A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus

For each irreducible module of the symmetric group SN there is a set of parametrized nonsymmetric Jack polynomials in N variables taking values in the module. These polynomials are simultaneous eigenfunctions of a commutative set of operators, self-adjoint with respect to two Hermitian forms, one ca...

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Дата:	2017
Автор:	Dunkl, C.F.
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Мова:	English
Опубліковано:	Інститут математики НАН України 2017
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/148638
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Цитувати:	A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus / C.F. Dunkl // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 11 назв. — англ.

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spelling	irk-123456789-1486382019-02-19T01:31:47Z A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus Dunkl, C.F. For each irreducible module of the symmetric group SN there is a set of parametrized nonsymmetric Jack polynomials in N variables taking values in the module. These polynomials are simultaneous eigenfunctions of a commutative set of operators, self-adjoint with respect to two Hermitian forms, one called the contravariant form and the other is with respect to a matrix-valued measure on the N-torus. The latter is valid for the parameter lying in an interval about zero which depends on the module. The author in a previous paper [SIGMA 12 (2016), 033, 27 pages] proved the existence of the measure and that its absolutely continuous part satisfies a system of linear differential equations. In this paper the system is analyzed in detail. The N-torus is divided into (N−1)! connected components by the hyperplanes xi=xj, i<j, which are the singularities of the system. The main result is that the orthogonality measure has no singular part with respect to Haar measure, and thus is given by a matrix function times Haar measure. This function is analytic on each of the connected components. 2017 Article A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus / C.F. Dunkl // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 11 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33C52; 32W50; 35F35; 20C30; 42B05 DOI:10.3842/SIGMA.2017.040 http://dspace.nbuv.gov.ua/handle/123456789/148638 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
description	For each irreducible module of the symmetric group SN there is a set of parametrized nonsymmetric Jack polynomials in N variables taking values in the module. These polynomials are simultaneous eigenfunctions of a commutative set of operators, self-adjoint with respect to two Hermitian forms, one called the contravariant form and the other is with respect to a matrix-valued measure on the N-torus. The latter is valid for the parameter lying in an interval about zero which depends on the module. The author in a previous paper [SIGMA 12 (2016), 033, 27 pages] proved the existence of the measure and that its absolutely continuous part satisfies a system of linear differential equations. In this paper the system is analyzed in detail. The N-torus is divided into (N−1)! connected components by the hyperplanes xi=xj, i<j, which are the singularities of the system. The main result is that the orthogonality measure has no singular part with respect to Haar measure, and thus is given by a matrix function times Haar measure. This function is analytic on each of the connected components.
format	Article
author	Dunkl, C.F.
spellingShingle	Dunkl, C.F. A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Dunkl, C.F.
author_sort	Dunkl, C.F.
title	A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus
title_short	A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus
title_full	A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus
title_fullStr	A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus
title_full_unstemmed	A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus
title_sort	linear system of differential equations related to vector-valued jack polynomials on the torus
publisher	Інститут математики НАН України
publishDate	2017
url	http://dspace.nbuv.gov.ua/handle/123456789/148638
citation_txt	A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus / C.F. Dunkl // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 11 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT dunklcf alinearsystemofdifferentialequationsrelatedtovectorvaluedjackpolynomialsonthetorus AT dunklcf linearsystemofdifferentialequationsrelatedtovectorvaluedjackpolynomialsonthetorus
first_indexed	2025-07-12T19:50:57Z
last_indexed	2025-07-12T19:50:57Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 040, 41 pages A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus 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/ Received December 11, 2016, in final form June 02, 2017; Published online June 08, 2017 https://doi.org/10.3842/SIGMA.2017.040 Abstract. For each irreducible module of the symmetric group SN there is a set of para- metrized nonsymmetric Jack polynomials in N variables taking values in the module. These polynomials are simultaneous eigenfunctions of a commutative set of operators, self-adjoint with respect to two Hermitian forms, one called the contravariant form and the other is with respect to a matrix-valued measure on the N -torus. The latter is valid for the parameter lying in an interval about zero which depends on the module. The author in a previous paper [SIGMA 12 (2016), 033, 27 pages] proved the existence of the measure and that its absolutely continuous part satisfies a system of linear differential equations. In this paper the system is analyzed in detail. The N -torus is divided into (N−1)! connected components by the hyperplanes xi = xj , i < j, which are the singularities of the system. The main result is that the orthogonality measure has no singular part with respect to Haar measure, and thus is given by a matrix function times Haar measure. This function is analytic on each of the connected components. Key words: nonsymmetric Jack polynomials; matrix-valued weight function; symmetric group modules 2010 Mathematics Subject Classification: 33C52; 32W50; 35F35; 20C30; 42B05 Contents 1 Introduction 2 2 Modules of the symmetric group 4 2.1 Jack polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 The differential system 7 3.1 The adjoint operation on Laurent polynomials and L(x) . . . . . . . . . . . . . . . . . . . 10 4 Integration by parts 11 5 Local power series near the singular set 12 5.1 Behavior on boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 6 Bounds 18 7 Sufficient condition for the inner product property 22 8 The orthogonality measure on the torus 27 8.1 Maximal singular support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 8.2 Boundary values for the measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 9 Analytic matrix arguments 38 References 40 mailto:cfd5z@virginia.edu http://people.virginia.edu/~cfd5z/ https://doi.org/10.3842/SIGMA.2017.040 2 C.F. Dunkl 1 Introduction The Jack polynomials form a parametrized basis of symmetric polynomials. A special case of these consists of the Schur polynomials, important in the character theory of the symmetric groups. By means of a commutative algebra of differential-difference operators the theory was extended to nonsymmetric Jack polynomials, again a parametrized basis but now for all poly- nomials in N variables. These polynomials are orthogonal for several different inner products, and in each case they are simultaneous eigenfunctions of a commutative set of self-adjoint op- erators. These inner products are invariant under permutations of the coordinates, that is, the symmetric group. One of these inner products is that of L2 ( TN ,Kκ(x)dm(x) ) , where TN := { x ∈ CN : \|xj \| = 1, 1 ≤ j ≤ N } , dm(x) = (2π)−Ndθ1 · · · dθN , xj = exp(iθj), −π < θj ≤ π, 1 ≤ j ≤ N, Kκ(x) = ∏ 1≤i<j≤N \|xi − xj \|2κ, κ > − 1 N ; defining the N -torus, the Haar measure on the torus, and the weight function respectively. Beerends and Opdam [1] discovered this orthogonality property of symmetric Jack polyno- mials. Opdam [9] established orthogonality structures on the torus for trigonometric polynomials associated with Weyl groups; the nonsymmetric Jack polynomials form a special case. Griffeth [7] constructed vector-valued Jack polynomials for the family G(n, p,N) of complex reflection groups. These are the groups of permutation matrices (exactly one nonzero entry in each row and each column) whose nonzero entries are nth roots of unity and the product of these entries is a (n/p)th root of unity. The symmetric groups and the hyperoctahedral groups are the special cases G(1, 1, N) and G(2, 1, N) respectively. The term “vector-valued” means that the polynomials take values in irreducible modules of the underlying group, and the action of the group is on the range as well as the domain of the polynomials. The author [2] together with Luque [5] investigated the symmetric group case more intensively. The basic setup is an irreducible representation of the symmetric group, specified by a partition τ of N , and a parameter κ restricted to an interval determined by the partition, namely −1/hτ < κ < 1/hτ where hτ is the maximum hook-length of the partition τ . More recently [3] we showed that there does exist a positive matrix measure on the torus for which the nonsymmetric vector-valued Jack polynomials (henceforth NSJP’s) form an orthogonal set. The proof depends on a matrix-version of Bochner’s theorem about the relation between positive measures on a compact abelian group and positive-definite functions on the dual group, which is a discrete abelian group. In the present situation the torus is the compact (multiplicative) group and the dual is ZN . By using known properties of the NSJP’s we produced a positive-definite matrix function on ZN and this implied the existence of the desired orthogonality measure. Additionally we showed that the part of the measure supported by TNreg := TN\ ⋃ i<j{x : xi = xj} is absolutely continuous with respect to the Haar measure dm and satisfies a first-order differential system. In this paper we complete the description of the measure by proving there is no singular part. The idea is to use the functional equations satisfied by the inner product to establish a correspondence to the differential system. The main reason for the argument being so complicated is that the “obvious” integration-by-parts argument which works smoothly for the scalar case with κ > 1 has great difficulty with the singularities of the measure of the form \|xi − xj \|−2\|κ\|. We use a Cauchy principal-value argument based on a weak continuity condition across the faces {x : xi = xj} (as an over-simplified one-dimensional example consider the integral ∫ 1 −1 d dxf(x)dx with f(x) = \|2x + x2\|−1/4: the integral is divergent but the principal value lim ε→0+ { ∫ −ε −1 + ∫ 1 ε } f ′(x)dx = f(1)− f(−1) + lim ε→0+ {f(−ε)− f((ε))} and f(−ε)− f(ε) = O ( ε3/4 ) hence the limit exists). A System of Differential Equations on the Torus 3 The differential system is a two-sided version of a Knizhnik–Zamolodchikov equation (see [6]) modified to have solutions homogeneous of degree zero, that is, constant on circles {(ux1, . . ., uxN ) : \|u\| = 1}. The purpose of the latter condition is to allow solutions analytic on connected components of TNreg. Denote the degree of τ by nτ . The solutions of the differential system are locally analytic nτ ×nτ matrix functions with initial condition given by a constant matrix. That is, the solution space is of dimension n2 τ but only one solution can provide the desired weight function. Part of the analysis deals with conditions specifying this solution – they turn out to be commutation relations involving certain group elements. In the subsequent discussion it is shown that the weight function property holds for a very small interval of κ values if these relations are satisfied. This is combined with the existence theorem of the positive-definite matrix measure to finally demonstrate that the measure has no singular part for any κ in −1/hτ < κ < 1/hτ . In a subsequent development [4] it is shown that the