Orthogonality Measure on the Torus for Vector-Valued Jack Polynomials

For each irreducible module of the symmetric group on $N$ objects 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 certain Hermitian forms. These polynomials were studied by the author and J.-G. Luque using a Yang-Baxter graph technique. This paper constructs a matrix-valued measure on the $N$-torus for which the polynomials are mutually orthogonal. The construction uses Fourier analysis techniques. Recursion relations for the Fourier-Stieltjes coefficients of the measure are established, and used to identify parameter values for which the construction fails. It is shown that the absolutely continuous part of the measure satisfies a first-order system of differential equations.


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 polynomials 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 operators. These inner products are invariant under permutations of the coordinates, that is, the symmetric group. One of these inner products is that of L 2 T N , K κ (x)dm(x) , where T N := x ∈ C N : |x j | = 1, 1 ≤ j ≤ N , dm(x) = (2π) −N dθ 1 · · · dθ N , x j = exp(iθ j ), −π < θ j ≤ π, 1 ≤ j ≤ 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 polynomials. Opdam [12] established orthogonality structures on the torus for trigonometric polynomials associated with Weyl groups; the nonsymmetric Jack polynomials form a special case. Details on the derivation of the norm formulae can be found in the treatise by Xu and the author [6, Section 10.6.3]. The weight function K κ turned out to be the square of the base state for the Calogero-Sutherland quantum mechanical model of N identical particles located at x 1 , x 2 , . . . , x N on the circle with a 1/r 2 potential. This means that the particles repel each other with a force corresponding to a potential C|x i − x j | −2 . See Lapointe and Vinet [10] for the construction of wavefunctions in terms of Jack polynomials for this model. More recently Griffeth [8] 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 n th 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 [3] together with Luque [5] investigated the symmetric group case more intensively. The results from these two papers are the foundation for the present work.
Since the torus structure is such an important aspect of the theory of Jack polynomials it seemed like an obvious research topic to find the role of the torus in the vector-valued Jack case. Is there a matrix-valued weight function on the torus for which the vector-valued Jack polynomials are mutually orthogonal? Some explorations in the N = 3 and N = 4 situation showed that the theory is much more complicated than the ordinary (scalar) case. For two-dimensional representations the weight function has hypergeometric function entries (see [4]); this is quite different from the rather natural product 1≤i<j≤N |x i − x j | 2κ , a power of the discriminant.
In this paper we will produce a matrix-valued measure on the torus for which the vectorvalued nonsymmetric polynomials are mutually orthogonal. The result applies to arbitrary irreducible representations of the symmetric groups. In each case there is a permitted range of the parameter. We start with a concise outline of the definitions and construction of the polynomials using the Yang-Baxter graph technique in Section 2, based on [3] and [5]. Section 3 contains the construction of the abstract Hermitian form which is designed to act like an integral over the torus; that is, multiplication by a coordinate function x j is an isometry. The method is algebraic and based on the Yang-Baxter graph. In Section 4 we use techniques from Fourier analysis to produce the desired measure. The Section begins by using the formulae from the previous sections to define the hypothetical Fourier-Stieltjes coefficients, defined on Z N which is the dual group of the torus, a multiplicative group, and then applying a matrix version of a theorem of Bochner about positive-definite functions to get the measure. There is an application of approximate-identity theory using a Cesàro kernel to construct a sequence of positive matrix-valued Laurent polynomials which converges to the orthogonality measure.
Section 5 develops a recurrence relation satisfied by the Fourier-Stieltjes coefficients of the orthogonality measure. The relation allows an inductive calculation for the coefficients (but actual work, even with symbolic computation software, may not be feasible unless the dimensions are reasonably small), and it describes the list of parameter values (certain rational numbers) for which the construction fails.
The scalar weight function on the torus satisfies a first-order differential system, In Section 6 we show that there is an analogous matrix differential system which is solved in a distribution sense by the orthogonality measure. We outline a result asserting that the orthogonality measure restricted to the complement of 1≤i<j≤N {x : x i = x j } is equal to an analytic solution of the differential system times the Haar measure dm. Finally there is Appendix A containing some technical background results.
