, g ∈ HV,κ,n. These relations are transferred to the Gaussian inner product 〈f, g〉τ = 〈 e−∆κ/2f, e−∆κ/2g 〉 G by computing e−∆κ/2|x|2mf(x, t) = m∑ j=0 ( −1 2 )j 4j j! (−m)j(−m− n)j |x|2m−2jf(x, t) = (−1)m2mm!L(n) m ( |x|2 2 ) f(x, t), for f ∈ HV,κ,n and m ≥ 0; where L (n) m denotes the Laguerre polynomial of degree m and index n (orthogonal for sne−sds on R+). Denote 〈f, f〉G by νG(f); recall ν(f) = 〈f, f〉τ for f ∈ PV . Proposition 8. The polynomials L (n) m ( |x|2 2 ) pn,i(x, t) for m,n ≥ 0 and 1 ≤ i ≤ 4 (except i = 1, 2 when n = 0) are mutually orthogonal in 〈·, ·〉G and νG(L (n) m (|x|2/2)pn,i(x, t)) = (n+1)m m! ν(pn,i). The factor with ν(pn,i) results from a simple calculation. Note νG(f) = ν(f) for any har- monic f . Because here γ(κ; τ) = 0 the harmonic decomposition formula (5) is valid for any pa- rameter values. This is a notable difference from the scalar case τ = 1 where γ(κ; 1) = 2k0 +2k1 and 2k0 + 2k1 6= −1,−2, . . . is required for validity. Using the same arguments as in the scalar case (see [5, Section 5.7]) we define the Fourier transform (for suitably integrable functions f). To adapt E(x, y) to vector notation write E(x, y) = 2∑ i,j=1 Eij(x, y)sitj and set Ff(y)l := ∫ R2 2∑ i,j=1 Eli(x,−iy)K(x)ijfj(x)dx1dx2, l = 1, 2, Ff(y) := Ff(y)1s1 + Ff(y)2s2. B2 Matrix Weight Function 23 For m,n ≥ 0 and 1 ≤ i ≤ 4 let φm,n,i(x) = L (n) m ( |x|2 ) pn,i(x)e−|x| 2/2. Proposition 9. Suppose m,n ≥ 0 and 1 ≤ i ≤ 4 then Fφm,n,i(y) = (−i)m+2nφm,n,i(y)∗. If f(x) = e−|x| 2/2g(x) for g ∈ PV then F(Djf)(y) = iyjFf(y), for j = 1, 2. This establishes a Plancherel theorem for F by use of the density (from Hamburger’s theorem) of span{φm,n,i} in L2(K(x)dx,R2). 6 Closing remarks The well-developed theory of the hypergeometric function allowed us to find the weight function which satisfies both a differential equation and geometric conditions. The analogous problem can be stated for any real reflection group and there are some known results about the differential system (13) (see [2, 3]); it appears some new insights are needed to cope with the geometric conditions. The fact that the Gaussian inner product 〈·, ·〉G is well-defined supports speculation that Gaussian-type weight functions exist in general settings. However it has not been shown that K can be produced as a product LTL, and the effect of the geometry of the mirrors (walls) on the solutions of the differential system is subtle, as seen in the B2-case. Acknowledgements This is the expanded version of an invited lecture presented at the Conference on Harmonic Analysis, Convolution Algebras, and Special Functions, TU München, September 10, 2012. References [1] Carter R.W., Finite groups of Lie type. Conjugacy classes and complex characters, Wiley Classics Library, John Wiley & Sons Ltd., Chichester, 1993. [2] Dunkl C.F., Differential-difference operators and monodromy representations of Hecke algebras, Pacific J. Math. 159 (1993), 271–298. [3] Dunkl C.F., Monodromy of hypergeometric functions for dihedral groups, Integral Transform. Spec. Funct. 1 (1993), 75–86. [4] Dunkl C.F., Opdam E.M., Dunkl operators for complex reflection groups, Proc. London Math. Soc. 86 (2003), 70–108, math.RT/0108185. [5] Dunkl C.F., Xu Y., Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, Vol. 81, Cambridge University Press, Cambridge, 2001. [6] Etingof P., Stoica E., Unitary representations of rational Cherednik algebras, Represent. Theory 13 (2009), 349–370, arXiv:0901.4595. [7] Griffeth S., Orthogonal functions generalizing Jack polynomials, Trans. Amer. Math. Soc. 362 (2010), 6131–6157, arXiv:0707.0251. [8] Olver F.W.J., Lozier D.W., Boisvert R.F., Clark C.W. (Editors), NIST handbook of mathematical functions, U.S. Department of Commerce National Institute of Standards and Technology, Washington, DC, 2010. http://dx.doi.org/10.1080/10652469308819011 http://dx.doi.org/10.1112/S0024611502013825 http://arxiv.org/abs/math.RT/0108185 http://dx.doi.org/10.1017/CBO9780511565717 http://dx.doi.org/10.1017/CBO9780511565717 http://dx.doi.org/10.1090/S1088-4165-09-00356-2 http://arxiv.org/abs/0901.4595 http://dx.doi.org/10.1090/S0002-9947-2010-05156-6 http://arxiv.org/abs/0707.0251 1 Introduction 2 General results 3 The group B2 4 The reproducing kernel 5 The Gaussian-type weight function 6 Closing remarks References

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