square root of the matrix weight function multiplied by vector-valued symmetric Jack polynomials provides novel wavefunctions of the Calogero–Sutherland quantum mechanical model of identical particles on a circle with 1/r2 interactions. Here is an outline of the contents of the individual sections: • Section 2: a short description of the representation of the symmetric group associated to a partition; the definition of Dunkl operators for vector-valued polynomials and the definition of nonsymmetric Jack polynomials (NSJP’s) as simultaneous eigenvectors of a commutative set of operators; and the Hermitian form given by an integral over the torus, for which the NSJP’s form an orthogonal basis. • Section 3: the definition of the linear system of differential equations which will be demon- strated to have a unique matrix solution L(x) such that L(x)∗L(x)dm(x) is the weight function for the Hermitian form; the proof that the system is Frobenius integrable and the analyticity and monodromy properties of the solutions on the torus. • Section 4: the use of the differential equation to relate the Hermitian form to L(x)∗L(x) by means of integration by parts; the result of this is to isolate the role of the singularities in the process of proving the orthogonality of the NSJP’s with respect to L∗Ldm. • Section 5: deriving power series expansions of L(x) near the singular set ⋃ i<j { x ∈ TN : xi = xj } , in particular near the set {x : xN−1 = xN}; description of commutation proper- ties of the coefficients with respect to the reflection (N − 1, N); the behavior of L across the mirror {x : xN−1 = xN}. • Section 6: the derivation of global bounds on L(x) and local bounds on the coefficients of the power series, needed to analyze convergence properties of the integration by parts. • Section 7: the proof of a sufficient condition for the validity of the Hermitian form; the condition is partly that κ lies in a small interval around 0 and that the boundary value of L(x) satisfies a commutativity condition; the proof involves very detailed analysis of bounds on L, since the local bounds have to be integrated over the entire torus. • Section 8: further analysis of the orthogonality measure constructed in [3], in particular the proof of the formal differential system satisfied by the Fourier–Stieltjes (Laurent) series of the measure; this is used to show that the measure has no singular part on the open faces, such as{( eiθ1 , eiθ2 , . . . , eiθN−1 , eiθN−1 ) : θ1 < θ2 < · · · < θN−2 < θN−1 < θ1 + 2π } ; in turn this property is shown to imply the validity of the sufficient condition set up in Section 7. 4 C.F. Dunkl • Section 9: analyticity properties of the solutions of matrix equations with analytic coeffi- cients; the results are used to extend the validity of the Hermitian form to the desired interval −1/hτ < κ < 1/hτ from the smaller interval found in Section 7. 2 Modules of the symmetric group The symmetric group SN , the set of permutations of {1, 2, . . . , N}, acts on CN by permutation of coordinates. For α ∈ ZN the norm is \|α\| := N∑ i=1 \|αi\| and the monomial is xα := N∏ i=1 xαii . Denote N0 := {0, 1, 2, . . .}. The space of polynomials P := spanC { xα : α ∈ NN0 } . Elements of spanC { xα : α ∈ ZN } are called Laurent polynomials. The action of SN is extended to polynomials by wp(x) = p(xw) where (xw)i = xw(i) (consider x as a row vector and w as a permutation matrix, [w]ij = δi,w(j), then xw = x[w]). This is a representation of SN , that is, w1(w2p)(x) = (w2p)(xw1) = p(xw1w2) = (w1w2)p(x) for all w1, w2 ∈ SN . Furthermore SN is generated by reflections in the mirrors {x : xi = xj} for 1 ≤ i < j ≤ N . These are transpositions, denoted by (i, j), so that x(i, j) denotes the result of interchanging xi and xj . Define the SN -action on α ∈ ZN so that (xw)α = xwα (xw)α = N∏ i=1 xαiw(i) = N∏ j=1 x αw−1(j) j , that is (wα)i = αw−1(i) (take α as a column vector, then wα = [w]α). The simple reflections si := (i, i+ 1), 1 ≤ i ≤ N − 1, generate SN . They are the key devices for applying inductive methods, and satisfy the braid relations: sisj = sjsi, \|i− j\| ≥ 2; sisi+1si = si+1sisi+1. We consider the situation where the group SN acts on the range as well as on the domain of the polynomials. We use vector spaces, called SN -modules, on which SN has an irreducible unitary (orthogonal) representation: τ : SN → Om(R) ( τ(w)−1 = τ ( w−1 ) = τ(w)T ) . See James and Kerber [8] for representation theory, including a modern discussion of Young’s methods. Denote the set of partitions NN,+0 := { λ ∈ NN0 : λ1 ≥ λ2 ≥ · · · ≥ λN } . We identify τ with a partition of N given the same label, that is τ ∈ NN,+0 and \|τ \| = N . The length of τ is `(τ) := max{i : τi > 0}. There is a Ferrers diagram of shape τ (also given the same label), with boxes at points (i, j) with 1 ≤ i ≤ `(τ) and 1 ≤ j ≤ τi. A tableau of shape τ is a filling of the boxes with numbers, and a reverse standard Young tableau (RSYT) is a filling with the numbers {1, 2, . . . , N} so that the entries decrease in each row and each column. We exclude the one-dimensional representations corresponding to one-row (N) or one- column (1, 1, . . . , 1) partitions (the trivial and determinant representations, respectively). We need the important quantity hτ := τ1 + `(τ)− 1, the maximum hook-length of the diagram (the hook-length of the node (i, j) ∈ τ is defined to be τi − j + #{k : i < k ≤ `(τ)&j ≤ τk} + 1). Denote the set of RSYT’s of shape τ by Y(τ) and let Vτ = span{T : T ∈ Y(τ)} (the field is C(κ)) with orthogonal basis Y(τ). For 1 ≤ i ≤ N and T ∈ Y(τ) the entry i is at coordinates (rw(i, T ), cm(i, T )) and the content is c(i, T ) := cm(i, T ) − rw(i, T ). Each A System of Differential Equations on the Torus 5 T ∈ Y(τ) is uniquely determined by its content vector [c(i, T )]Ni=1. Let S1(τ) = N∑ i=1 c(i, T ) (this sum depends only on τ) and γ := S1(τ)/N . The SN -invariant inner product on Vτ is defined by 〈T, T ′〉0 := δT,T ′ × ∏ 1≤i<j≤N, c(i,T )≤c(j,T )−2 ( 1− 1 (c(i, T )− c(j, T ))2 ) , T, T ′ ∈ Y(τ). It is unique up to multiplication by a constant. The Jucys–Murphy elements N∑ j=i+1 (i, j) satisfy N∑ j=i+1 τ((i, j))T = c(i, T )T and thus the central element ∑ 1≤i<j≤N (i, j) satisfies ∑ 1≤i<j≤N τ((i, j))T = S1(τ)T for each T ∈ Y(τ). The basis is ordered such that the vectors T with c(N − 1, T ) = −1 appear first (that is, cm(N − 1, T ) = 1, rw(N − 1, T ) = 2). This results in the matrix representation of τ((N − 1, N)) being[ −Imτ O O Inτ−mτ ] , where nτ := dimVτ = #Y(τ) and mτ is given by tr(τ((N − 1, N))) = nτ − 2mτ . From the sum∑ i<j τ((i, j)) = S1(τ)I it follows that ( N 2 ) tr(τ((N −1, N))) = S1(τ)nτ and mτ = nτ ( 1 2 − S1(τ) N(N−1) ) . (The transpositions are conjugate to each other implying the traces are equal.) 2.1 Jack polynomials The main concerns of this paper are measures and matrix functions on the torus associated to Pτ := P ⊗ Vτ , the space of Vτ -valued polynomials, which is equipped with the SN action: w (xα ⊗ T ) = (xw)α ⊗ τ(w)T, α ∈ NN0 , T ∈ Y(τ), wp(x) = τ(w)p(xw), p ∈ Pτ , extended by linearity. There is a parameter κ which may be generic/transcendental or complex. Definition 2.1. The Dunkl and Cherednik–Dunkl operators are (1 ≤ i ≤ N , p ∈ Pτ ) Dip(x) := ∂ ∂xi p(x) + κ ∑ j 6=i τ((i, j)) p(x)− p(x(i, j)) xi − xj , Uip(x) := Di(xip(x))− κ i−1∑ j=1 τ((i, j))p(x(i, j)). The commutation relations analogous to the scalar case hold: DiDj = DjDi, UiUj = UjUi, 1 ≤ i, j ≤ N ; wDi = Dw(i)w, ∀w ∈ SN ; sjUi = Uisj , j 6= i− 1, i; siUisi = Ui+1 + κsi, Uisi = siUi+1 + κ, Ui+1si = siUi − κ. The simultaneous eigenfunctions of {Ui} are called (vector-valued) nonsymmetric Jack poly- nomials (NSJP). For generic κ these eigenfunctions form a basis of Pτ (this property fails for certain rational numbers outside the interval −1/hτ < κ < 1/hτ ). There is a partial order on NN0 × Y(τ) for which the NSJP’s have a triangular expression with leading term indexed by 6 C.F. Dunkl (α, T ) ∈ NN0 ×Y(τ). The polynomial with this label is denoted by ζα,T , homogeneous of degree N∑ i=1 αi and satisfies Uiζα,T = (αi + 1 + κc (rα(i), T )) ζα,T , 1 ≤ i ≤ N, rα(i) := #{j : αj > αi}+ #{j : 1 ≤ j ≤ i, αj = αi}; the rank function rα ∈ SN and rα = I if and only if α is a partition. The vector [αi + 1 + κc(rα(i), T )]Ni=1 is called the spectral vector for (α, T ). The NSJP structure can be extended to Laurent polyno- mials. Let eN := N∏ i=1 xi and 1 := (1, 1, . . . , 1) ∈ NN0 , then rα+m1 = rα for any α ∈ NN0 and m ∈ Z. The commutation Ui(emNp) = emN (m + Ui)p for 1 ≤ i ≤ N and p ∈ Pτ imply that emNζα,T and ζα+m1,T have the same spectral vector for any m ∈ N0. They also have the same leading term (see [3, Section 2.2]) and hence emNζα,T = ζα+m1,T for α ∈ NN0 . This fact allows the definition of ζα,T for any α ∈ ZN : let m = −mini αi then α+m1 ∈ NN0 and set ζα,T := e−mN ζα+m1,T . For a complex vector space V a Hermitian form is a mapping 〈·, ·〉 : V ⊗ V → C such that 〈u, cv〉 = c〈u, v〉, 〈u, v1 + v2〉 = 〈u, v1〉 + 〈u, v2〉 and 〈u, v〉 = 〈v, u〉 for u, v1, v2 ∈ V , c ∈ C. The form is positive semidefinite if 〈u, u〉 ≥ 0 for all u ∈ V . The concern of this paper is with a particular Hermitian form on Pτ which has the properties (for all f, g ∈ Pτ , T, T ′ ∈ Y(τ), w ∈ SN , 1 ≤ i ≤ N): 〈1⊗ T, 1⊗ T ′〉 = 〈T, T ′〉0, (2.1) 〈wf,wg〉 = 〈f, g〉, 〈xiDif, g〉 = 〈f, xiDig〉, 〈xif, xig〉 = 〈f, g〉. The commutation Ui = Dixi − κ ∑ j<i (i, j) = xiDi + 1 + κ ∑ j>i (i, j) together with 〈(i, j)f, g〉 = 〈f, (i, j)g〉 show that 〈Uif, g〉 = 〈f,Uig〉 for all i. Thus uniqueness of the spectral vectors (for all but a certain set of rational κ values) implies that 〈ζα,T , ζβ,T ′〉 = 0 whenever (α, T ) 6= (β, T ′). In particular polynomials homogeneous of different degrees are mutually orthogonal, by the basis property of {ζα,T }. For this particular Hermitian form, multiplication by any xi is an isometry for all 1 ≤ i ≤ N . This involves an integral over the torus. The equations (2.1) determine the form uniquely (up to a multiplicative constant if the first condition is removed). Denote C× := C\{0} and CNreg := CN×\ ⋃ i<j {x : xi = xj}. The torus is a compact multiplicative abelian group. The notations for the torus and its Haar measure in terms of polar coordinates are TN := { x ∈ CN : \|xj \| = 1, 1 ≤ j ≤ N } , dm(x) = (2π)−Ndθ1 · · · dθN , xj = exp(iθj), −π < θj ≤ π, 1 ≤ j ≤ N. Let TNreg := TN ∩CNreg, then TNreg has (N − 1)! connected components and each component is homotopic to a circle (if x is in some component then so is ux = (ux1, . . . , uxN ) for each u ∈ T). Definition 2.2. Let ω := exp 2πi N and x0 := ( 1, ω, . . . , ωN−1 ) . Denote the connected component of TNreg containing x0 by C0. A System of Differential Equations on the Torus 7 Thus C0 is the set consisting of ( eiθ1 , . . . , eiθN ) with θ1 < θ2 < · · · < θN < θ1 + 2π. In [3] we showed that if −1/hτ < κ < 1/hτ then there exists a positive matrix-valued measure dµ on TN such that for f, g ∈ C(1) ( TN ;Vτ ) , w ∈ SN , 1 ≤ i ≤ N ,∫ TN f(x)∗dµ(x)g(x) = ∫ TN f(xw)∗τ(w)−1dµ(x)τ(w)g(xw),∫ TN (xiDif(x))∗dµ(x)g(x) = ∫ TN f(x)∗dµ(x)xiDig(x),∫ TN (1⊗ T )∗dµ(x)(1⊗ T ) = 〈T, T 〉0, T ∈ Y(τ). We introduced the notation f(x)∗dµ(x)g(x) := ∑ T,T ′∈Y(τ) f(x)T g(x)T ′dµT,T ′(x), where f, g ∈ Pτ have the components (fT ), (gT ) with respect to the orthonormal basis{ 〈T, T 〉−1/2 0 T : T ∈ Y(τ) } . Thus the Hermitian form 〈f, g〉 = ∫ TN f(x)∗dµ(x)g(x) satisfies (2.1). Furthermore we showed that dµ(x) = dµs(x) + L(x)∗H(C)L(x)dm(x), where the singular part µs is the restriction of µ to TN\TNreg, H(C) is constant and positive- definite on each connected component C of TNreg and L(x) is a matrix function solving a system of differential equations. That system is the subject of this paper. In a way the main problem is to show that µ has no singular part. 3 The differential system Consider the system (with ∂i := ∂ ∂xi , 1 ≤ i ≤ N) for nτ × nτ matrix functions L(x) ∂iL(x) = κL(x) ∑ j 6=i 1 xi − xj τ((i, j))− γ xi I  , 1 ≤ i ≤ N, (3.1) γ := S1(τ) N = 1 2N `(τ)∑ i=1 τi(τi − 2i+ 1). The effect of the term γ xi I is to make L(x) homogeneous of degree zero, that is, N∑ i=1 xi∂iL(x) = 0. The differential system is defined on CNreg, Frobenius integrable and analytic, thus any local solution can be continued analytically to any point in CNreg. Different paths may produce different values; if the analytic continuation is done along a closed path then the resultant solution is a constant matrix multiple of the original solution, called the monodromy matrix, however if the closed path is contained in a simply connected subset of CNreg then there is no change. Integrability means that ∂i(κL(x)Aj(x)) = ∂j(κL(x)Ai(x)) for i 6= j, writing the system as ∂iL(x) = κL(x)Ai(x), 1 ≤ i ≤ N (where Ai(x) is defined by equation (3.1)). The condition becomes κ2L(x)Ai(x)Aj(x) + κL(x)∂iAj(x) = κ2L(x)Aj(x)Ai(x) + κL(x)∂jAi(x). 