2 Vector-valued Jack polynomials and the Yang-Baxter graph This is a summary of the definitions and results from [3] and [5]. For and α is called a multi-index or a composition of |α|. We denote two distinguished elements by 0 = (0, 0, . . . , 0), and 1 = (1, 1, . . . , 1). The degree of x α is |α|, and a polynomial is a finite linear combination of monomials. The linear space of all polynomials is denoted by P, and P n = span{x α : |α| = n} is the subspace of polynomials homogeneous of degree n. The specific polynomials considered here have coefficients in Q(κ) where κ is transcendental (indeterminate) but which will also take on certain real values. The multi-indices α have an important partial order: let α + denote the nonincreasing rearrangement of α, for example if α = (1, 2, 1, 4) then α + = (4, 2, 1, 1). Let N N,+ 0 denote the set of partition multi-indices, that is, λ ∈ N N 0 : λ 1 ≥ λ 2 ≥ · · · ≥ λ N .
For example (3, 2, 1) (0, 2, 4) (4, 0, 2), while (4, 1, 1), (3, 3, 0) are not -comparable. The symmetric group S N , the set of permutations of {1, 2, . . . , N }, acts on C N by permutation of coordinates. The action is extended to polynomials by wp(x) = p(xw) where (xw) i = x w(i) (consider x as a row vector and w as a permutation matrix, [w] ij = δ i,w(j) , then xw = x[w]). This is a representation of S N , that is, w 1 (w 2 p)(x) = (w 2 p)(xw 1 ) = p(xw 1 w 2 ) = (w 1 w 2 )p(x) for all w 1 , w 2 ∈ S N . Furthermore S N is generated by reflections in the mirrors {x : x i −x j = 0} for 1 ≤ i < j ≤ N . These are transpositions, denoted by (i, j), interchanging x i and x j . Define the S N -action on The simple reflections s i := (i, i + 1), 1 ≤ i ≤ N − 1, generate S N . They are the key devices for applying inductive methods, and satisfy the braid relations: We consider the situation where the group S N acts on the range as well as on the domain of the polynomials. We use vector spaces (called S N -modules) on which S N has an irreducible unitary (orthogonal) representation: τ : . See James and Kerber [9] for representation theory, including a modern discussion of Young's methods. We will specify an orthogonal basis and the images τ (s i ) for each i, which suffices for our purposes. Identify τ with a partition of N : (τ 1 , τ 2 , . . .) ∈ N N,+ 0 such that |τ | = N . The length of τ is (τ ) = max{i : τ i > 0}. There is a Ferrers diagram of shape τ (this diagram is given the same name), with boxes at points (i, j) with 1 ≤ i ≤ (τ ) and 1 ≤ j ≤ τ i . 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, that is, we require dim V τ ≥ 2. The hook-length of the node (i, j) ∈ τ is defined to be We will need the key quantity h τ := hook(τ ; 1, 1) = τ 1 + (τ ) − 1, the maximum hook-length of the diagram. Denote the set of RSYT's of shape τ by Y(τ ) and let V τ := span{T : For 1 ≤ i ≤ N and T ∈ Y(τ ) the entry i is at coordinates (rw(i, T ), cm(i, T )) and the content is  The representation theory can be developed using the content vectors in place of tableaux; this is due to Okounkov and Vershik [14].

Description of the representation τ
The formulae for the action of τ (s i ) on the basis Y(τ ) are from Murphy [11,Theorem 3.12].
3. if rw(i, T ) < rw(i + 1, T ) (then cm(i, T ) > cm(i + 1, T )), position To eliminate extra parentheses we will write τ (i, j) for τ ((i, j)); where (i, j) is a transposition. There is a (unique up to constant multiple) positive Hermitian form on V τ for which τ is unitary (real orthogonal), that is τ (w)S 1 , The verification of the unitary property is based on the relation is an orthogonal matrix with respect to the orthonormal basis T, T (i, j) (with ω N = 0), which commute pairwise and τ (ω i )T = c(i, T )T, for 1 ≤ i ≤ N (see [11,Lemma 3.6]); as usual, τ is extended to a homomorphism of the group algebra CS N by τ ( w b w w) = w b w τ (w).

Vector-valued nonsymmetric Jack polynomials
The main concern of this paper is P τ = P ⊗ V τ , the space of V τ valued polynomials in x, which is equipped with the S N action: extended by linearity to Definition 2.5. The Dunkl and Cherednik-Dunkl operators are (1 ≤ i ≤ N , p ∈ P τ ) The commutation relations analogous to the scalar case hold, that is, The simultaneous eigenfunctions of {U i } are called (vector-valued) nonsymmetric Jack polynomials (NSJP). For generic κ these eigenfunctions form a basis of P τ (we will specify the excluded rational values in the sequel). They have a triangularity property with respect to the partial order . However the structure does not merely rely on leading terms of the type x α ⊗ T . We need the rank function: A consequence is that r α α = α + , the nonincreasing rearrangement of α, for any α ∈ N N 0 . For example if α = (1, 2, 1, 4) then r α = [3, 2, 4, 1] and r α α = α + = (4, 2, 1, 1) (recall wα i = α w −1 (i) ). Also r α = I if and only if α is a partition (α 1 ≥ α 2 ≥ · · · ≥ α N ).