8 C.F. Dunkl Since ∂iAj(x) = τ((i,j)) (xi−xj)2 = ∂jAi(x) it suffices to show that Ai(x)′ := ∑ k 6=i τ((i,k)) xi−xk and Aj(x)′ :=∑̀ 6=j τ((j,`)) xj−x` commute with each other. The product Ai(x)′Aj(x)′ is a sum of − 1 (xi−xj)2 I, terms of the form τ((i,k)(j,`)) (xi−xk)(xj−x`) + τ((i,`)(j,k)) (xi−x`)(xj−xk) for {i, k} ∩ {j, `} = ∅, and terms involving the 3-cycles (i, j, k) and (j, i, k) occurring as τ((i, j)(j, k)) (xi − xj)(xj − xk) + τ((i, k)(j, i)) (xi − xk)(xj − xi) + τ((i, k)(j, k)) (xi − xk)(xj − xk) = τ((i, j, k)) (xi − xk)(xj − xk) + τ((j, i, k)) (xi − xk)(xj − xk) , (because (i, j)(j, k) = (i, k)(j, i) = (i, j, k) and (i, k)(j, k) = (j, i, k)) and the latter two terms are symmetric in i, j. We consider only fundamental solutions, that is, detL(x) 6= 0. Recall Jacobi’s identity: ∂ ∂t detF (t) = tr ( adj(F (t)) ∂ ∂t F (t) ) , where F (t) is a differentiable matrix function and adj(F (t))F (t) = detF (t)I, that is, adj(F (t)) = {detF (t)}F (t)−1 when F (t) is invertible; thus ∂i detL(x) = tr ( {detL(x)}L(x)−1∂iL(x) ) = κdetL(x) trAi(x). This can be solved: from ∑ i<j τ((i, j)) = S1(τ)I it follows that tr(τ((i, j))) = ( N 2 )−1 S1(τ)nτ = 2 N−1γnτ (and nτ = #Y(τ)). We obtain the system ∂i detL(x) = κγnτ  2 N − 1 ∑ j 6=i 1 xi − xj − 1 xi detL(x), 1 ≤ i ≤ N. By direct verification detL(x) = c ∏ 1≤i<j≤N ( −(xi − xj)2 xixj )κλ/2 , λ := γnτ 2(N − 1) = tr(τ((1, 2))), is a local solution for any c ∈ C×, if xk = eiθk , 1 ≤ k ≤ N then − (xi−xj)2 xixj = 4 sin2 θi−θj 2 (with the principal branch of the power function, positive on positive reals). This implies detL(x) 6= 0 for x ∈ CNreg (and of course detL(x) is homogeneous of degree zero). Proposition 3.1. If L(x) is a solution of (3.1) in some connected open subset U of CNreg then L(xw)τ(w)−1 is a solution in Uw−1. Proof. First let w = (j, k) for some fixed j, k. If i 6= j, k then replace x by x(j, k) in ∂iL to obtain ∂iL(x(j, k))τ((j, k)) = κL(x(j, k))  ∑ `6=i,j,k τ((i, `)) xi − x` + τ((i, j)) xi − xk + τ((i, k)) xi − xj − γ xi I  τ(j, k) = κL(x(j, k))τ(j, k)  ∑ `6=i,j,k τ((i, `)) xi − x` + τ((i, k)) xi − xk + τ((i, j)) xi − xj − γ xi I  , A System of Differential Equations on the Torus 9 because (i, j)(j, k) = (j, k)(i, k). Next let w = (i, j), then ∂i[L(x(i, j))] = (∂jL)(x(i, j)) and ∂i[L(x(i, j))τ((i, j))] = κL(x(i, j)) ∑ `6=i,j τ((j, `)) xi − x` + τ((i, j)) xi − xj − γ xi I  τ((i, j)) = κL(x(i, j))τ((i, j)) ∑ `6=i,j τ((i, `)) xi − x` + τ((i, j)) xi − xj − γ xi I  , by use of (j, `)(i, j) = (i, j)(i, `). Arguing by induction suppose L(xw)τ(w)−1 is a solution then so is L(x(j, k)w)τ(w)−1τ((j, k)) = L(x(j, k)w)τ((j, k)w)−1, for any(j, k), that is, the statement holds for w′ = (j, k)w. � Let w0 := (1, 2, 3, . . . , N) = (12)(23) · · · (N − 1, N) denote the N -cycle and let 〈w0〉 denote the cyclic group generated by w0. There are two components of TNreg which are set-wise invariant under 〈w0〉 namely C0 and the reverse {θN < θN−1 < · · · < θ1 < θN + 2π}. Indeed 〈w0〉 is the stabilizer of C0 as a subgroup of SN . Henceforth we use L(x) to denote the solution of (3.1) in C0 which satisfies L(x0) = I. Proposition 3.2. Suppose x ∈ C0 and m ∈ Z then L(xwm0 ) = τ(w0)−mL(x)τ(w0)m. Proof. Consider the solution L(xw0)τ(w0)−1 which agrees with ΞL(x) for all x ∈ C0 for some fixed matrix Ξ. In particular for x = x0 where x0w0 = ( ω, . . . , ωN−1, 1 ) = ωx0 (recall (xw)i = xw(i)) we obtain ΞL(x0) = L(x0w0)τ(w0)−1 = L(ωx0)τ(w0)−1 = L(x0)τ(w0)−1; because L(x) is homogeneous of degree zero. Thus Ξ = τ(w0)−1. Repeated use of the relation shows L(xwm0 ) = τ(w0)−mL(x)τ(w0)m. � Because of its frequent use denote υ := τ(w0) (the letter υ occurs in the Greek word cycle). Definition 3.3. For w ∈ SN set ν(w) := υ1−w(1). For any x ∈ TNreg there is a unique wx such that wx(1) = 1 and xw−1 x ∈ C0. Set M(w, x) := ν(wxw). As a consequence ν(wm0 w) = υ−mν(w) for any w ∈ SN and m ∈ Z; since wm0 w(1) − 1 = (w(1) + m − 1) modN . Also M(I, x) = I. There is a 1-1 correspondence w 7→ C0w between {w ∈ SN : w(1) = 1} and the connected components of TNreg. Proposition 3.4. For any w1, w2 ∈ SN and x ∈ TNreg M(w1w2, x) = M(w2, xw1)M(w1, x). Proof. By definition M(w1w2, x) = ν(wxw1w2) and M(w1, x) = ν(wxw1) = υ−m where wxw1(1) = m+1. Let w3 = wxw1 , that is, w3(1) = 1 and xw1w −1 3 ∈ C0. From ( xw−1 x )( wxw1w −1 3 ) ∈ C0 it follows that wxw1w −1 3 ∈ 〈w0〉, in particular wxw1w −1 3 = wm0 because wxw1w −1 3 (1) = wxw1(1) = m + 1 = wm0 (1). Thus M(w2, xw1) = ν(w3w2) = ν ( w−m0 wxw1w2 ) = υmν(wxw1w2), and υm = M(w1, x)−1. This completes the proof. � Corollary 3.5. Suppose w ∈ SN and x ∈ TNreg then M ( w−1, xw ) = M(w, x)−1. Proof. Indeed M ( w−1, xw ) M(w, x) = M ( ww−1, x ) = I. � We can now extend L(x) to all of TNreg from its values on C0. Definition 3.6. For x ∈ TNreg let L(x) := L ( xw−1 x ) τ(wx). 10 C.F. Dunkl Proposition 3.7. For any x ∈ TNreg and w ∈ SN L(xw) = M(w, x)L(x)τ(w). Proof. Let w1 = wxw, that is, w1(1) = 1 and xww−1 1 ∈ C0, then by definition L(xw) = L ( xww−1 1 ) τ(w1) and L ( xw−1 x ) = L(x)τ(wx)−1. Let m = wxw(1) − 1. Since wxww −1 1 fixes C0 and wxww −1 1 (1) = wxw(1) = m+ 1 it follows that wxww −1 1 = wm0 . Thus w1 = w−m0 wxw, L ( xww−1 1 ) τ(w1) = L ( xw−1 x wm0 ) τ ( w−m0 wxw ) = υ−mL ( xw−1 x ) υmτ ( w−m0 wxw ) = υ−mL ( xw−1 x ) τ(wxw) = υ−mL(x)τ(w) and M(w, x) = ν(wxw) = υ−m. � 3.1 The adjoint operation on Laurent polynomials and L(x) The purpose is to define an operation which agrees with taking complex conjugates of functions and Hermitian adjoints of matrix functions when restricted to TN , and which preserves ana- lyticity. The parameter κ is treated as real in this context even where it may be complex (to preserve analyticity in κ). For x ∈ CN× define φx := ( x−1 1 , x−1 2 , . . . , x−1 N ) , then φ(xw) = (φx)w for all w ∈ SN . Definition 3.8. (1) If f(x) = ∑ α∈ZN cαx α is a Laurent polynomial then f∗(x) := ∑ α∈ZN cαx −α. (2) If f(x) = ∑ α∈ZN Aαx α is a Laurent polynomial with matrix coefficients then f∗(x) :=∑ α∈ZN A∗αx −α. (3) if F (x) is a matrix-valued function analytic in an open subset U of CN× then F ∗(x) := (F (φx)) T and F ∗ is analytic on φU , that is, if F (x)=[Fij(x)]Ni,j=1 then F ∗(x)=[Fji(φx)]Ni,j=1 (for example if F12(x) = c1κx1x −1 3 + c2x 2 2x −1 3 x−1 4 then F ∗21(x) = c1κx −1 1 x3 + c2x −2 2 x3x4). Loosely speaking F ∗(x) is obtained by replacing x by φx, conjugating the complex constants and transposing. The fundamental chamber C0 is mapped by φ onto {( eiθj )N j=1 : θ1 > θ2 > · · · > θN > θ1− 2π } , again set-wise invariant under w0. Using d dt { f ( 1 t )} = − 1 t2 ( d dtf )( 1 t ) we obtain the system ∂iL(φx) = κL(φx) ∑ j 6=i xj xi τ((i, j)) xi − xj + γ xi  , 1 ≤ i ≤ N. Transposing this system leads to (note τ(w)T = τ(w)∗ = τ ( w−1 ) ) ∂iL(φx)T = κ ∑ j 6=i xj xi τ((i, j)) xi − xj + γ xi L(φx)T , 1 ≤ i ≤ N. Now use part (3) of Definition 3.8 and set up the system whose solution of ∂iL ∗(x) = κ ∑ j 6=i xj xi τ((i, j)) xi − xj + γ xi L∗(x), 1 ≤ i ≤ N. (3.2) A System of Differential Equations on the Torus 11 satisfying L∗(x0) = I is denoted by L∗(x). The constants in the system are all real so replacing complex constants by their complex conjugates preserves solutions of the system. The effect is that L(x)∗ agrees with the Hermitian adjoint of L(x) for x ∈ C0 (for real κ). The goal here is to establish conditions on a constant Hermitian matrix H so that K(x) := L∗(x)HL(x) has desirable properties, such as K(xw) = τ(w)−1K(x)τ(w) and K(x) ≥ 0 (i.e., positive definite). Similarly to the above τ((i, j))L∗(x(i, j)) is also a solution of (3.2), implying that τ(w)L∗(xw) is a solution for any w ∈ SN , the inductive step is τ((i, j))τ(w)L(x(i, j)w)∗ = τ((i, j)w)L(x(i, j)w)∗. Also L∗(x0w0) = L∗ ( ω−1x0 ) = L∗(x0) = I (thus there is a matrix Ξ̃ such that τ(w0)L∗(xw0) = L∗(φx)Ξ̃ for all x ∈ C0, and Ξ̃ = τ(w0) = υ. In analogy to L for x ∈ TNreg and the same wx as above let L(φx0)T = I, L(φx)T := τ(wx)−1L ( φxw−1 x )T (since φxw−1 x ∈ φC0). Then L(φxw)T = τ(w)−1L(φx)TM(w, x)−1. For any nonsingular constant matrix C the function CL(x) also satisfies (3.1) and the function K(x) := L∗(x)C∗CL(x) satisfies the system xi∂iK(x) = κ ∑ j 6=i { xj xi − xj τ((i, j))K(x) +K(x)τ((i, j)) xi xi − xj } , 1 ≤ i ≤ N. (3.3) This formulation can be slightly generalized by replacing C∗C by a Hermitian matrix H (not necessarily positive-definite) without changing the equation. For the purpose of realizing the form (2.1) we want K to satisfy K(xw) = τ(w)−1K(x)τ(w), that is, K(xw) = τ(w)−1L∗(x)M(w, x)−1HM(w, x)L(x)τ(w) = τ(w)−1L∗(x)υmHυ−mL(x)τ(w) (from Proposition 3.7), where m = wxw(1)− 1. The condition is equivalent to υH = Hυ, which is now added to the hypotheses, summarized here: Condition 3.9. L(x) is the solution of (3.1) such that L(x0) = I and L(x) = L ( xw−1 x ) τ(wx) for x ∈ TNreg where wx(1) = 1 and xw−1 x ∈ C0; L∗(x) is the solution of (3.2) satisfying L∗(x0) = I, K(x) = L∗(x)HL(x) is a solution of (3.3) and H satisfies H∗ = H, υH = Hυ. 4 Integration by parts In this section we establish the relation between the differential system and the abstract relation 〈xiDif, g〉 = 〈f, xiDig〉 holding for 1 ≤ i ≤ N and f, g ∈ C1 ( TN ;Vτ ) . We demonstrate how close L is to providing the desired inner product, by performing an integration-by-parts over an SN -invariant closed set ⊂ TNreg. Here L(x) and H satisfy the hypotheses listed in Condition 3.9 above. We use the identity xi∂if ∗(x) = −(xi∂if)∗(x). For δ > 0 let Ωδ := { x ∈ TN : min 1≤i<j≤N \|xi − xj \| ≥ δ } . This set is invariant under SN and K(x) is bounded and smooth on it. Thus the following integrals exist. 12 C.F. Dunkl Proposition 4.1. Suppose H satisfies Condition 3.9 then for f, g ∈ C1 ( TN ;Vτ ) and 1 ≤ i ≤ N∫ Ωδ { −(xiDif(x))∗K(x)g(x) + f(x)∗K(x)xiDig(x) } dm(x) = ∫ Ωδ xi∂i{f(x)∗K(x)g(x)}dm(x). Proof. By definition xiDig(x) = xi∂ig(x) + κ ∑ j 6=i xi xi − xj τ((i, j))(g(x)− g(x(i, j))), (xiDif(x))∗ = −xi∂if(x)∗ + κ ∑ j 6=i xj xj − xi (f(x)∗ − f(x(i, j))∗)τ((i, j)). Thus −(xiDif(x))∗K(x)g(x) + f(x)∗K(x)xiDig(x) = xi∂if(x)∗ + xi∂ig(x) + κf(x)∗ ∑ j 6=i { xj xi − xj τ((i, j))K(x) +K(x)τ((i, j)) xi xi − xj } g(x) − κ ∑ j 6=i 1 xi − xj { xjf(x(i, j))∗τ((i, j))K(x)g(x) + xif(x)∗K(x)τ((i, j))g(x(i, j)) } = xi∂i{f(x)∗K(x)g(x)} (4.1) − κ ∑ j 6=i 1 xi − xj { xjf(x(i, j))∗τ((i, j))K(x)g(x) + xif(x)∗K(x)τ((i, j))g(x(i, j)) } . For each pair {i, j} the term inside {·} is invariant under x 7→ x(i, j), because K(x(i, j)) = τ((i, j))K(x)τ((i, j)), and xi − xj changes sign under this transformation. Thus∫ Ωδ xjf(x(i, j))∗τ((i, j))K(x)g(x) + xif(x)∗K(x)τ((i, j))g(x(i, j)) xi − xj dm(x) = 0 for each j 6= i because Ωδ and dm are invariant under (i, j). � Observe the value of κ is not involved in the proof. Since xj∂j = −i ∂ ∂θj when xj = eiθj and dm(x) = (2π)−Ndθ1 · · · dθN one step of integration can be directly evaluated. Consider the case i = N and for a fixed (N − 1)-tuple (θ1, . . . , θN−1) with θ1 < θ2 < · · · < θN−1 < θ1 + 2π such that ∣∣eiθj − eiθi ∣∣ ≥ δ the integral over θN is over a union of closed intervals. These are the complement of ⋃ 1≤j≤N−1 {θ : θj − δ′ < θ < θj + δ′} in the circle, where sin δ′ 2 = δ 2 . This results in an alternating sum of values of f∗Kg at the end-points of the closed intervals. Analyzing the resulting integral (over (θ1, . . . , θN−1) with respect to dθ1 · · · dθN−1) is one of the key steps in showing that a given K provides the desired inner product. In other parts of this paper we find that H must satisfy another commuting relation. 