For each α ∈ N N 0 and T ∈ Y(τ ) there is a NSJP ζ α,T with leading term x α ⊗ τ r −1 α T , that is,

The Yang-Baxter graph
The NSJP's can be constructed by means of a Yang-Baxter graph. The details are in [5]; this paper has several figures illustrating some typical graphs. A node consists of (α, T, ξ α.T , r α , ζ α,T ), . . , 1 column-bycolumn in the Ferrers diagram, for example τ = (3, 3, 1) There is an adjacency relation in Y(τ ) based on the positions of the pairs {i, i + 1} and an inversion counter.
Recall from Section 2.1 that there are four types of positions of a given pair {i, i + 1} in T , and in case (3) it is straightforward to check that inv(T (i) ) = inv(T ) + 1.
If α i = α i+1 then r s i α = r α s i . The cycle w 0 := (123 . . . N ) and the affine transformation Φ(a 1 , a 2 , . . . , a N ) := (a 2 , a 3 , . . . , a N , a 1 + 1) are fundamental parts of the construction; and r Φα = r α w 0 for any α, that is, The jumps in the graph, which raise the degree by one, are There are two types of steps, labeled by s i : Observe that this construction is valid provided ξ α, 2. If α i = α i+1 , and the positions of j := r α (i), j + 1 in T are of type (3), that is, c(j, T ) − c(j + 1, T ) ≥ 2 (the definition of r α implies r α (i + 1) = j + 1 and s i r −1 is the result of interchanging j and j + 1 in T ). The leading term is transformed There are two other possibilities corresponding to (1) and (2) for the action of s i on ζ α,T when The proofs that these formulae are mutually compatible for different paths in the graph from the root (0, T 0 ) to a given node (α, T ), use inductive arguments based on the fact that these paths have the same length. The number of jumps is clearly |α| and the number of steps is

Hermitian forms
For a complex vector space V a Hermitian form is a mapping ·, · : 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 P τ which has the properties (for all f, g ∈ P τ ): Thus the uniqueness of the spectral vectors discussed above implies that ζ α,T , ζ β,T = 0 whenever (α, T ) = (β, T ). In particular polynomials homogeneous of different degrees are mutually orthogonal, by the basis property of {ζ α,T }. We can deduce contiguity relations corresponding to the steps described above and implied by the properties of the form. Consider step type (1) with .
A necessary condition that the form be positive-definite (f = 0 implies f, f > 0) is that −1 < b < 1 in each of the possible steps. Since (with j = r α (i) and = r α (i + 1)) the extreme values of (c( , T ) − c(j, T ) ± 1) are ±h τ , and Similarly the steps of type (2) It was shown in [3] (this is a special case of a result of Griffeth [8, Theorem 6.1]) that the definition for λ ∈ N N,+ together with formula (3.2) produce a Hermitian form (called the covariant form) satisfying (3.1) and the additional property x i f, g = f, D i g for all f, g ∈ P τ and 1 ≤ i ≤ N (the bound −1/h τ < κ < 1/h τ for positivity of this form was found by Etingof and Stoica [7]).
Here we want a Hermitian form for which multiplication by any x i is an isometry, that is, x i f, x i g = f, g for all f, g ∈ P τ and 1 ≤ i ≤ N . Heuristically this should involve an integral over the N -torus. The isometry postulate, and the equations (3.1) determine the form uniquely, as will be shown. The postulate x N f, g = f, D N g in the covariant form is used to compute the effect of a jump ζ Φα,T = x N w −1 0 ζ α,T (see (2.1)), that is, to evaluate ζ Φα,T , ζ Φα,T / ζ α,T , ζ α,T . From the proofs in [3, Appendix, Corollary 5, Theorem 10] and [5] we see that the factor (1 + κc(i, T )) λ i arises from ratios of this type. This aspect (here we need the ratio to be 1) motivates the following: the form is extended to all of P τ by linearity in the second variable and Hermitian symmetry, Observe that the formula is invariant when λ is replaced by λ + m 1 = (λ 1 + m, λ 2 + m, . . ., λ N + m) for any m ∈ N. This follows easily from the commutation (where e N := thus U i (e m N ζ α,T ) = (m + α i + κc(r α (i), T ))e m N ζ α,T , and e m N ζ α,T is a simultaneous eigenfunction of {U i } with the same eigenvalues and the same leading term as ζ α+m1,T . Hence ζ α+m1,T = e m N ζ α,T . We now extend the structure of NSJP's to V τ -valued Laurent polynomials, thereby producing a basis: The proof that the form satisfies the properties (3.1) with respect to steps is the same as the one in [3,Propositions 8 and 9], and it suffices to verify the effect of a jump.