5 Local power series near the singular set In this section assume κ /∈ Z+1 2 . We consider the system (3.1) in a neighborhood of the face {x : xN−1 = xN} of C0. We use a coordinate system which treats the singularity in a simple way. For a more concise notation define x(u, z) = (x1, x2, . . . , xN−2, u− z, u+ z) ∈ CN× A System of Differential Equations on the Torus 13 We consider the system in terms of the variable x(u, z) subject to the conditions that the points x1, x2, . . . , xN−2, u are pairwise distinct and \|z\| < min 1≤j≤N−2 \|xj−u\|, also \|z\| < \|u\|, Im z u > 0 (these conditions imply arg(u− z) < arg(u+ z)). This allows power series expansions in z. For z1, z2 ∈ C× let ρ(z1, z2) := [ z1Imτ O O z2Inτ−mτ ] . Let σ := τ((N − 1, N)) = ρ(−1, 1). We analyze the local solution L(x(u − z, u + z)) with an initial condition specified later. We obtain the differential system (using ∂z := ∂ ∂z , ∂u := ∂ ∂u) ∂zL(x) = ∂NL− ∂N−1L = κL  N−2∑ j=1 ( τ((j,N)) u− xj + z − τ((j,N − 1)) u− xj − z ) + τ(N − 1, N) z − γ u+ z I + γ u− z I  , ∂uL(x) = ∂NL+ ∂N−1L = κL  N−2∑ j=1 ( τ((j,N)) u− xj + z + τ((j,N − 1)) u− xj − z ) − γ u+ z I − γ u− z I  , ∂jL(x) = κL(x)  N−2∑ i=1,i 6=j τ((i, j)) xj − xi − γ xj I + τ((j,N − 1)) xj − u+ z + τ((j,N)) xj − u− z  , 1 ≤ j ≤ N − 2. Using the expansion 1 t−z = ∞∑ n=0 zn tn+1 for \|z\| < \|t\| we let βn(x(u, 0)) := N−2∑ j=1 τ((j,N)) (u− xj)n+1 for n = 0, 1, 2, . . .and express the equations as (since στ((j,N))σ = τ((j,N − 1))) ∂zL(x) = κL(x) { ∞∑ n=0 { (−1)nβn(x(u, 0))− σβn(x(u, 0))σ } zn + σ z − γ u+ z I + γ u− z I } , ∂uL(x) = κL(x) { ∞∑ n=0 { (−1)nβn(x(u, 0)) + σβn(x(u, 0))σ } zn − γ u+ z I − γ u− z I } , ∂jL(x) = κL(x)  N−2∑ i=1,i 6=j τ((i, j)) xj − xi − γ xj I − ∞∑ n=0 τ((j,N − 1)) + (−1)nτ((j,N)) (u− xj)n+1 zn  , 1 ≤ j ≤ N − 2. Set Bn(x(u, 0)) = (−1)nβn(x(u, 0))− σβn(x(u, 0))σ, n = 0, 1, 2, . . . . Note σBnx(u, 0)σ = (−1)n+1Bn(x(u, 0)). Suggested by the relation ∂ ∂z ρ ( z−κ, zκ ) = κ z ρ ( − z−κ, zκ ) = κ z ρ ( z−κ, zκ ) σ 14 C.F. Dunkl we look for a solution of the form L(x) = (u2 − z2 )N−2∏ j=1 xj −γκ ρ(z−κ, zκ) ∞∑ n=0 αn(x(u, 0))zn, (5.1) where each an(x(u, 0)) is matrix-valued and analytic in x(u, 0), and the initial condition is α0 ( x(0) ) = I, where x(0) is a base point, chosen as ( 1, ω, ω2, . . . , ωN−3, ω−3/2, ω−3/2 ) (that is, u = ω−3/2, z = 0), where ω := e2πi/N . Implicitly restrict (x1, . . . , xN−1, u) to a simply connected open subset of CN−1 reg containing ( 1, ω, ω2, . . . , ωN−3, ω−3/2 ) . Substitute (5.1) in the ∂z equation (suppressing the x(u, 0) argument in the αn’s) ∂zL = κγ ( 1 u+ z − 1 u− z )(u2 − z2 )N−2∏ j=1 xj −γκ ρ(z−κ, zκ) ∞∑ n=0 αnz n + (u2 − z2 )N−2∏ j=1 xj −γκ κ z ρ ( z−κ, zκ ) σ ∞∑ n=0 αnz n + (u2 − z2 )N−2∏ j=1 xj −γκ ρ(z−κ, zκ) ∞∑ n=1 nαnz n−1 = κ (u2 − z2 )N−2∏ j=1 xj −γκ ρ(z−κ, zκ) × ∞∑ n=0 αnz n { ∞∑ m=0 Bm(u)zm + σ z − γ ( 1 u+ z − 1 u− z )} , which simplifies to κ z ∞∑ n=0 (σαn − αnσ)zn + ∞∑ n=1 nαnz n−1 = κ ∞∑ n=0 αnz n ∞∑ m=0 Bm(x(u, 0))zm. (5.2) The equations for ∂u and ∂j simplify to(u2 − z2 )N−2∏ j=1 xj −γκ{ ∞∑ n=0 ∂uαnz n − κγ ( 1 u+ v + 1 u− v ) ∞∑ n=0 αnz n } = κ (u2 − z2 )N−2∏ j=1 xj −γκ ∞∑ n=0 αnz n × { ∞∑ m=0 { (−1)mβm(x(u, 0)) + σβm(x(u, 0))σ } zm − γ u+ v I − γ u− v I } , leading to (with 1 ≤ j ≤ N − 2) ∞∑ n=0 ∂uαn(x(u, 0))zn = κ ∞∑ n=0 αn(x(u, 0))zn ∞∑ m=0 { (−1)mβm(x(u, 0)) + σβm(x(u, 0))σ } zm, ∞∑ n=0 ∂jαn(x(u, 0))zn = κ ∞∑ n=0 αn(x(u, 0))zn A System of Differential Equations on the Torus 15 ×  N−2∑ i=1,i/∈j τ((i, j)) xj − xi − ∞∑ m=0 τ((j,N − 1)) + (−1)mτ((j,N)) (u− xj)m+1 zm  . We only need the equations for α0(x(u, 0)) (that is, the coefficient of z0) to initialize the ∂z equation (this is valid because the system is Frobenius integrable): ∂uα0(x(u, 0)) = κα0(x(u, 0)) { β0(x(u, 0)) + σβ0(x(u, 0))σ } , (5.3) ∂jα0(x(u, 0)) = κα0(x(u, 0))  N−2∑ i=1,i/∈j τ((i, j)) xj − xi − τ((j,N − 1)) + τ((j,N)) (u− xj)  , 2 ≤ j ≤ N − 2. Lemma 5.1. σα0(x(u, 0))σ = α0(x(u, 0)) and α0(x(u, 0)) is invertible. Proof. By hypothesis α0 ( x(0) ) = I. The right hand sides of the system are invariant un- der the transformation Q 7→ σQσ thus α0(x(u, 0)) and σα0(x(u, 0))σ satisfy the same system. They agree at the base-point x(0), hence everywhere in the domain. By Jacobi’s identity the determinant satisfies (where λ := tr(σ) = nτ − 2mτ ) ∂u detα0(x(u, 0)) = κdetα0(x(u, 0)) tr{β0(x(u, 0)) + σβ0(x(u, 0))σ} = 2κdetα0(x(u, 0))λ N−2∑ j=1 1 (u− xj) , ∂j detα0(x(u, 0)) = κλdetα0(x(u, 0))  N−2∑ i=1,i/∈j 1 xj − xi − 2 u− xj  , 1 ≤ j ≤ N − 2, detα0(x(u, 0)) = ∏ 1≤i<j≤N−2 ( xi − xj x (0) i − x (0) j )λκ N−2∏ i=1 ( xi − u x (0) i − x (0) N−1 )2λκ , the multiplicative constant follows from α0 ( x(0) ) = I. Thus α0(x(u, 0)) is nonsingular in its domain. � We turn to the inductive definition of {αn(x(u, 0)) : n ≥ 1}. In terms of the block decomposition (mτ + (nτ −mτ ))× (mτ + (nτ −mτ )) (henceforth called the σ-block decomposition) of a matrix α = [ α11 α12 α21 α22 ] σασ = α if and only if α12 = O = α21 and σασ = −α if and only if α11 = O = α22. Write the σ-block decomposition of αn(u) as αn = [ αn,11 αn,12 αn,21 αn,22 ] then the coefficient of zn−1 on the left side of equation (5.2) is κ(σαn − αnσ) + nαn = [ nαn,11 (n− 2κ)αn,12 (n+ 2κ)αn,21 nαn,22 ] , 16 C.F. Dunkl and on the right side it is κSn(x(u, 0)) := κ n−1∑ i=0 αn−1−iBi(x(u, 0)), for n ≥ 1. Arguing inductively suppose σαmσ = (−1)mαm for 0 ≤ m ≤ n, then σSnσ = n−1∑ i=0 (σαn−1−iσ)(σBiσ) = n−1∑ i=0 (−1)n−1−i+i−1αn−1−iBi and thus σSn(u)σ = (−1)nSn(u). In terms of the σ-block decomposition [ Sn,11 Sn,12 Sn,21 Sn,22 ] of Sn(x(u, 0)) this condition implies Sn,12 = O = Sn,21 when n is even, and Sn,11 = O = Sn,22 when n is odd. This implies (for n = 1, 2, 3, . . .) α2n(x(u, 0)) = κ 2n S2n(x(u, 0)), (5.4) α2n−1(x(u, 0)) = ρ ( κ 2n− 1− 2κ , κ 2n− 1 + 2κ ) S2n−1(x(u, 0)), and thus σαn(x(u, 0))σ = (−1)nαn(x(u, 0)). In particular α1(x(u, 0)) = ρ ( κ 1− 2κ , κ 1 + 2κ ) α0(x(u, 0))B0(x(u, 0)), and all the coefficients are determined; by hypothesis κ /∈ Z+1 2 and the denominators are of the form 2m+ 1± 2κ. Henceforth denote the series (5.1), solving (3.1) with the normalization α0 ( x(0) ) = I by L1(x). It is defined for all x(u, z) ∈ C0 subject to \|z\| < min 1≤j≤N−2 \|xj − u\|, also \|z\| < \|u\|, Im z u > 0. The radius of convergence depends on x(u, 0). Return to using L(x) to denote the solution from Definition 3.6 (on all of TNreg and L(x0) = I). In terms of x(u, z) the point x0 corresponds to u = 1 2 ( ω−1 +ω−2 ) , z = 1 2 ( ω−1−ω−2 ) , x(u, z) = ( 1, ω, . . . , ωN−3, u−z, u+z ) , then min 1≤j≤N−2 \|u− xj \| = sin π N ( 5+4 cos 2π N ) and \|z\| = sin π N (also z u = i tan π N ) and x0 is in the domain of convergence of the series L1(x). Thus the relation L1(x) = L1(x0)L(x) holds in the domain of L1 in C0. This implies the important fact that L1(x0) is an analytic function of κ, to be exploited in Section 9. 5.1 Behavior on boundary The term ρ(z−κ, zκ) implies that L1(x) is not continuous at z = 0, that is, on the boundary {x : xN−1 = xN}. However there may be a weak type of continuity, specifically lim xN−1−xN→0 (K(x)−K(x(N − 1, N))) = 0. With the aim of expressing the desired K(x) in the form L(x)∗C∗CL(x) (and C is unknown at this stage) we consider CL(x) in series form, that is CL1(x0)−1L1(x) (recall detL(x) 6= 0 in C0). We analyze the effect of C on the weak continuity condition. Denote C ′ := CL1(x0)−1. From Proposition 3.7 L(x(N − 1, N)) = ν((N − 1, N))L(x)τ((N − 1, N)) = L(x)σ, because w(1) = 1 for w = (N−1, N), [for the special case N = 3, τ = (2, 1), T3 reg has two components and we define L(x) = L(x(2, 3))σ for the component 6= C0]. By use of x(u, z)(N − 1, N) = x(u,−z) it follows that CL(x(u, z)(N − 1, N)) = CL(x(u, z))σ = C ′ (xNxN−1)−γκ ρ ( z−κ, zκ ) ∞∑ n=0 αn(u)znσ A System of Differential Equations on the Torus 17 = C ′σ(xNxN−1)−γκρ ( z−κ, zκ ) ∞∑ n=0 αn(u)(−1)nzn, because σαn(u)σ = (−1)nαn(u) and σ = ρ(−1, 1). Recall L∗(x) is defined as L(φx)T with complex constants replaced by their conjugates. Then φx(u, z) = ( x−1 1 , x−1 2 , . . . , x−1 N−2, 1 u−z , 1 u+z ) . To compute L1(φx(u, z)) replace u by u′ = u (u+z)(u−z) and replace z by z′ = − z (u+z)(u−z) . When restricted to the torus u′ = 1 2 ( 1 xN−1 + 1 xN ) = u and z′ = 1 2 ( 1 xN−1 − 1 xN ) = z. The terms βn(u) := N−2∑ j=1 τ((j,N)) (u−xj)n+1 in the intermediate formulae for L1 are replaced by their complex conjugates when x(u, z) ∈ TN . Similarly β̃k := ∞∑ m=0 N−2∑ j=1 τ((j,N)) (u0−xj)k+1 transforms to (β̃k) because the constant u0 is conjugated. Thus for x(u, z) ∈ TNreg L1(x(u, z))∗ = ∞∑ m=0 αm(u)∗zmρ ( z−κ, zκ ) C ′∗(xNxN−1)−γκ; αm(u)∗ denotes the adjoint of the matrix αm(u). Then L1(x(u, z)(N − 1, N))∗ = ∞∑ m=0 (−1)mαm(u)∗zmρ(z−κ, zκ)σC ′∗(xNxN−1)−γκ. Furthermore (recall K(x(N − 1, N)) = σK(x)σ by definition) K(x(u, z)) = ∞∑ m,n=0 zmznαm(u)∗ρ ( z−κ, zκ ) C ′∗C ′ρ ( z−κ, zκ ) αn(u), K(x(u,−z)) = ∞∑ m,n=0 zmzn(−1)m+nαm(u)∗ρ ( z−κ, zκ ) σC ′∗C ′σρ ( z−κ, zκ ) αn(u). The term of lowest order in z in K(x(u, z))−K(x(u,−z)) is α0(u)∗ρ ( z−κ, zκ ){ C ′∗C ′ − σC ′∗C ′σ } ρ ( z−κ, zκ ) α0(u). In terms of the σ-block decomposition, with C ′∗C ′ = [ c11 c12 c∗12 c22 ] , α0(u) = [ a11(u) O O a22(u) ] the expression equals 2  O (z z )κ a11(u)∗c12a22(u)( z z )κ a22(u)∗c∗12a11(u) O  , which tends to zero as z → 0 if and only if c12 = 0, that is, σC∗Cσ = C∗C. Proposition 5.2. Suppose C ′∗C ′ commutes with σ then K(x(u, z))−K(x(u, z)(N − 1, N)) = O ( \|z\|1−2\|κ\|). 18 C.F. Dunkl Proof. The hypothesis implies C ′∗C ′ commutes with ρ(z−κ, zκ), thus K(x(u, z))−K(x(u, z)(N − 1, N)) = ∞∑ m,n=0 zmzn ( 1− (−1)m+n ) αm(u)∗ρ ( \|z\|−2κ, \|z\|2κ ) C ′∗C ′αn(u) = 2zα0(u)∗ρ ( \|z\|−2κ, \|z\|2κ ) C ′∗C ′α1(u) + 2zα1(u)∗ρ ( \|z\|−2κ, \|z\|2κ ) C ′∗C ′α0(u) + ∞∑ m+n≥2 zmzn ( 1− (−1)m+n ) αm(u)∗ρ ( \|z\|−2κ, \|z\|2κ ) C ′∗C ′αn(u). The dominant terms come from m = 0, n = 1 and m = 1, n = 0; both of order O ( \|z\|1−2\|κ\|). � We will see later for purpose of integration by parts, that the change in K between the points( x1, . . . , xN−2, e iθ, ei(θ−ε)) and ( x1, . . . , xN−2, e iθ, ei(θ+ε) ) is a key part of the analysis; this uses the relation K (( x1, . . . , xN−2, e iθ, ei(θ−ε))) = σK (( x1, . . . , xN−2, e i(θ−ε), eiθ )) σ. 6 Bounds In this section we derive bounds on L(x) of global and local type. Throughout we adopt the normalization L(x0) = I. The operator norm on nτ × nτ complex matrices is defined by ‖M‖ = sup{\|Mv\| : \|v\| = 1}. Theorem 6.1. There is a constant c depending on κ such that ‖L(x)‖ ≤ c ∏ 1≤i<j≤N \|xi− xj \|−\|κ\| for each x ∈ TNreg. The proof is a series of steps starting with a general result which applies to matrix functions satisfying a linear differential equation in one variable. Lemma 6.2. Suppose M(0) = I, d dtM(t) = M(t)F (t) and ‖F (t)‖ ≤ f(t) for 0 ≤ t ≤ 1 then ‖M(t)− I‖ ≤ exp ∫ t 0 f(s)ds− 1 and ‖M(1)‖ ≤ exp ∫ 1 0 f(s)ds. Proof from [11, Theorem 7.1.11]. Let `(t) := ‖M(t) − I‖ then the equation M(t) − I =∫ t 0 M(s)F (s)ds and the inequalities ‖M(t)‖ ≤ ‖M(t) − I‖ + ‖I‖ (and ‖I‖ = 1) imply that `(t) ≤ ∫ t 0 (`(s) + 1)f(s)ds. Define differentiable functions b(t) and h(t) by h(t) := exp ∫ t 0 f(s)ds, b(t)h(t) = ∫ t 0 (`(s) + 1)f(s)ds+ 1. Apply d dtto the latter equation: b′(t)h(t) + b(t)f(t)h(t) = (`(t) + 1)f(t), b′(t)h(t) = f(t) { `(t) + 1− ∫ t 0 (`(s) + 1)f(s)ds− 1 } = f(t) { `(t)− ∫ t 0 (`(s) + 1)f(s)ds } ≤ 0. Hence b′(t) ≤ 0 and b(t) ≤ b(0) = 1 which implies `(t) ≤ ∫ t 0 (`(s) + 1)f(s)ds = b(t)h(t)− 1 ≤ h(t)− 1. Finally ‖M(1)‖ ≤ ‖M(1)− I‖+ 1 ≤ exp ∫ 1 0 f(s)ds. � A System of Differential Equations on the Torus 19 Next we set up a differentiable path p(t) = (p1(t), . . . , pN (t)) in CNreg starting at x0 and obtain the equation d dt L(p(t)) = κL(p(t)) N∑ i=1 ∑ j 6=i p′i(t) pi(t)− pj(t) τ((i, j))− γ p ′ i(t) pi(t) I  = κL(p(t))  ∑ 1≤i<j≤N p′j(t)− p′i(t) pj(t)− pi(t) τ((i, j))− γ N∑ i=1 p′i(t) pi(t) I  . Suppose x = ( eiθ1 , . . . , eiθN ) ∈ C0 and θ1 < θ2 < · · · < θN < θ1 + 2π. Define the path p(t) = ( eig1(t), . . . , eigN (t) ) where gj(t) = (1 − t)2(j−1)π N + tθj for 1 ≤ j ≤ N . Then p(t) ∈ C0 for 0 ≤ t ≤ 1 because gi+1(t) − gi(t) = (1 − t)2π N + t(θi+1 − θi) > 0 for 1 ≤ i < N and 2π + g1(t) − gN (t) = 2π + tθ1 − (1 − t)2(N−1)π N − tθN = (1 − t)2π N + t(2π + θ1 − θN ) > 0. The factor of τ((i, j)) in the equation is i g′j(t)e igj(t) − g′i(t)eigi(t) eigj(t) − eigi(t) = 1 2 { (g′j(t)− g′i(t)) cos ( 1 2(gj(t)− gi(t)) ) sin ( 1 2(gj(t)− gi(t)) ) + i(g′j(t) + g′i(t)) } . We will apply Lemma 6.2 to L̃(x) = N∏ j=1 xγκj L(x); this only changes the phase and removes the N∑ i=1 p′i(t) pi(t) term. In the notation of Lemma 6.2 f(t) = \|κ\| ∑ i<j ∣∣∣∣∣ig′j(t)eigj(t) − g′i(t)eigi(t) eigj(t) − eigi(t) ∣∣∣∣∣ (since ‖τ((i, j))‖ = 1). To set up the integral ∫ 1 0 f(t)dt let φij(t) = 1 2 (gj(t)− gi(t)) = 1 2 { (1− t)2(j − i)π N + t(θj − θi) } so that φ ′ ij(t) = 1 2 ( θj − θi + 2(j−i)π N ) and 0 < φij(t) < π for i < j and 0 ≤ t ≤ 1. The terms \|i(g′j(t) + g′i(t))\| ≤ 4π provide a simple bound (no singularities off TNreg). The dominant terms come from ∫ 1 0 \|φ′ij cosφij(t)\| sinφij(t) dt. There are two cases. Let φ0, φ1 satisfy 0 < φ0, φ1 < π and let φ(t) = (1 − t)φ0 + tφ1. The antiderivative ∫ φ′ cosφ(t) sinφ(t) dt = log sinφ(t). The first case applies when either 0 < φ0, φ1 ≤ π 2 or π 2 ≤ φ0, φ1 < π (assign φ0 = π 2 = φ1 to the first interval); then φ′ cosφ(t) ≥ 0 if 0 < φ0 ≤ φ1 ≤ π 2 or π 2 ≤ φ1 ≤ φ0 < π and φ′ cosφ(t) < 0 otherwise. These imply∫ 1 0 \|φ′ cosφ(t)\| sinφ(t) dt = ∣∣∣∣log sinφ1 sinφ0 ∣∣∣∣ . The second case applies when either 0 < φ0 < π 2 < φ1 < π (thus φ′ > 0) or 0 < φ1 < π 2 < φ0 < π. Let φ(t0) = π 2 (that is, t0 = π/2−φ0 φ1−φ0 ). In the first situation∫ 1 0 \|φ′ cosφ(t)\| sinφ(t) dt = ∫ t0 0 φ′ cosφ(t) sinφ(t) dt− ∫ 1 t0 φ′ cosφ(t) sinφ(t) dt = − log sinφ0 − log sinφ1, 20 C.F. Dunkl since log sin π 2 = 0; and the same value holds for the second situation. We obtain∫ 1 0 f(t)dt ≤ \|κ\| ∑ 1≤i<j≤N { − log sin θj − θi 2 − log sin (j − i)π N + 4π } . Taking exponentials and using the lemma (recall ∣∣eiφ1 − eiφ2 ∣∣ = 2 sin ∣∣φ1−φ2 2 ∣∣) we obtain ‖L(x)‖ ≤ c ∏ 1≤i<j≤N \|xi − xj \|−\|κ\|. This bound applies to all of TNreg when L(x0) commutes with υ and L is extended to TNreg as in Definition 3.6. This completes the proof of Theorem 6.1. Next we find bounds on the series expansion from (5.1) L1(x) = (u2 − z2 )N−2∏ j=1 xj −γκ ρ(z−κ, zκ) ∞∑ n=0 αn(x(u, 0))zn, where \|z\| < δ0 := min 1≤j≤N−2 \|u− xj \| and Im z u > 0. Recall the recurrence (5.4) Sn := n−1∑ i=0 αn−1−i { (−1)iβi − σβiσ } , βi := N−2∑ j=0 τ((j,N)) (u− xj)i+1 , α2n(x(u, 0)) = κ 2n S2n, α2n+1(x(u, 0)) = ρ ( κ 2n+ 1− 2κ , κ 2n+ 1 + 2κ ) S2n+1. Proposition 6.3. Suppose \|κ\| ≤ κ0 < 1 2 and λ := (N − 2)κ0 then for n ≥ 0 ‖α2n(x(u, 0))‖ ≤ ‖α0(x(u, 0))‖ (λ)n ( λ+ 1 2 − κ0 ) n n! ( 1 2 − κ0 ) n δ−2n 0 , (6.1) ‖α2n+1(x(u, 0))‖ ≤ ‖α0(x(u, 0))‖ (λ)n+1 ( λ+ 1 2 − κ0 ) n n! ( 1 2 − κ0 ) n+1 δ−2n−1 0 . Proof. Suppose n ≥ 1 then ‖Sn‖ ≤ n−1−i∑ i=0 ‖αn−1−i‖(2N − 4)δ−i−1 0 (since ‖τ((j,N − 1))‖ = 1). Furthermore, since ∣∣ κ n±2κ ∣∣ ≤ κ0 n−2κ0 for n ≥ 2, we find ‖α2n+1‖ ≤ 2λ 2n+ 1− 2κ0 2n∑ i=0 ‖α2n−i‖δ−i−1 0 , ‖α2n‖ ≤ λ n 2n−1∑ i=0 ‖α2n−1−i‖δ−i−1 0 . To set up an inductive argument let tn denote the hypothetical bound on ‖αn(x(u, 0))‖ and set vn = n−1∑ i=0 tn−1−iδ −i−1 0 ; then vn = δ−1 0 (tn−1 + vn−1) for n ≥ 2. Setting t2n = λ nv2n and t2n+1 = 2λ 2n+1−2κ0 v2n+1 the recurrence relations become t2n = λ n ( t2n−1 + 2n− 1− 2κ0 2λ t2n−1 ) δ−1 0 = 2λ+ 2n− 1− 2κ0 2n t2n−1δ −1 0 , A System of Differential Equations on the Torus 21 t2n+1 = 2λ 2n+ 1− 2κ0 ( t2n + n λ t2n ) δ−1 0 = 2λ+ 2n 2n+ 1− 2κ0 t2nδ −1 0 . Starting with ‖α1‖ ≤ 2λ 1−2κ0 ‖α0‖δ−1 0 = t1 the stated bounds are proved inductively. � By use of Stirling’s formula for Γ(n+a) Γ(n+b) ∼ na−b we see that tn behaves like (a multiple of) n2λ−1 for large n. Also there is a constant c′ depending on N and κ0 such that ∞∑ n=2 ‖αn(x(u, 0))‖\|z\|n ≤ c′‖α0(x(u, 0))‖ ( \|z\| δ0 )2( 1− \|z\| δ0 )−2λ−2 . (6.2) We also need to analyze the effect of small changes in u. Fix a point x(ũ, 0) and consider series expansions of αn(x(ũ, 0)) in powers of (u− ũ). Let δ1 := min 1≤j≤N−2 \|ũ−xj \|. Recall equation (5.3) ∂uα0(x(u, 0)) = κα0(x(u, 0)){β0(x(u, 0)) + σβ0(x(u, 0))σ}, and solve this in the form α0(x(u, 0)) = ∞∑ n=0 α0,n(x(ũ, 0))(u− ũ)n. This leads to the recurrence (suppressing the arguments x(ũ, 0)) ∞∑ n=1 nα0,n(u− ũ)n−1 = κ ∞∑ n=0 α0,n(u− ũ)n ∞∑ m=0 (−1)m(u− ũ)m N−2∑ j=0 τ((j,N − 1)) + τ((j,N)) (ũ− xj)m+1 , (n+ 1)α0,n+1 = κ n∑ m=0 α0,n−mβ̃m(x(ũ, 0)), β̃m(x(ũ, 0)) := (−1)m N−2∑ j=0 τ((j,N − 1)) + τ((j,N)) (ũ− xj)m+1 . Thus ‖β̃m‖ ≤ 2(N−2) δm+1 1 and by a similar method as above we find ‖α0,n(x(ũ, 0))‖ ≤ (2λ)n n! δ−n1 ‖α0(x(ũ, 0))‖, (6.3) where λ = (N − 2)κ0. From α1(x(u, 0)) = ρ ( κ 1− 2κ , κ 1 + 2κ ) α0(x(u, 0)) N−2∑ j=0 τ((j,N − 1))− τ((j,N)) (u− xj) = ρ ( κ 1− 2κ , κ 1 + 2κ ) ∞∑ n=0 α0,n(u− ũ)n × ∞∑ m=0 (−1)m(u− ũ)m N−2∑ j=0 τ((j,N − 1))− τ((j,N)) (ũ− xj)m+1 , 22 C.F. Dunkl we can derive a recurrence for the coefficients in α1(x(u, 0)) = ∞∑ n=0 α1,n(x(ũ, 0))(u− ũ)n. Also ‖α1,n(x(ũ, 0))‖ ≤ 2λ 1− 2κ0 ‖α0(x(ũ, 0))‖ n∑ j=0 (2λ)n−j (n− j)! δj−n1 δ−j−1 1 = 2λ(2λ+ 1)n (1− 2κ0)n! δ−n−1 1 ‖α0(x(ũ, 0))‖; note 2λ(2λ+ 1)n = (2λ)n+1. Essentially we are setting up bounds on behavior of L(x(u, z)) for points near x(ũ, 0) in terms of ‖α0(x(ũ, 0))‖ which is handled by the global bound. In the series ∞∑ n=0 αn(x(u, 0))zn = ∞∑ m,n=0 αn,m(x(ũ, 0))zn(u− ũ)m the first order terms are α00(x(ũ, 0)) + α0,1(x(ũ, 0))(u− ũ) + α1,0(x(ũ, 0))z, and the bounds (6.1) for the omitted terms ∞∑ n=2 ‖αn(x(u, 0))‖\|z\|n ≤ c′‖α0(x(u, 0))‖ ( \|z\| δ0 )2( 1− \|z\| δ0 )−2λ−2 , ∞∑ n=2 ‖α0,n(x(ũ, 0))‖\|u− ũ\|n ≤ (2λ)2 ( \|u− ũ\| δ1 )2( 1− \|u− ũ\| δ1 )−2λ−2 ‖α0(x(ũ, 0))‖, (6.4) \|z\| ∞∑ n=1 ‖α1,n(x(ũ, 0))‖\|u− ũ\|n ≤ (2λ)2 1− 2κ0 ( \|z(u− ũ)\| δ2 1 )( 1− \|u− ũ\| δ1 )−2λ−2 ‖α0(x(ũ, 0))‖. Note there is a difference between δ0 and δ1: δ0, δ1 are the distances from the nearest xj (1 ≤ j ≤ N − 2) to u, ũ respectively; thus the double series converges in \|z\| + \|u − ũ\| < δ1 because this implies \|z\| < δ1 − \|u− ũ\| ≤ δ0, by the triangle inequality: δ1 ≤ \|u− ũ\|+ δ0 . 7 Sufficient condition for the inner product property In this section we will use the series L1(y, u− z, u+ z) =  N∏ j=1 xj −γκ ρ(z−κ, zκ) ∞∑ n=0 αn(x(u, 0))zn, normalized by α0 ( x(0) ) = I where x(0) = ( 1, ω, ω2, . . . , ωN−3, ω−3/2, ω−3/2 ) , ω = e2πi/N . The hypothesis is that there exists a Hermitian matrix H such that υH = Hυ (recall υ := τ(w0)) and the matrix H1 defined by L1(x)∗H1L1(x) = L(x)∗HL(x) (7.1) commutes with σ = τ(N − 1, N) (recall L(x0) = I). Setting x = x0 we find that H = L1(x0)∗H1L1(x0). The analogous condition has to hold for each face of C0 and any such face can be obtained from {xN−1 = xN} by applying x 7→ xwm0 with suitable m. For notational simplicity we will work out only the {xN−2 = xN−1} case. From the general relation w(i, j)w−1 = A System of Differential Equations on the Torus 23 (w(i), w(j)) we obtain w−1 0 (N − 1, N)w0 = (N − 2, N − 1). A matrix M commutes with τ(N − 2, N − 1) if and only if υMυ−1 commutes with σ. Let x′ = ( x′1, . . . , x ′ N−3, u− z, u+ z, x′N ) , x′w−1 0 = ( x′N , x ′ 1, . . . , x ′ N−3, u− z, u+ z ) = x, L2(x′) := υ−1L1 ( x′w−1 0 ) υ =  N∏ j=1 xj −γκ υ−1ρ ( z−κ, zκ ) ∞∑ n=0 αn(y, u)υzn. This is a solution of (3.1) by Proposition 3.1. This has the analogous behavior to L1; writing υ−1ρ ( z−κ, zκ ) αn(y, u)υ = { υ−1ρ ( z−κ, zκ ) υ }{ υ−1αn(y, u)υ } implies the relations τ(N − 2, N − 1) { υ−1ρ ( z−κ, zκ ) υ } = { υ−1ρ ( z−κ, zκ ) υ } τ(N − 2, N − 1), τ(N − 2, N − 1) { υ−1αn(y, u)υ } τ(N − 2, N − 1) = (−1)n { υ−1αn(y, u)υ } , n ≥ 0. We claim that the Hermitian matrix H2 defined by L2(x)∗H2L2(x) = L(x)∗HL(x) (7.2) commutes with τ(N−2, N−1). There is a subtle change: the base point x(0) = ( 1, ω, . . . , ωN−2, ω−3/2, ω−3/2 ) is replaced by ( ω, . . . , ωN−2, ω−3/2, ω−3/2, 1 ) and now ωx0 = ( ω, . . . , ωN−1, 1 ) is in the domain of convergence of L2. Set x = ωx0 in (7.2) to obtain L2(ωx0) = υ−1L1 ( ωx0w −1 0 ) υ = υ−1L1(x0)υ, H2 = (L2(ωx0)∗)−1HL2(ωx0)−1 = υ−1(L1(x0)∗)−1υHυ−1L1(x0)−1υ = υ−1(L1(x0)∗)−1HL1(x0)−1υ = υ−1H1υ, because H commutes with υ (and L(ωx0) = L(x0) = I by the homogeneity). Thus H2 commutes with τ(N − 2, N − 1). From Theorem 6.1 we have the bound ‖L(x)∗HL(x)‖ ≤ c ∏ i<j \|xi − xj \|−2\|κ\|. Denote K(x) = L(x)∗HL(x). We will show that there is an interval −κ1 < κ < κ1 where κ1 depends on N such that∫ TN { (xNDNf)∗(x)K(x)g(x)− f∗(xt)K(x)xNDNg(x) } dm(x) = 0, for each f, g ∈ C1 ( TN ;Vτ ) . Consider the Haar measure of { x : min i<j \|xi − xj \| < ε } ; let sin ε′ 2 = ε 2 and i < j then m { x : \|xi − xj \| ≤ ε } = 1 πε ′, thus m { x : min i<j \|xi − xj \| < ε } ≤ ( N 2 ) ε′ π . The integral is broken up into three pieces. The aim is to let δ → 0; where δ satisfies an upper bound δ < min (( 2 sin π N )2 , 1 9 ) ; the first term comes from the maximum spacing of N points on T and the second is equivalent to 3δ < δ1/2. Also δ′ := 2 arcsin δ 2 . 1. min i<j \|xi − xj \| < δ, done with the integrability of ∏ i<j \|xi − xj \|−2\|κ\| for \|κ\| < 1 N (from the Selberg integral ∫ TN ∏ i<j \|xi−xj \|−2\|κ\|dm(x) = Γ1−N \|κ\|) Γ(1−\|κ\|)N ), and the measure of the set is O(δ). The limit as δ → 0 is zero by the dominated convergence theorem. 24 C.F. Dunkl 2. δ ≤ min 1≤i<j≤N−1 \|xi − xj \| < δ1/2 and δ ≤ min 1≤i≤N−1 \|xi − xN \|; this case uses the same bound on K and the TN−1-Haar measure of { x ∈ TN−1 : min 1≤i<j≤N−1 \|xi − xj \| < δ1/2 } ; 3. min 1≤i<j≤N−1 \|xi−xj \| ≥ δ1/2 and min 1≤i≤N−1 \|xi−xN \| ≥ δ. This is done with a detailed analysis using the double series from (6.4). The total of parts (2) and (3), that is, the integral over Ωδ, equals∫ Ωδ xN∂N{f(x)∗K(x)g(x)}dm(x). We use the coordinates xj = eiθj , 1 ≤ j ≤ N ; thus xN∂N = −i ∂ ∂θN . For fixed (θ1, . . . , θN−1) the condition x ∈ Ωδ implies that the set of θN -values is a union of disjoint closed inter- vals (it is possible there is only one, in the extreme case θj = jδ′ for 1 ≤ j ≤ N − 1 the interval is Nδ′ ≤ θN ≤ 2π). In case (2) the θN -integration results in a sum of terms (f∗Kg) ( eiθ1 , . . . , eiθN−1 , eiφ ) with coefficients ±1 where min 1≤i≤N−1 \|φ− θi\| = δ. Each such sum is bounded by 2(N − 1)c‖f‖∞‖g‖∞δ−N(N−1)\|κ\|, because ∏ i<j \|xi−xj \| ≥ δN(N−1)/2 on Ωδ. Thus the integral for part (2) is bounded by 2(N − 1)‖f‖∞‖g‖∞δ−N(N−1)\|κ\| ( N − 1 2 )( 2 arcsin δ1/2 2 ) ≤ c′δ1/2−N(N−1)\|κ\|, for some finite constant c′ (depending on f , g). This term tends to zero as δ → 0 if \|κ\| < 1 2N(N−1) . In part (3) the intervals [θi − δ′, θi + δ′] are pairwise disjoint because \|θi − θj \| ≥ 3δ′ for i 6= j (recall √ δ > 3δ). To simplify the notation assume θ1 < θ2 < · · · (the other cases follow from the group invariance of the setup). Then the θN -integration yields (2π)1−N N−1∑ j=1 ∫ Rδ { (f∗Kg) ( eiθ1 , . . . , eiθj , . . . , ei(θj−δ′) ) −(f∗Kg) ( eiθ1 , . . . , eiθj , . . . , ei(θj+δ ′) )}dθ1 · · · dθN−1, where Rδ := { (θ1, . . . , θN−1) : θ1 < θ2 < · · · < θN−1 < θ1+2π,min ∣∣eiθj−eiθk ∣∣ ≥ √δ}. It suffices to deal with the term with j = N − 1; this allows the use of the double series. It is fairly easy to show that (f∗Kg) ( eiθ1 , . . . , eiθN−1 , ei(θN−1−δ′) ) − (f∗Kg) ( eiθ1 , . . . , eiθN−1 , ei(θN−1+δ′) ) tends to zero with δ but this is not enough to control the integral. The idea is to show that∣∣(f∗Kg) ( eiθ1 , . . . , eiθN−1 , ei(θN−1−δ′) ) − (f∗Kg) ( eiθ1 , . . . , eiθN−1 , ei(θN−1+δ′) )∣∣ ≤ c′′δ1/2−2\|κ\|∣∣(f∗Kg) ( eiθ1 , . . . , eiθN−1 , ei(θN−1+δ′) )∣∣ for some constant c′′. This can then be bounded using the ‖K‖ bound for sufficiently small \|κ\|. Fix y = ( eiθ1 , . . . , eiθN−2 ) and let x(y, u− v, u+ v) denote ( eiθ1 , . . . , eiθN−2 , u− z, u+ z ) . We will use the form K = L∗1H1L1 from (7.1) with two pairs of values along with ũ = eiθN−1 , and set ζ = eiδ′ 1) η(1) = x(y, u1−z1, u1 +z1) = x ( y, eiθN−1 , ei(θN−1+δ′) ) , then u1 = 1 2 ũ(1+ ζ), z1 = 1 2 ũ(ζ−1), u1 − ũ = z1, \|z1\| = δ, 2) η(2) = x(y, u2 − z2, u2 + z2) = x ( y, ei(θN−1−δ′), eiθN−1 ) , then u2 = 1 2 ũ ( 1 + ζ−1 ) , z2 = 1 2 ũ ( 1− ζ−1 ) = ζ−1z1, u2 − ũ = −z2, \|z2\| = δ. A System of Differential Equations on the Torus 25 Let η(3) =x ( y, eiθN−1 , ei(θN−1−δ′) ) =η(2)(N−1, N), then by construction K ( η(3) ) =σK ( η(2) ) σ. We start by disposing of the f and g factors: by uniform continuous differentiability there is a constant c′′′ such that ∥∥f(η(1) ) − f ( η(3) )∥∥ ≤ c′′′δ′ and ∥∥g(η(1) ) − g ( η(3) )∥∥ ≤ c′′′δ′ (same constant for all of TN ). So the error made by assuming f and g are constant is bounded by c′′′δ′ (∥∥K(η(1) )∥∥ + ∥∥K(η(2) )∥∥). The problem is reduced to bounding K ( η(1) ) − σK ( η(2) ) σ. To add more detail about the effect of the ∗-operation on u and z we compute z∗ = 1 2 ( 1 u+ z − 1 u− z ) = − z u2 − z2 , u∗ = 1 2 ( 1 u+ z + 1 u− z ) = u u2 − z2 and if u− z = eiθN−1 , u+ z = eiθN then z = 1 2 ei(θN−1+θN )/2 ( ei(θN−θN−1)/2 − ei(θN−1−θN )/2 ) = iei(θN−1+θN )/2 sin θN − θN−1 2 , z∗ = ie−i(θN−1+θN )/2 sin θN−1 − θN 2 = z, u = 1 2 ei(θN−1+θN )/2 ( ei(θN−θN−1)/2 + ei(θN−1−θN )/2 ) = ei(θN−1+θN )/2 cos θN − θN−1 2 , u∗ = e−i(θN−1+θN )/2 cos θN−1 − θN 2 = u; the ∗-operation agrees with complex conjugate on the torus and ρ(z−κ, zκ)∗ = ρ(z−κ, zκ). The reason for