The terms in the product for i = 1 and 2 We summarize the key results. We say κ is generic if (α, T ) = (β, T ) implies the spectral vectors ξ α,T = ξ β,T .
This lays the abstract foundation for the next developments.

Fourier-Stieltjes coef f icients on the torus
The torus T N := x ∈ C N : |x i | = 1, 1 ≤ i ≤ N is a multiplicative compact abelian group with dual group Z N . We will use this property to find the measure of orthogonality for the NSJP's on the torus. First we produce the Fourier-Stieltjes coefficients of the hypothetical measure and then use a matrix version of a theorem of Bochner to deduce the existence of the measure.
Introduce the index set Z N and its graded components by Formula (A.1) for #Z N,n is in Appendix A.
Definition 4.2. For γ ∈ Z N the matrix A γ (of size #Y(τ ) × #Y(τ )) is given by |β| let ζ i = min(α i , β i ) for 1 ≤ i ≤ N then x ζ is a factor of both x α and x β ; by Proposition 3.4 For a formal Laurent series h(x) = α∈Z N c α x α let CT(h(x)) = c 0 , the constant term. Then In the next section we investigate analytical properties of the formal series, but first we consider algebraic properties, that is, those not needing any convergence results.
There is a slight abuse of notation here: if x ∈ T N then (x α ) = x −α and f (x) * agrees with the adjoint of the matrix f (x). Proposition 4.6. Suppose n = 0, 1, 2, . . . and w ∈ S N then H n (xw) = τ (w) −1 H n (x)τ (w) and H n (x) * = H n (x).

Proof . Compute
As a consequence we find an important commutation satisfied by a particular point value of H n (x) (recall the N -cycle w 0 = (1, 2, . . . , N )).
We turn to the harmonic analysis significance of the matrices {A α }. For an integrable function f on T N the Fourier transform (coefficient) is where x := (exp(iθ 1 ), . . . , exp(iθ N )) and dm(x) = (2π) −N dθ 1 · · · dθ N ; and for a Baire measure µ on T N the Fourier-Stieltjes transform is We will show that there is a matrix-valued measure µ, positive in a certain sense, such that µ(α) = A α for all α ∈ Z N provided that −1/h τ < κ < 1/h τ . There is a version of a theorem of Bochner about positive-definite functions on a locally compact abelian group which proves this claim. The details of the proof and some consequences are in Appendix A. Let n = dim V τ and identify V τ with C n whose elements are considered as column vectors (in effect we use indices 1 ≤ i ≤ n instead of {T ∈ Y(τ )}). The inner product on C n is u, v := n i=1 u i v i , and the norm is |v| = v, v . Note that u, Av = u * Av. A positive-definite matrix P satisfies u, P u ≥ 0 for all u ∈ C n (this implies P * = P ).
for any finitely supported C n -valued function f on Z N .
Furthermore each µ jj is positive and defines a positive-semidefinite Hermitian form on C T N ; C n (continuous C n -valued functions on T N ) satisfying | f, g F | ≤ B f ∞ g ∞ for f, g ∈ C T N ; C n with B < ∞.
The proof is in Appendix A.1 and Theorem A.3. (In general the measures µ jk are not real-valued for j = k.) For notational simplicity we introduce Let µ = [µ T,T ] be the matrix of measures produced by the theorem, that is, on T N such that for all Laurent polynomials f , g with coefficients in V τ , in particular for all NSJP's f , g.
Of course we want more detailed information about these measures. The first step is to apply an approximate identity, a tool from the convolution structure for measures and functions on the torus. We consider Cesàro summation of the series α A α x α based on summing first over each Z N,n . Set a Laurent polynomial, and the corresponding (C, δ)-kernel (for δ > 0) is defined to be (the The point is that lim n→∞ (−n) k (−n−δ) k = 1 for fixed k. In terms of convolution and σ δ n (α) = (−n) k (−n−δ) k for α ∈ Z N,k for 0 ≤ k ≤ n and = 0 for |α| > 2n (or α / ∈ Z N ). Thus Theorem 4.11. For n ≥ 0 Proof . The product is a sum of terms x α−β with α, β ∈ N N 0 and |α| = n = |β|. For example the term x 0 = 1 appears exactly (N )n n! times, because the number of terms in h n is (N )n n! . Consider a fixed γ ∈ Z N,m for some m with 0 ≤ m ≤ n. The term x γ appears in the product for each pair (α, β) with To finish the proof multiply this relation by n! (N )n and compute Here are approximate identity properties of σ N −1 n ; we use T N /D to refer to functions homogeneous of degree zero. There is a standard formula: Proof . The left side equals (by Fubini's theorem) which equals the right side.
The following is a standard result on approximate identities. Proof . For ε > 0 there exists a Laurent polynomial p on T N /D such that f − p ∞ < ε. Then c α x α for some coefficients c α (and finite M ); thus which tends to zero in norm as n → ∞.
Proof . By Lemma 4.13   (1) K n (x) is positive semi-definite for each x ∈ T N , dm(x) = f, g T for all f, g ∈ P τ ; the limit exists for any f, g ∈ C T N ; V τ and defines a · ∞ -bounded positive Hermitian form.
Proof . Part (1) is a consequence of Theorem A.4. Parts (2) and (3)  Clearly ρ(p) ∞ ≤ p ∞ ; in effect ρ is the projection onto Fourier series supported by Z N . Then To extend this to the form ·, · T express the typical sum n i,j=1 Then (ρ is applied to matrices entry-wise) The convergence properties of Proposition 4.14 imply part (4).

Recurrence relations
As a simple illustration consider Z N,1 where it suffices to find A 1,−1,0...,0 . Introduce the unit basis vectors ε i for Z N (with (ε i ) j = δ ij ), so that (1, −1, 0, . . .) = ε 1 −ε 2 . The relation ( recall the Jucys-Murphy elements ω i := N j=i+1 (i, j) and the action τ (ω i )T = c(i, T )T for T ∈ Y(τ ). Next the equation and A 0 = I. This holds for arbitrary T , T , and τ (ω 1 ) is diagonal with the entry at (T, T ) being c(1, T ) thus Proof . This follows from performing the division in Proof . The statement follows from the equation and the lemma.
The following is one of the main results of this section. Note that it is important to involve the multiplicity of the first part of γ. Theorem 5.3. Suppose γ ∈ Z N,n such that γ π is a partition and γ π 1 = γ π m > γ m+1 then Each of the coefficients A δ appearing on the second line of the equation satisfies δ ∈ Z N,s for some s < n and for A δ on the right-hand side of the first line δ = δ π − γ ν where δ π γ π .
The proof is technical and is presented as Proposition A.5.
Theorem 5.5. The coefficients A α are rational functions of κ and are finite provided Also A −α = A * α and τ (w) * A wα τ (w) = A α for all α ∈ Z N , w ∈ S N and permitted values of κ. Proof . The NSJP ζ α,T is a rational function of κ with no poles in −1/h τ < κ < 1/h τ . The coefficients A α are defined in terms of all the NSJP's and are also rational in κ. In equation (5.2) the operator on the left of A γ is where ω m is the Jucys-Murphy element The largest interval around 0 without poles is − 1 τ 1 −1 < κ < 1 (τ )−1 . As illustration we describe A γ for γ ∈ Z N,2 . Above we showed Next for α = ε 1 + ε 2 and β = 2ε j for 3 ≤ j ≤ N we find j).
The other coefficients for n = 2 are obtained using the relations The dif ferential equation We will show that µ satisfies a differential system in a distributional sense. Let T N reg := T N \ 1≤i<j≤N {x : x i = x j } (this avoids the singularities of the system) and ∂ j := ∂ ∂x j for 1 ≤ j ≤ N . The system is Any solution of this system satisfies N i=1 x i ∂ i K (x) = 0 and thus is homogeneous of degree zero.