this is to emphasize that L(x)∗ is an analytic function agreeing with the (Hermitian) adjoint of L(x). Thus K ( η(1) ) = ∞∑ n,m=0 αn(x(u1, 0))∗ρ ( z−κ1 , zκ1 )∗ H1ρ ( z−κ1 , zκ1 ) αm(x(u1, 0))(z∗1)mzn1 = ∞∑ n,m=0 αn(x(u1, 0))∗H1ρ ( \|z1\|−2κ, \|z1\|2κ ) αm(x(u1, 0))z1 mzn1 , (7.3) because H1 commutes with σ and hence with ρ(z−κ1 , zκ1 ), and K ( η(3) ) = σK ( η(2) ) σ = ∞∑ n,m=0 (−1)m+nαn(x(u2, 0))∗H1ρ ( \|z2\|−2κ, \|z2\|2κ ) αm(x(u2, 0))z2 mzn2 , (7.4) because σαn(x(u2, 0))σ = (−1)nαn(x(u2, 0)) for n ≥ 0. Now we use the expansion in powers of (u− ũ)n to evaluate K ( η(1) ) −K ( η(3) ) . From the inequality (6.2) ∞∑ n=2 ‖αn(x(u, 0))‖\|z\|n ≤ c′‖α0(x(u, 0))‖ ( \|z\| δ0 )2( 1− \|z\| δ0 )−2λ−2 = c′δ‖α0(x(u, 0))‖ ( 1− δ1/2 )−2λ−2 (7.5) with δ0 = min 1≤j≤N−2 \|u − xj \| = δ1/2 we can restrict the problem to 0 ≤ n,m ≤ 1. The omitted terms in K ( η(1) ) −K ( η(3) ) are bounded by c′′δ1−2\|κ\|‖B1‖‖α0(x(u2, x))‖2, for some constant c′′. Then L1 ( η(1) ) =  N∏ j=1 x (1) j −γκ ρ(z−κ1 , zκ1 ){α00(x(ũ, 0)) + α0,1(x(ũ, 0))(u1 − ũ) +α1,0(x(ũ, 0))z1 +O(δ) } , 26 C.F. Dunkl σL1 ( η(2) ) σ =  N∏ j=1 x (2) j −γκ ρ(z−κ2 , zκ2 ){α00(x(ũ, 0)) + α0,1(x(ũ, 0))(u2 − ũ) −α1,0(x(ũ, 0))z2 +O(δ) } , because σα1(x(u, 0))σ = (−1)nα1(x(u, 0)). The terms O(δ) correspond to the bound in (7.5). Drop the argument x(ũ, 0) for brevity. Combining these with (7.3) and (7.4) we obtain K ( η(1) ) −K ( η(3) ) = {α0,1(u1 − ũ) + α1,0z1}∗H1ρ ( \|z1\|−2κ, \|z1\|2κ ) α00 + α∗00H1ρ ( \|z1\|−2κ, \|z1\|2κ ) {α0,1(u1 − ũ) + α1,0z1} + {α0,1(u1 − ũ) + α1,0z1}∗H1ρ ( \|z1\|−2κ, \|z1\|2κ ) × {α0,1(u1 − ũ) + α1,0z1} − {α0,1(u2 − ũ)− α1,0z2}∗H1ρ ( \|z2\|−2κ, \|z2\|2κ ) α00 − α∗00H1ρ ( \|z2\|−2κ, \|z2\|2κ ) {α0,1(u1 − ũ)− α1,0z1} − {α0,1(u2 − ũ) + α1,0z2}∗H1ρ ( \|z2\|−2κ, \|z2\|2κ ) × {α0,1(u2 − ũ) + α1,0z2}+O(δ). The key fact is that the α∗00H1ρ ( \|z1\|−2κ, \|z1\|2κ ) α00 terms cancel out (\|z1\| = \|z2\|). From ‖α1,0(x(ũ, 0))‖ ≤ (2λ)2 (1−2κ0)δ −1 1 ‖α0(x(ũ, 0))‖ and ‖α0,1(x(ũ, 0))‖ ≤ 2λδ−1 1 ‖α0(x(ũ, 0))‖ (from (6.3)) δ1 = δ0 − \|u\| = δ1/2 − δ = δ1/2(1 − δ1/2). Thus the sum of the first order terms in K ( η(1) ) −K ( η(3) ) is bounded by c′′′‖α0(x(ũ, 0))‖2δ1/2−2\|κ\|(1− δ1/2 )−1‖H1‖, where the con- stant c′′′ is independent of x(ũ, 0) (but is dependent on κ0 and N). Note \|u1 − ũ\| = \|u2 − ũ\| = \|z1\| = \|z2\| = δ. The second last step is to relate ‖α0(x(ũ, 0))‖ to ‖L1 ( η(1) ) ‖; indeed L1 ( η(1) ) =  N∏ j=1 x (1) j −γκ ρ(z−κ1 , zκ1 ){ α0(x(ũ, 0)) + ∞∑ n=1 αn(x(ũ, 0))zn1 } . Similarly to (7.5) ∞∑ n=1 ‖αn(x(ũ, 0))‖\|z\|n ≤ c′‖α0(x(ũ, 0))‖ ( \|z\| δ0 )( 1− \|z\| δ0 )−2λ−1 = c′‖α0(x(ũ, 0))‖δ1/2 ( 1− δ1/2 )−2λ−1 ≤ c′′‖α0(x(ũ, 0))‖δ1/2, (if δ < 1 9 then 1− δ1/2 > 2 3); thus ‖α0(x(ũ, 0))‖ ( 1− c′′δ1/2 ) ≤ δ−\|κ\| ∥∥L1 ( η(1) )∥∥. By Theorem 6.1 ∥∥L(η(1) )∥∥ ≤ c ∏ 1≤i<j≤N−1 \|xi − xj \|−\|κ\| N−2∏ j=1 ∣∣eiθj − ei(θN−1+δ′) ∣∣−\|κ\|∣∣eiθN−1 − ei(θN−1+δ′) ∣∣−\|κ\| ≤ cδ−\|κ\|{(N+1)(N−2)/2+1}, because the first two groups of terms satisfy the bound \|xi − xj \| ≥ δ1/2. Combining everything we obtain the bound∥∥K(η(1) ) −K ( η(3) )∥∥ ≤ c′′′‖α0(x(ũ, 0))‖2δ1/2−2\|κ\|(1− δ1/2 )−1‖B1‖ ≤ c′′‖H1‖δ1/2−2\|κ\|−\|κ\|{(N+1)(N−2)+2}. A System of Differential Equations on the Torus 27 The constant is independent of η(1) and the exponent on δ is 1 2 − \|κ\| ( N2 − N + 2 ) . Thus the integral of part (3) goes to zero as δ → 0 if \|κ\| < ( 2 ( N2 − N + 2 ))−1 . This is a crude bound, considering that we know everything works for −1/hτ < κ < 1/hτ , but as we will see, an open interval of κ values suffices. Theorem 7.1. If there exists a Hermitian matrix H such that υH = Hυ and (L1(x0)∗)−1HL1(x0)−1 commutes with σ, and − ( 2 ( N2 −N + 2 ))−1 < κ < ( 2 ( N2 −N + 2 ))−1 then∫ TN {(xiDif(x))∗L(x)∗HL(x)g(x)− f(x)∗L(x)∗HL(x)xiDig(x)}dm(x) = 0 for f, g ∈ C(1) ( TN ;Vτ ) and 1 ≤ i ≤ N . It is important that we can derive uniqueness of H from the relation, because the conditions 〈wf,wg〉 = 〈f, g〉, 〈xif, xig〉 = 〈f, g〉, and 〈xiDif, g〉 = 〈f, xiDig〉 for w ∈ SN and 1 ≤ i ≤ N determine the Hermitian form uniquely up to multiplication by a constant. Thus the measure K(x)dm(x) is similarly determined, by the density of Laurent polynomials. 8 The orthogonality measure on the torus At this point there are two logical threads in the development. On the one hand there is a sufficient condition implying the desired orthogonality measure is of the form L∗HLdm, specifically if H commutes with υ, (L1(x0)∗)−1HL1(x0)−1 commutes with σ, and \|κ\| < (2(N2− N + 2))−1. However we have not yet proven that H exists. On the other hand in [3] we showed that there does exist an orthogonality measure of the form dµ = dµS +L∗HLdm where sptµS ⊂ TN\TNreg, H commutes with υ, and −1/hτ < κ < 1/hτ (the support of a Baire measure ν, denoted by spt ν, is the smallest compact set whose complement has ν-measure zero). In the next sections we will show that (L1(x0)∗)−1HL1(x0)−1 commutes with σ and that H is an analytic function of κ in a complex neighborhood of this interval. Combined with the above sufficient condition this is enough to show that there is no singular part, that is, µS = 0. The proof involves the formal differential equation satisfied by the Fourier–Stieltjes series of µ, which is used to show µS = 0 on { x ∈ TN : #{xj}Nj=1 = N − 1 } (that is, x has at least N − 1 distinct components). In turn this implies (L∗1(x0))−1HL1(x0)−1 commutes with σ. The proofs unfortunately are not short. In the sequel H refers to the Hermitian matrix in the formula for dµ and K denotes L∗HL. Also H is positive-definite since the measure µ is positive (else there exists a vector v with Hv = 0 and then the C(1) ( TNreg;Vτ ) function given by f(x) := L(x)−1vg(x) where g is a smooth scalar nonnegative function with support in a sufficiently small neighborhood of x0, has norm 〈f, f〉 = 0, a contradiction). Thus H has a positive-definite square root C which commutes with υ. Now extend CL(x) from C0 to all of TNreg by Definition 3.6 and so K(x) = L∗(x)C∗CL(x) for all x ∈ TNreg (this follows from K(xw) = τ(w)−1K(x)τ(w)). Furthermore ∫ TN ‖K(x)‖dm(x) < ∞ because Kdm is the absolutely continuous part of the finite Baire measure µ. We will show that (L∗1(x0))−1C∗CL1(x0)−1 commutes with σ. The proof begins by establish- ing a recurrence relation for the Fourier coefficients of K(x), which comes from equation (3.3). For F (x) integrable on TN , possibly matrix-valued, and α ∈ ZN let F̂α = ∫ TN F (x)x−αdm(x). Clearly ∫ TN x βF (x)x−αdm(x) = F̂α−β; and if ∂iF (x) is also integrable then (integration-by- parts)∫ TN xi∂iF (x)x−αdm(x) = αi ∫ TN F (x)x−αdm(x). (8.1) 28 C.F. Dunkl For a subset J ⊂ {1, 2, . . . , N} let εJ ∈ NN0 be defined by (εJ)i = 1 if i ∈ J and = 0 otherwise; also εi := ε{i}. For 1 ≤ i ≤ N let Ei := {1, 2, . . . , N}\{i}, Eij := Ei\{j}, pi(x) := ∏ j 6=i (xi − xj) = N−1∑ `=0 (−1)`xN−1−` i ∑ J⊂Ei,#J=` xεJ . Equation (3.3) can be rewritten as pi(x)xi∂iK(x) = κ ∑ j 6=i ∏ ` 6=i,j (xi − x`){xjτ((i, j))K(x) +K(x)τ((i, j))xi}; (8.2) this is a polynomial relation which shows that pi(x)xi∂iK(x) is integrable and which has impli- cations for the Fourier coefficients of K. Proposition 8.1. For 1 ≤ i ≤ N and α ∈ ZN the Fourier coefficients K̂ satisfy N−1∑ `=0 (−1)`(αi + `) ∑ J⊂Ei,#J=` K̂α+`εi−εJ = κ ∑ j 6=i N−2∑ `=0 (−1)` ∑ J⊂Eij ,#J=` { τ((i, j))K̂α+`εi−εj−εJ + K̂α+(l+1)εi−εJ τ((i, j)) } . (8.3) Proof. Multiply both sides of (8.2) by x1−N i ; this makes the terms homogeneous of degree zero. Suppose j 6= i then x1−N i ∏ 6̀=i,j (xi − x`) = ∏ ` 6=i,j ( 1− x` xi ) = N−2∑ `=0 (−1)`x−`i ∑ J⊂Eij ,#J=` xεJ . Multiply the right side by x−αdm(x) and integrate over TN to obtain κ ∑ j 6=i N−2∑ `=0 (−1)` ∑ J⊂Eij ,#J=` { τ((i, j))K̂α+`εi−εj−εJ + K̂α+(l+1)εi−εJ τ((i, j)) } . The sum is zero unless α ∈ ZN where ZN := { α ∈ ZN : N∑ j=1 αj = 0 } , by the homogeneity. For the left side start with (8.1) applied to x1−N i pi(x)xi∂iK(x) (αi +N − 1) ∫ TN pi(x)K(x)x1−N i x−αdm(x) = ∫ TN { (xi∂ipi(x))K(x) + pi(x)(xi∂iK(x)) } x1−N i x−αdm(x),∫ TN pi(x)(xi∂iK(x))x1−N i x−αdm(x) = ∫ TN ((αi +N − 1)pi(x)− xi∂ipi(x))K(x)x1−N i x−αdm(x) = ∫ TN N−1∑ `=0 (−1)`(αi + `)x−`i ∑ J⊂Ei,#J=` xεJK(x)x−αdm(x) A System of Differential Equations on the Torus 29 = N−1∑ `=0 (−1)`(αi + `) ∑ J⊂Ei,#J=` K̂α+`εi−εJ . Combining the two sides finishes the proof. If α /∈ ZN then both sides are trivially zero. � This system of recurrences has the easy (and quite undesirable) solution K̂α = I for all α ∈ ZN and 0 otherwise. The right side becomes 2κ ∑ j 6=i τ((i, j)) N−2∑̀ =0 (−1)` ( N−2 ` ) = 0 (for N ≥ 3, an underlying assumption), and the left side is N−1∑̀ =0 (−1)`(αi+`) ( N−1 ` ) I = 0. This K̂ corresponds to the measure 1 2πdθ on the circle { eiθ(1, . . . , 1) : − π < θ ≤ π } . Next we show that µ̂α :=∫ TN x −αdµ(x) satisfies the same recurrences. Proposition 5.2 of [3] asserts that if α, β ∈ NN0 and N∑ j=1 (αj − βj) = 0 then (αi − βi)µ̂α−β = κ ∑ αj>αi αj−αi∑ `=1 τ((i, j))µ̂α+`(εi−εj)−β − κ ∑ αi>αj αi−αj−1∑ `=0 τ((i, j))µ̂α+`(εj−εi)−β − κ ∑ βj>βi βj−βi∑ `=1 µ̂α−`(εi−εj)−βτ((i, j)) + κ ∑ βi>βj βi−βj−1∑ `=0 µ̂α−`(εj−εi)−βτ((i, j)). (8.4) The relation τ(w)∗µ̂wατ(w) = µ̂α is shown in [3, Theorem 4.4]. Introduce Laurent series∑ α∈ZN B (i,j) α xα (i 6= j) satisfying B(i,j) α −B(i,j) α+εi−εj = µ̂α, B (i,j) α−αj(εj−εi) = 0, note α− αj(εj − εi) = ( . . . , i αi + αj , . . . , j 0, . . . , ` α`, . . . ) , ` 6= i, j. The purpose of the definition is to produce a formal Laurent series satisfying( 1− xj xi )∑ α B(i,j) α xα = ∑ α µ̂αx α. The ambiguity in the solution is removed by the second condition (note that ∑ α ( B (i,j) α − cI ) xα also solves the first equation for any constant c). Proposition 8.2. Suppose i 6= j and α ∈ ZN then B (i,j) α τ((i, j)) = τ((i, j))B (j,i) (i,j)α. Proof. Start with µ̂ατ((i, j)) = τ((i, j))µ̂(i,j)α and the defining relations B(i,j) α τ((i, j))−B(i,j) α+εi−εjτ((i, j)) = µ̂ατ((i, j)), τ((i, j))B (j,i) (i,j)α − τ((i, j))B (j,i) (i,j)α+εj−εi = τ((i, j))µ̂(i,j)α; 30 C.F. Dunkl subtract the second equation from the first: B(i,j) α τ((i, j))− τ((i, j))B (j,i) (i,j)α = B (i,j) α+εi−εjτ((i, j))− τ((i, j))B (j,i) (i,j)α+εj−εi . By two-sided induction B(i,j) α τ((i, j))− τ((i, j))B (j,i) (i,j)α = B (i,j) α+s(εi−εj)τ((i, j))− τ((i, j))B (j,i) (i,j)α+s(εj−εi) for all s ∈ Z, in particular for s = αj where the right hand side vanishes by definition. � Theorem 8.3. For γ ∈ ZN and 1 ≤ i ≤ N γiµ̂γ = κ ∑ j 6=i { −τ((i, j))B(j,i) γ +B(i,j) γ τ((i, j)) } . (8.5) Proof. The proof involves a number of cases (for each (i, j) whether γi ≥ 0 or γi < 0, γj ≥ 0 or γj < 0). Consider equation (8.4), in the terms on the first line (with τ((i, j)) acting on the left) use the substitution µ̂δ = B (j,i) δ − B(j,i) δ−εi+εj , and for the terms on the second line (with τ((i, j)) acting on the right) use the substitution µ̂δ = B (i,j) δ − B(i,j) δ+εi−εj . Set α` = max(γ`, 0) and β` = max(0,−γ`) for 1 ≤ ` ≤ N , thus γ = α−β. The left hand side is (αi−βi)µ̂α−β = γiµ̂γ . We consider two possibilities separately: (i) αi ≥ 0, βi = 0; (ii) αi = 0, βi > 0; and describe the typical τ((i, j)) terms. The sums over ` telescope. In the following any term of the form τ((i, j))µ̂· or µ̂·τ((i, j)) not mentioned explicitly is zero. Proposition 8.2 is used in each case. For case (i) and αj > αi τ((i, j)) αj−αi∑ `=1 µ̂α+`(εi−εj)−β = τ((i, j)) αj−αi∑ `=1 ( B (j,i) γ+`(εi−εj) −B (j,i) γ+`(εi−εj)−εi+εj ) = τ((i, j)) αj−αi∑ `=1 ( B (j,i) γ+`(εi−εj) −B (j,i) γ+(`−1)(εi−εj) ) = τ((i, j)) ( B (j,i) γ+(αj−αi)(εi−εj) −B (j,i) γ ) = τ((i, j)) ( B (j,i) (i,j)γ −B (j,i) γ ) = −τ((i, j))B(j,i) γ +B(i,j) γ τ((i, j)) For case (i) and αi > αj ≥ 0 = βj −τ((i, j)) αi−αj−1∑ `=0 µ̂α+`(εj−εi)−β = −τ((i, j)) αi−αj−1∑ `=0 ( B (j,i) γ+`(εj−εi) −B (j,i) γ+(`+1)(εi−εj) ) = −τ((i, j)) ( B(j,i) γ −B(j,i) (i,j)γ ) = −τ((i, j))B(j,i) γ +B(i,j) γ τ((i, j)), note γ + (αi − αj)(εj − εi) = (i, j)γ. For case (i) and αi > αj = 0 > −βj −τ((i, j)) αi−1∑ `=0 µ̂α+`(εj−εi)−β = −τ((i, j)) αi−1∑ `=0 ( B (j,i) γ+`(εj−εi) −B (j,i) γ+(`+1)(εj−εi) ) = −τ((i, j)) ( B(j,i) γ −B(j,i) γ+γi(εj−εi) ) , − βj∑ `=1 µ̂α−`(εi−εj)−βτ((i, j)) = − βj∑ `=1 ( B (i,j) γ−`(εi−εj) −B (i,j) γ−`(εi−εj)+εi−εj ) τ((i, j)) A System of Differential Equations on the Torus 31 = − βj∑ `=1 ( B (i,j) γ−`(εi−εj) −B (i,j) γ−(`−1)(εi−εj) ) τ((i, j)) = ( B(i,j) γ −B(i,j) γ+γj(εi−εj) ) τ((i, j)) let δ = γ+γi(εj−εi) then δk = γk for k 6= i, j, δi = 0, and δj = γi+γj ; also (i, j)δ = γ+γj(εi−εj). Thus the sum of the terms for this case is −τ((i, j))B(j,i) γ +B(i,j) γ τ((i, j)) + τ((i, j))B (j,i) δ +B (i,j) (i,j)δτ((i, j)) = −τ((i, j))B(j,i) γ +B(i,j) γ τ((i, j)). For case (ii) and βj = −γj > βi = −γi > 0 − βj−βi∑ `=1 µ̂α−`(εi−εj)−βτ((i, j)) = − βj−βi∑ `=1 ( B (i,j) γ−`(εi−εj) −B (i,j) γ−(`−1)(εi−εj) ) τ((i, j)) = ( −B(i,j) γ−(γi−γj)(εi−εj) +B(i,j) γ ) τ((i, j)) = ( −B(i,j) (i,j)γ +B(i,j) γ ) τ((i, j)) = −τ((i, j))B(j,i) γ +B(i,j) γ τ((i, j)). For case (ii) and βi > βj = −γj ≥ 0 (and αj = 0) βi−βj−1∑ `=0 µ̂α−`(εj−εi)−βτ((i, j)) = βi−βj−1∑ `=0 ( B (i,j) γ−`(εj−εi) −B (i,j) γ−(`+1)(εj−εi) ) τ((i, j)) = ( B(i,j) γ −B(i,j) γ−(γj−γi)(εj−εi) ) τ((i, j)) = ( B(i,j) γ −B(i,j) (i,j)γ ) τ((i, j)) = −τ((i, j))B(j,i) γ +B(i,j) γ τ((i, j)). For case (ii) and −βi = γi < 0 < γj = αj (and βj = 0) τ((i, j)) αj∑ `=1 µ̂α+`(εi−εj)−β + βi−1∑ `=0 µ̂α−`(εj−εi)−βτ((i, j)) = τ((i, j)) αj∑ `=1 ( B (j,i) γ+`(εi−εj) −B (j,i) γ+(`−1)(εi−εj) ) + βi−1∑ `=0 ( B (i,j) γ−`(εj−εi) −B (i,j) γ−(`+1)(εj−εi) ) τ((i, j)) = τ((i, j)) ( B (j,i) γ+γj(εi−εj) −B (j,i) γ ) + ( B(i,j) γ −B(i,j) γ+γi(εj−εi) ) τ((i, j)) = −τ((i, j))B(j,i) γ +B(i,j) γ τ((i, j)), because (i, j)(γ+γj(εi−εj)) = γ+γi(εj−εi). In the trivial case γi = γj so that (i, j)γ = γ where are no nonzero τ((i, j)) terms the equation −τ((i, j))B (j,i) γ − B(i,j) γ τ((i, j)) = 0 applies. Thus in each case and for each j 6= i the right hand side contains the expression −κ ( τ((i, j))B (j,i) γ − B (i,j) γ τ((i, j)) ) . � In the following there is no implied claim about convergence, because any term xα appears only a finite number of times in the equation. 32 C.F. Dunkl Theorem 8.4. For 1 ≤ i ≤ N the formal Laurent series F (x) := ∑ α∈ZN µ̂αx α satisfies the equation pi(x)xi∂iF (x) = κ ∑ j 6=i ∏ ` 6=i,j (xi − x`){xjτ((i, j))F (x) + F (x)τ((i, j))xi}. (8.6) Proof. Start with multiplying equation (8.5) by x1−N i pi(x)xγ and sum over γ ∈ ZN to obtain N∏ j=1, j 6=i ( 1− xj xi ) ∑ γ∈ZN γiµ̂γx γ = κ ∑ j 6=i ∏ k 6=i,j ( 1− xk xi )( 1− xj xi ) × −τ((i, j)) ∑ γ∈ZN B(j,i) γ xγ + ∑ γ∈ZN B(i,j) γ xγτ((i, j))  . By construction( 1− xj xi ) ∑ γ∈ZN B(i,j) γ xγ = ∑ γ∈ZN ( B(i,j) γ −B(i,j) γ+εi−εj ) xγ = ∑ γ∈ZN µ̂γx γ and ( 1− xj xi ) ∑ γ∈ZN B(j,i) γ xγ = −xj xi ( 1− xi xj ) ∑ γ∈ZN B(j,i) γ xγ = −xj xi ∑ γ∈ZN µ̂γx γ . Thus the equation becomes N∏ j=1, j 6=i ( 1− xj xi ) ∑ γ∈ZN γiµ̂γx γ = κ ∑ j 6=i ∏ k 6=i,j ( 1− xk xi )xjxi τ((i, j)) ∑ γ∈ZN µ̂γx γ + ∑ γ∈ZN µ̂γx γτ((i, j))  . This completes the proof. � Corollary 8.5. The coefficients {µ̂α} satisfy the same recurrences as { K̂α } in (8.3). 8.1 Maximal singular support Above we showed that µ and K satisfy the same Laurent series differential systems (8.2) and (8.6), thus the singular part µS also satisfies this relation. The singular part µS is the restriction of µ to ⋃ i<j { x ∈ TN : xi = xj } , a closed set. For each pair {k, `} let Ek` = { x ∈ TN : xk 6= x` } , an open subset of TN . For i 6= j let Ti,j = { x ∈ TN : xi = xj } ∩ ⋂ {k,`}∩{i,j}=∅ {Ek` ∩ Eik ∩ Ejk}; this is an intersection of a closed set and an open set, hence Ti,j is a Baire set and the restric- tion µi,j of µ to Ti,j is a Baire measure. Informally Ti,j = { x ∈ TN : xi = xj ,#{xk} = N − 1 } . We will prove that µi,j = 0 for all i 6= j. That is, µS is supported by { x ∈ TN : #{xk} ≤ N −2 } A System of Differential Equations on the Torus 33 (the number of distinct coordinate values is ≤ N − 2). In [3, Corollary 4.15] there is an approx- imate identity σN−1 n (x) := n∑ k=0 (−n)k (1− n−N)k ∑ α∈ZN , \|α\|=2k xα, which satisfies σN−1 n (x) ≥ 0 and σN−1 n ∗ν → ν as n→∞, in the weak-∗ sense for any finite Baire measure ν on TN/D (referring to functions and measures on TN homogeneous of degree zero as Laurent series). The set Ti,j is pointwise invariant under (i, j) thus dµi,j(x) = dµi,j(x(i, j)) = τ((i, j))dµi,j(x)τ((i, j)). Remark 8.6. The density of Laurent polynomials in C(1) ( TN ) can be shown by using an approximate identity, for example: un(x) = { 1 n+1 n∑ j=−n (n− \|j\|+ 1)xj1 } σN−1 n (x); for any α ∈ ZN the coefficient of xα in un(x) tends to 1 as n→∞ (express α = (α1−m)ε1 + (−m,α2, . . . , αN ) where m = N∑ j=2 αj). Then f ∗ un → f in the C(1) ( TN ) norm. Let Ks n = σN−1 n ∗ µS (convolution), a Laurent polynomial, fix ` in 1 ≤ ` ≤ N , and consider the functionals F`,n, G`,n on scalar functions p ∈ C(1) ( TN ) F`,n(p) := ∫ TN p(x) ∏ j 6=` ( 1− xj x` ) x`∂`K s n(x)dm(x), G`,n(p) := κ ∑ i 6=` ∫ TN p(x) ∏ j 6=`,i ( 1− xj x` ){ xi x` τ (`, i)Ks n(x) +Ks n(x)τ(`, i) } dm(x). By construction the functionals annihilate xα for α /∈ ZN . For a fixed α ∈ ZN the value F`,n(x−α)−G`,n(x−α) is α`Aαbn(α) + N−1∑ i=1 (−1)i ∑ J⊂E`,#J=i (α` + i)Aα+iε`−εJ bn(α+ iε` − εJ) −κ N∑ j=1, j 6=` N−2∑ i=0 (−1)` ∑ J⊂E`,j ,#J=i { τ(`, j)Aα+(i+1)ε`−εj−εJ bn(α+ (i+ 1)ε` − εj − εJ) +Aα+iε`−εJ τ(`, j)bn(α+ iε` − εJ) } , where bn(γ) := (−n)\|γ\|/2 (1−N−n)\|γ\|/2 (from the Laurent series of σN−1 n ), and Aγ := ∫ TN x −γdµS . Thus for fixed α the coefficients bn(·) → 1 as n → ∞ and the expression tends to the differential system 8.2 and lim n→∞ ( F`,n(x−α)−G`,n(x−α) ) = 0. This result extends to any Laurent polynomial by linearity. From the approximate identity property lim n→∞ G`,n(p) = κ ∑ i 6=` ∫ TN p(x) ∏ j 6=`,i ( 1− xj x` ){ xi x` τ(`, i)dµS(x) + dµS(x)τ(`, i) } , and ‖G`,n(p)‖ ≤M sup x, i ∣∣∣∣∣∣p(x) ∏ j 6=`,i ( 1− xj x` )∣∣∣∣∣∣ , 34 C.F. Dunkl where M depends on µS . Also F`,n(p) = − ∫ TN x`∂` p(x) ∏ j 6=` ( 1− xj x` )Kn(x)dm(x), and lim n→∞ F`,n(p) = − ∫ TN x`∂` p(x) ∏ j 6=` ( 1− xj x` ) dµS(x) for Laurent polynomials p. By density of Laurent polynomials in C(1) ( TN/D ) (D={(u, u, . . . , u) : \|u\| = 1} thus functions homogeneous of degree zero on TN can be considered as functions on the quotient group TN/D) we obtain − ∫ TN x`∂` p(x) ∏ j 6=` ( 1− xj x` ) dµS(x) = κ ∑ i 6=` ∫ TN p(x) ∏ j 6=`,i ( 1− xj x` ){ xi x` τ (`, i) dµS(x) + dµS(x)τ(`, i) } , (8.7) for all p ∈ C(1) ( TN/D ) . Theorem 8.7. For 1 ≤ i < j ≤ N the restriction µS \|Ti,j = 0. Proof. It suffices to take i = 1, j = 2. Let E be an open neighborhood of a point in T1,2 such that if x ∈ E (the closure) and xi = xj for some pair i < j then i = 1 and j = 2. Let f(x) ∈ C(1) ( TN/D ) have support ⊂ E. Thus f(x) = 0 = ∂1f(x) at each point x such that xi = xj for some pair {i, j} 6= {1, 2} (f = 0 on a neighborhood of ⋃ i<j {x : xi = xj}\T1,2). Then in formula (8.7) (with ` = 1) applied to f the measure µS can be replaced with µ1,2. Evaluate the derivative x1∂1 f(x) ∏ j 6=1 ( 1− xj x1 ) = ( 1− x2 x1 ) f(x)x1∂1 ∏ j>2 ( 1− xj x1 ) + f(x) x2 x1 ∏ j>2 ( 1− xj x1 ) + (x1∂1f(x)) ( 1− x2 x1 )∏ j>2 ( 1− xj x1 ) . Each term vanishes on ⋃ i<j {x : xi = xj}\T1,2, and restricted to T1,2 the value is f(x) ∏ j>2 ( 1− xj x1 ) . Thus − ∫ TN x1∂1 f(x) ∏ j 6=1 ( 1− xj x1 )dµS(x) = − ∫ TN x1∂1 f(x) ∏ j 6=1 ( 1− xj x1 )dµ1,2(x) = − ∫ TN f(x) ∏ j>2 ( 1− xj x1 ) dµ1,2(x). The right hand side of the formula reduces to κ ∫ TN f(x) ∏ j>2 ( 1− xj x1 ){ x2 x1 τ(1, 2)dµ1,2(x) + dµ1,2(x)τ(1, 2) } A System of Differential Equations on the Torus 35 = 2κτ(1, 2) ∫ TN f(x) ∏ j>2 ( 1− xj x1 ) dµ1,2(x), since dµ1,2(x)τ(1, 2) = τ(1, 2)dµ1,2(x). Thus the integral is a matrix F (f) such that (I + 2κτ(1, 2))F (f) = 0, which implies F (f) = 0 provided κ 6= ±1 2 . Replacing f(x) by f(x) ∏ j>2 ( 1 − xj x1 )−1 shows that µ1,2 = 0, since E was arbitrarily chosen. � 8.2 Boundary values for the measure In this subsection we will show that K satisfies the weak continuity condition lim xN−1−xN→0 (K(x)−K(x(N − 1, N))) = 0 at the faces of C0 and then deduce that H1 commutes with σ (as described in Theorem 7.1). The idea is to use the inner product property of µ on functions supported in a small enough neighborhood of x(0) = ( 1, ω, . . . , ωN−3, ω−3/2, ω−3/2 ) where µS vanishes, so that only K is involved, then argue that a failure of the continuity condition leads to a contradiction. Let 0 < δ ≤ 2π 3N and define the boxes Ωδ = { x ∈ TN : ∣∣xj − x(0) j ∣∣ ≤ 2 sin δ 2 , 1 ≤ j ≤ N } , Ω′δ = { x ∈ TN−1 : ∣∣xj − x(0) j ∣∣ ≤ 2 sin δ 2 , 1 ≤ j ≤ N − 1 } (so if xj = eiθj then ∣∣θj − 2π(j−1) N ∣∣ ≤ δ, for 1 ≤ j ≤ N −2 and ∣∣θj − (2N−3)π 2 ∣∣ for N −1 ≤ j ≤ N). Then x ∈ Ωδ implies \|xi − xj \| ≥ 2 sin δ 2 for 1 ≤ i < j ≤ N except for i = N − 1, j = N (that is, \|θi − θj \| ≥ δ). Further Ωδ is invariant under (N − 1, N), while Ωδ ∩ Ωδ(i,N) = ∅ for 1 ≤ i ≤ N − 2. For brevity set φ0 = (2N−3)π 2 , eiφ0 = ω−3/2. We consider the identity∫ TN (xNDNf(x))∗dµ(x)g(x)− ∫ TN f(x)∗dµ(x)xNDNg(x) = 0 for f, g ∈ C(1) ( TN ;Vτ ) whose support is contained in Ωδ. Then spt((xNDNf(x))∗g(x)) ⊂ Ωδ and spt(f(x)∗xNDNg(x)) ⊂ Ωδ. The support hypothesis and the construction of Ωδ imply that Ωδ∩ ( TN\TNreg ) ⊂ TN−1,N and thus dµ can be replaced by K(x)dm(x) in the formula. Recall the general identity (4.1) −(xNDNf(x))∗K(x)g(x) + f(x)∗K(x)xNDNg(x) = xN∂N{f(x)∗K(x)g(x)} − κ ∑ 1≤j≤N−1 1 xN − xj { xjf(x(j,N))∗τ((j,N))K(x)g(x) + xNf(x)∗K(x)τ((j,N))g(x(j,N)) } . Specialize to spt(f) ⊂ Ωδ and spt(g) ⊂ Ωδ and x ∈ Ωδ then only the j = N − 1 term in the sum remains, and this term changes sign under x 7→ x(N − 1, N). For ε > 0 let Ωδ,ε = { x ∈ Ωδ : \|xN−1 − xN \| ≥ 2 sin ε 2 } , then∫ Ωδ,ε { xN∂N (f(x)∗K(x)g(x)) + (xNDNf(x))∗K(x)g(x)− f(x)∗K(x)xNDNg(x) } dm(x) = 0, 36 C.F. Dunkl because Ωδ,ε is (N − 1, N)-invariant (similar argument to Proposition 4.1). By integrability lim ε→0+ ∫ Ωδ,ε { (xNDNf(x))∗K(x)g(x)− f(x)∗K(x)xNDNg(x)dm(x) } = 0, hence lim ε→0+ ∫ Ωδ,ε xN∂N (f(x)∗K(x)g(x))dm(x) = 0. Now we use iterated integration. For fixed θN−1 the ranges for θN are obtained by inserting suitable gaps into the interval [φ0 − δ, φ0 + δ] (as usual, x = ( eiθ1 , . . . , eiθN ) ): 1) φ0 − δ ≤ θN−1 ≤ φ0 − δ + ε : [θN−1 + ε, φ0 + δ], 2) φ0 − δ + ε ≤ θN−1 ≤ φ0 + δ − ε : [φ0 − δ, θN−1 − ε] ∪ [θN−1 + ε, φ0 + δ], 3) φ0 + δ − ε ≤ θN−1 ≤ φ0 + δ : [φ0 − δ, θN−1 − ε]. From xN∂N = −i ∂ ∂θN it follows that 1 2π ∫ b a xN∂N (f∗Kg) (( eiθ1 , . . . , eiθN )) dθN = 1 2πi { (f∗Kg) (( eiθ1 , . . . , eiθN−1 , eib )) − (f∗Kg) (( eiθ1 , . . . , eiθN−1 , eia ))} . Since f and g are at our disposal we can take their supports contained in Ωδ/2 then for 0 < ε ≤ δ 4 the dθN -integrals for (1) and (3) vanish and the integrals in (2) have the value 1 2πi { (f∗Kg) (( eiθ1 , . . . , eiθN−1 , ei(θN−1−ε) )) − (f∗Kg) (( eiθ1 , . . . , eiθN−1 , ei(θN−1+ε) ))} . We use the power series (from (5.1)) L1(x(u, z)) = (u2 − z2 )N−2∏ j=1 xj −γκ ρ(z−κ, zκ) ∞∑ n=0 αn(x(u, 0))zn, with the notation x(u, z) = (x1, . . . , xN−2, u− z, u+ z) for x ∈ Ωδ. Recall αn(x(u, 0)) is analytic for a region including Ωδ and σαn(x(u, 0))σ = (−1)nαn(x(u, 0)). Also α0(x(u, 0)) is invertible. As in Section 5 define C1 := CL1(x0)−1 so that L1(x)∗C∗1C1L1(x) = L(x)∗HL(x) on their common domain, and set H1 = C∗1C1. It suffices to use the approximation ∞∑ n=0 αn(x(u, 0))zn = α0(x(u, 0)) +O(\|z\|), uniformly in Ω2π/3N . Let η(1)(x, θ, ε) := ( x1, . . . , xN−2, e iθ, ei(θ+ε) ) , η(2)(x, θ, ε) := ( x1, . . . , xN−2, e i(θ−ε), eiθ ) , η(3)(x, θ, ε) := ( x1, . . . , xN−2, e iθ, ei(θ−ε)) with η(1), η(2) ∈ Ωδ ∩ C0 and η(3) = η(2)(N − 1, N). Set ζ = eiε. Then η(1) = x(u1 − z1, u1 + z1), u1 = 1 2 eiθ(1 + ζ), z1 = 1 2 eiθ(ζ − 1), η(2) = x(u2 − z2, u2 + z2), u2 = 1 2 eiθ ( 1 + ζ−1 ) , z2 = 1 2 eiθ ( 1− ζ−1 ) = ζ−1z1. A System of Differential Equations on the Torus 37 The invariance properties of K imply K ( η(3) ) = σK ( η(2) ) σ. Then K ( η(1) ) = α0(x(u1, 0))∗ρ ( z−κ1 , zκ1 )∗ H1ρ ( z−κ1 , zκ1 ) α0(x(u1, 0)) +O ( \|z1\|1−2\|κ\|), σK ( η(2) ) σ = α0(x(u2, 0))∗ρ ( z−κ2 , zκ2 )∗ σH1σρ ( z−κ2 , zκ2 ) α0(x(u2, 0)) +O ( \|z2\|1−2\|κ\|), because σα0(x(u, 0))σ = α0(x(u, 0)) and σ = ρ(−1, 1) commutes with ρ(z−κ2 , zκ2 ). To express K ( η(1) ) − σK ( η(2) ) σ let A1 = ρ ( z−κ1 , zκ1 )∗ H1ρ ( z−κ1 , zκ1 ) = O ( \|z1\|−2\|κ\|), A2 = ρ ( z−κ2 , zκ2 )∗ σH1σρ ( z−κ2 , zκ2 ) = O ( \|z2\|−2\|κ\|), then K ( η(1) ) − σK ( η(2) ) σ = α0(x(u1, 0))∗A1α0(x(u1, 0))− α0(x(u2, 0))∗A2α0(x(u2, 0)) +O ( \|z1\|1−2\|κ\|). Also u2−u1 = 1 2xN−1ξ1 ( ζ−1−ζ ) = O(\|z1\|) (since \|z1\| = \|1−ζ\|) thus α0(x(u1, 0))−α0(x(u2, 0)) = O(\|z1\|) and K ( η(1) ) − σK ( η(2) ) σ = α0(x(u1, 0))∗(A1 −A2)α0(x(u1, 0)) +O ( \|z1\|1−2\|κ\|). Using the σ-decomposition write H1 := [ H11 H12 H12 ∗ H11 ] , then A1 −A2 =  H11\|z1\|−2κ H12 ( z1 z1 )κ H12 ∗ ( z1 z1 )−κ H11\|z1\|2κ −  H11\|z2\|−2κ −H12 ( z2 z2 )κ −H12 ∗ ( z2 z2 )−κ H11\|z2\|2κ  =  O H12 {( z1 z1 )κ + ( z2 z2 )κ} H12 ∗ {( z1 z1 )−κ + ( z2 z2 )−κ} O  , because \|z2\| = \|z1\|. Next z1 z1 = e2iθ e iε − 1 e−iε − 1 = −ei(2θ+ε), z2 z2 = e2iθ 1− e−iε 1− eiε = −ei(2θ−ε),( z1 z1 )κ + ( z2 z2 )κ = ( −e2iθ )κ{ eiεκ + e−iεκ } = 2 ( −e2iθ )κ cos εκ, where some branch of the power function is used (the interval where it is applied is a small arc of the unit circle), and φ0 − δ1 < θ < φ0 − δ1. We show that H12 = O, equivalently H1 commutes with σ. By way of contradiction suppose some entry hij 6= 0 (1 ≤ i ≤ mτ < j ≤ nτ ). There exist r > 0, δ1 ≥ δ2 > 0 and c ∈ C with \|c\| = 1 such that Re ( 2c ( −e2iθ )κ hij ) > r 38 C.F. Dunkl for φ0 − δ2 ≤ θ ≤ φ0 + δ2. Let p(x) ∈ C(1) ( TN ) such that spt(p) ⊂ Ωδ2/2, 0 ≤ p(x) ≤ 1 and p(x) = 1 for x ∈ Ωδ2/4. Let f(x) = p(x)α0(x(u, 0))−1εi and g(x) = cp(x)α0(x(u, 0))−1εj (for x ∈ Ωδ2/2). Also impose the bound 0 < ε < δ2 4 . Then f ( η(1) )∗ K ( η(1) ) g ( η(1) ) = p ( η(1) )2 c ( z1 z1 )κ hij +O ( \|z1\|1−2\|κ\|), f ( η(3) )∗ σK ( η(2) ) σg ( η(3) ) = −p ( η(3) )2 c ( z2 z2 )κ hij +O ( \|z2\|1−2\|κ\|), Suppose x = ( eiθ1 , . . . , eiθN ) ∈ Ωδ2/2 then p(x) = 1 for φN−1− δ2 4 ≤ θN−1, θN ≤ φN−1− δ2 4 , thus p ( η(1)(x, θ, ε) ) = 1 for φ0− δ2 4 ≤ θ ≤ φ0− δ2 4 −ε and p ( η(3)(x, θ, ε) ) = 1 for φ0− δ2 4 +ε ≤ θ ≤ φ0− δ2 4 . By the continuous differentiability it follows that for φ0− δ2 4 ≤ θ ≤ φ0− δ2 4 both p ( η(1) ) = 1+O(ε) and p ( η(3) ) = 1 +O(ε). Thus p ( η(1) ) f ( η(1) )∗ K ( η(1) ) g ( η(1) ) p ( η(1) ) − p ( η(3) ) f ( η(3) )∗ σK ( η(2) ) σg ( η(3) ) p ( η(3) ) = p ( η(1) )2 c {( z1 z1 )κ + ( z2 z2 )κ} hij +O ( \|z1\|1−2\|κ\|)+O(ε). By construction Re ( c {( z1 z1 )κ + ( z2 z2 )κ} hij ) > r cos εκ, multiply the inequality by p (( eiθ1 , . . . , eiθN−1 , eiθN−1 ))2 and integrate over the (N − 1)-box Ω′δ2 with respect to dmN−1 = ( 1 2π )N−1 dθ1 · · · dθN−1. This integral dominates the integral over Ω′δ2/4, thus Re ∫ Ω′δ2 p ( η(1) )2 c {( z1 z1 )κ + ( z2 z2 )κ} hijdmN−1 ≥ r cos εκ ( δ2 2π )N−1 . This contradicts the limit of the integral being zero as ε→ 0. The ignored parts of the integral are O ( ε1−2\|κ\|) and \|κ\| < 1 2 . We have proven the following: Theorem 8.8. For −1/hτ < κ < 1/hτ the matrix H1 = (L∗1(x0))−1HL1(x0)−1 commutes with σ. 9 Analytic matrix arguments In this section we set up some tools from linear algebra dealing with matrices whose entries are analytic functions of one variable. The aim is to establish the existence of an analytic solution for the matrices described in Theorem 8.8. The key fact is that the solution L1(x;κ) of (3.1) is analytic in κ for \|κ\| < 1 2 , in fact for κ ∈ C\(Z+ 1 2); the series expansion in (5.1) does not apply to κ ∈ Z+ 1 2 and a logarithmic term has to be included for this case. Set bN := (2(N2−N + 2))−1, the bound from Section 7. One would like use analytic continuation to extend the inner product property of L∗HL from the interval −bN < κ < bN to −1/hτ < κ < 1/hτ but the Bochner theorem argument for the existence of µ does not allow κ to be a complex variable. The following arguments work around this obstacle. Theorem 9.1. Suppose M(κ) is an m×n complex matrix with m ≥ n− 1 such that the entries are analytic in κ ∈ Dr := {κ ∈ C : \|κ\| < r}, some r > 0 and rank(M(κ)) = n − 1 for a real interval −r1 < κ < r1 then rank(M(κ)) = n−1 for all κ ∈ Dr except possibly at isolated points λ where rank(M(λ)) < n − 1, and there is a nonzero vector function v(κ), analytic on Dr such that M(κ)v(κ) = 0 and v(κ) is unique up to multiplication by a scalar function. A System of Differential Equations on the Torus 39 Proof. Let M ′(κ) be any n × n submatrix of M(κ) (when m ≥ n), that is, M ′ is composed of n rows of M(κ), then detM ′(κ) is analytic for κ ∈ Dr and by hypothesis detM ′(κ) = 0 for −r1 < κ < r1. This implies detM ′(κ) ≡ 0 for all κ, by analyticity. Thus rank(M(κ)) ≤ n − 1 for all κ ∈ Dr. For each subset J = {j1, . . . , jn−1} with 1 ≤ j1 < · · · < jn−1 ≤ m let MJ,k(κ) be the (n − 1) × (n − 1) submatrix of M(κ) consisting of rows # j1, . . . , jn−1 and deleting column #k, and XJ(κ) := [detMJ,1(κ), . . . ,detMJ,n(κ)], an n-vector of analytic functions. There exists at least one set J for which XJ(0) 6= [0, . . . , 0] otherwise rank(M(0)) < n− 1. By continuity there exists δ > 0 such that at least one detMJ,k(κ) 6= 0 for \|κ\| < δ and v(κ) =[ (−1)k−1 detMJ,k(κ) ]n k=1 is a nonzero vector in the null-space of M(κ) (by Cramer’s rule and the rank hypothesis). The analytic equation M(κ)v(κ) = 0 holds in a neighborhood of κ = 0 and thus for all of Dr. If v(κ) = 0 for isolated points κ1, . . . , κ` in \|κ\| ≤ r2 < r then v(κ) can be multiplied by ∏̀ j=1 ( 1 − κ κj )−aj for suitable positive integers a1, . . . , a` to produce a solution never zero in \|κ\| ≤ r2 < r. (It may be possible that there are infinitely many zeros in the open set Dr.) � We include the parameter in the notations for L and L1. The ∗ operation replaces xj by x−1 j , the constants by their conjugates, and transposing, but κ is unchanged to preserve the analytic dependence, see Definition 3.8. For x ∈ TNreg and real κ the Hermitian adjoint of L1(x0;κ)) agrees with L1(x0;κ)∗. The matrix M(κ) is implicitly defined by the linear system with the unknown B1 B1 = σB1σ, υL∗1(x0;κ)B1L1(x0;κ) = L∗1(x0;κ)B1L1(x0;κ)υ. (9.1) (Recall υ = τ(w0).) The entries of M(κ) are analytic in \|κ\| < 1 2 . The equation B1 = σB1σ implies that B1 has n := m2 τ+(nτ−mτ )2 possible nonzero entries, by the σ-block decomposition. The number of equations m = n2 τ −dim{A : Aυ = υA}. Because w0 and (N −1, N) generate SN and τ is irreducible Aσ = σA and Aυ = υA imply A = cI for c ∈ C by Schur’s lemma. This implies n ≥ m − 1 (else there are two linearly independent solutions). By a result of Stembridge [10, Section 3] n can be computed from the following: (recall ω := exp 2πi N ) for 0 ≤ j ≤ N − 1 set ej equal to the multiplicity of ωj in the list of the nτ eigenvalues of υ and set Fτ (q) := qe0 + qe1 + · · ·+ qeN−1 then Fτ (q) = qn(τ) N∏ i=1 ( 1− qi ) ∏ (i,j)∈τ ( 1− qh(i,j) )−1 mod ( 1− qN ) , where n(τ) := `(τ)∑ i=1 (i − 1)τi and h(i, j) is the hook length at (i, j) in the diagram of τ (note Fτ (1) = nτ ). Thus dim{A : Aυ = υA} = N−1∑ j=0 e2 j . For example let τ = (4, 2) then nτ = 9, mτ = 3 and n = 45 while Fτ (q) = 2 + q + 2q2 + q3 + 2q4 + q5 and dim{A : Aυ = υA} = 15, m = 66. Theorem 9.2. For −1/hτ < κ < 1/hτ there exists a unique Hermitian matrix H such that dµ = L∗HLdm. Also (L1(x0)∗)−1HL1(x0)−1 commutes with σ. Proof. For any Hermitian nτ × nτ matrix B define the Hermitian form 〈f, g〉B := ∫ TN f(x)∗L(x)∗BL(x)g(x)dm(x) 40 C.F. Dunkl for f, g ∈ C(1) ( TN ;Vτ ) . If the form satisfies 〈wf,wg〉B = 〈f, g〉B for all w ∈ SN and 〈xiDif, g〉B = 〈f, xiDig〉B for 1 ≤ i ≤ N then B is determined up to multiplication by a constant. This follows from the density of the span of the nonsymmetric Jack (Laurent) polynomials in C(1) ( TN ;Vτ ) . By Theorem 8.8 there exists a nontrivial solution of the system (9.1) for −1/hτ < κ < 1/hτ . Thus rankM(κ) ≤ n − 1 in this interval. Now suppose that B1 is a non- trivial solution of (9.1) for some κ such that −bN < κ < bN . Then both B(1) := B1 + B∗1 and B(2) := i(B1 − B∗1) are also solutions (by the invariance of (9.1) under the adjoint operation). Let H(i) := L∗1(x0;κ)B(i)L1(x0;κ) for i = 1, 2 then by Theorem 7.1 the forms 〈·, ·〉H(1) and 〈·, ·〉H(2) satisfy the above uniqueness condition. Hence either B1 is Hermitian or B(1) = rB(2) for some r 6= 0 which implies B1 = 1 2(r − i)B(2), that is, B1 is a scalar multiple of a Hermitian matrix. Thus there is a unique (up to scalar multiplication) solution of (9.1) which implies rankM(κ) ≥ n− 1 in −bN < κ < bN . Hence the hypotheses of Theorem 9.1 are satisfied, and there exists a nontrivial solution B1(κ) which is analytic in \|κ\| < 1 2 . Since the Hermitian form is positive definite for −1/hτ < κ < 1/hτ we can use the fact that B1(κ) is a multiple of a positive-definite matrix when κ is real (in fact, of the matrix H1 arising from µ as in Section 8) and its trace is nonzero (at least on a complex neighborhood of {κ : − 1/hτ < κ < 1/hτ} by continuity). Set B′1(κ) := ( nt/ nτ∑ i=1 B1(κ)ii ) B1(κ), analytic and tr(B′1(κ)) = 1 thus the normalization produces a unique analytic (and Hermitian for real κ) matrix in the null-space of M(κ). Let H(κ) = L1(x0;κ)∗B1(κ)L1(x0;κ) then for fixed f, g ∈ C(1) ( TN ;Vτ ) and 1 ≤ i ≤ N∫ TN { (xiDif(x))∗L∗(x;κ)H(κ)L(x;κ)g(x) −f(x)∗L∗(x;κ)H(κ)L(x;κ)xiDig(x) } dm(x) is an analytic function of κ which vanishes for −bN < κ < bN hence for all κ in −1/hτ < κ < 1/hτ ; this condition is required for integrability. This completes the proof. � By very complicated means we have shown that the torus Hermitian form for the vector- valued Jack polynomials is given by the measure L∗HLdm. The orthogonality measure we constructed in [3] is absolutely continuous with respect to the Haar measure. We conjecture that L∗(x;κ)H(κ)L(x;κ) is integrable for −1/τ1 < κ < 1/`(τ) but H(κ) is not positive outside \|κ\| < 1/hτ (the length of τ is `(τ) := max{i : τi ≥ 1}). In as yet unpublished work we have found explicit formulas for L∗HL for the two-dimensional representations (2, 1) and (2, 2) of S3 and S4 respectively, using hypergeometric functions. It would be interesting to find the normalization constant, that is, determine the scalar multiple of H(κ) which results in 〈1 ⊗ T, 1 ⊗ T 〉H(κ) = 〈T, T 〉0 (see (2.1)) the “initial condition” for the form. In [3, Theorem 4.17(3)] there is an infinite series for H(κ) but it involves all the Fourier coefficients of µ. Acknowledgement Some of these results were presented at the conference “Dunkl operators, special functions and harmonic analysis” held at Universität Paderborn, Germany, August 8–12, 2016. References [1] Beerends R.J., Opdam E.M., Certain hypergeometric series related to the root system BC, Trans. Amer. Math. Soc. 339 (1993), 581–609. [2] Dunkl C.F., Symmetric and antisymmetric vector-valued Jack polynomials, Sém. Lothar. Combin. 64 (2010), Art. B64a, 31 pages, arXiv:1001.4485. [3] Dunkl C.F., Orthogonality measure on the torus for vector-valued Jack polynomials, SIGMA 12 (2016), 033, 27 pages, arXiv:1511.06721. https://doi.org/10.2307/2154288 https://doi.org/10.2307/2154288 http://arxiv.org/abs/1001.4485 https://doi.org/10.3842/SIGMA.2016.033 http://arxiv.org/abs/1511.06721 A System of Differential Equations on the Torus 41 [4] Dunkl C.F., Vector-valued Jack polynomials and wavefunctions on the torus, J. Phys. A: Math. Theor. 50 (2017), 245201, 21 pages, arXiv:1702.02109. [5] Dunkl C.F., Luque J.G., Vector-valued Jack polynomials from scratch, SIGMA 7 (2011), 026, 48 pages, arXiv:1009.2366. [6] Felder G., Veselov A.P., Polynomial solutions of the Knizhnik–Zamolodchikov equations and Schur–Weyl duality, Int. Math. Res. Not. 2007 (2007), rnm046, 21 pages, math.RT/0610383. [7] Griffeth S., Orthogonal functions generalizing Jack polynomials, Trans. Amer. Math. Soc. 362 (2010), 6131–6157, arXiv:0707.0251. [8] James G., Kerber A., The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, Vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981. [9] Opdam E.M., Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995), 75–121. [10] Stembridge J.R., On the eigenvalues of representations of reflection groups and wreath products, Pacific J. Math. 140 (1989), 353–396. [11] Stoer J., Bulirsch R., Introduction to numerical analysis, Springer-Verlag, New York – Heidelberg, 1980. https://doi.org/10.1088/1751-8121/aa6fb9 http://arxiv.org/abs/1702.02109 https://doi.org/10.3842/SIGMA.2011.026 http://arxiv.org/abs/1009.2366 https://doi.org/10.1093/imrn/rnm0046 http://arxiv.org/abs/math.RT/0610383 https://doi.org/10.1090/S0002-9947-2010-05156-6 http://arxiv.org/abs/0707.0251 https://doi.org/10.1007/BF02392487 https://doi.org/10.2140/pjm.1989.140.353 https://doi.org/10.2140/pjm.1989.140.353 https://doi.org/10.1007/978-0-387-21738-3 1 Introduction 2 Modules of the symmetric group 2.1 Jack polynomials 3 The differential system 3.1 The adjoint operation on Laurent polynomials and L(x) 4 Integration by parts 5 Local power series near the singular set 5.1 Behavior on boundary 6 Bounds 7 Sufficient condition for the inner product property 8 The orthogonality measure on the torus 8.1 Maximal singular support 8.2 Boundary values for the measure 9 Analytic matrix arguments References

A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus

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