The relation x i D i f, g T = f, x i D i g T extends to C (1) T N ; V τ , the continuously differentiable V τ -valued functions, because Laurent polynomials are dense in this space. We have shown That is, x j ∂ j is self-adjoint with respect to f, g := T N f * (x)g(x)dm. The result extends to Specialize to a closed S N -invariant subset E ⊂ T N reg which is the closure of its interior (for example E ε = x ∈ T N : min i<j |x i − x j | ≥ ε for ε > 0), and let f, g ∈ C (1) T N ; V τ have supports contained in E, that is, f (x) = 0 = g(x) for x / ∈ E. For fixed f , g, i let By using , and rearranging the sums we obtain Each integral in the third line is finite because f and g vanish on a neighborhood of 1≤i<j≤N {x : x i = x j }. The terms inside {·} are invariant under the change of variable x → x(i, j), because τ (i, j)K n (x(i, j)) = K n (x)τ (i, j), but the denominator (x i − x j ) changes sign; thus the integrand is odd under (i, j) and the integral vanishes. Because terms like τ (i, j) x i x i −x j g(x) are in C (1) T N ; V τ (assumption on the support of g) we can take the limit as n → ∞ in the second line. By the adjoint property of x i ∂ i we find Thus the fact that lim n→∞ I n = 0 implies (recall the matrix-valued integral notation (4.1)) This statement is valid for all f, g ∈ C (1) T N ; V τ that vanish on a neighborhood of 1≤i<j≤N {x : x i = x j }. The first line of the equation can be written as which expresses the distributional derivative of dµ. Thus the distribution-sense differential system is satisfied by dµ on closed subsets of T N reg . In the scalar case (τ = (N )) the orthogonality weight is known to be (due to [1] for the symmetric Jack polynomials) j κ and the differential equation system reduces to (note τ (w) = 1) The differential system (6.1) could be the subject for an article all by itself, but we can sketch a result about the absolutely continuous part of µ, namely that dµ(x) = L(x) * BL(x)dm(x), x ∈ T N reg where B is a locally constant positive matrix and L(x) is a fundamental solution of 3) The effect of the term γ x i I is to make L(x) homogeneous of degree zero, that is, (the sum of the contents in the diagram of τ ).
The differential system is Frobenius integrable; this means that in the system are equal to each other for all i, j (see [2]). The system is analytic and thus any local solution can be continued analytically to any point in C . . , e iθ N : − π < θ 1 < θ 2 < · · · < θ N < π by C 0 . Since x and ux are in the same component for |u| = 1 we see that x 0 ∈ C 0 (recall x 0 = 1, ω, . . . , ω N −1 , ω = e 2πi/N ). The components are {C 0 w : w ∈ S N , w(1) = 1} (corresponding to the (N − 1)! circular permutations of (1, 2, . . . , N )). Now fix the unique solution L(x) such that L(x 0 w) = I for each w with w(1) = 1. By differentiating L(x) −1 L(x) = I obtain the system satisfied by L −1 : The goal here is to replace f , g in formula (6.2) by L −1 f , L −1 g and deduce the desired result. By use of Substitute this relation in formula (6.2) The formula is valid because L −1 f, L −1 g ∈ C (1) T N reg ; V τ and vanish for x / ∈ E. The first line of the equation is the distributional derivative of µ so the equation is equivalent to Thus all the partial derivatives of the distribution L −1 * dµL −1 vanish and L −1 * dµL −1 = Bdm, where B is constant on each component of T N reg . We conclude dµ(x) = L(x) * BL(x)dm(x) for x ∈ T N reg . Part (1) of Theorem 4.17 implies B is a positive matrix. We have to point out that this result provides no information about the behavior of µ on the singular set i<j {x : x i = x j }. We conjecture that µ does not have a singular part. This question seems a worthy topic for further investigations. Some of the difficulties in this problem come from the behavior of the solutions of system (6.3) in neighborhoods of the sets {x : x i = x j }; there are singularities of order |x i − x j | ±κ . Both signs appear because the eigenvalues of τ (i, j) are 1 and −1; a consequence of the assumption that τ is not one-dimensional.
In particular, if F (0) = I then µ jk ≤ 2 + √ 2. For the space C T N ; C n (continuous functions on T N , values in C n ) define an inner product f, g F := n i,j=1 T N f (x) i g(x) j dµ ij (x), then for α, β ∈ Z N , and 1 ≤ i, j ≤ n (with the standard unit basis vectors ε i of C n ) The following summarizes the above results.
Theorem A.3. The Hermitian form ·, · F is bounded on C T N ; C n , that is | f, g F | ≤ B f ∞ g ∞ for some B < ∞, is positive-semidefinite, and if f , g are finitely supported functions on Z N with values in C n then for f (x) := α f (α)x α and g(x) := α g(α)x α f , g F = α,β f (α) * F (α − β)g(β).
Proof . The bound follows from the uniform bound on µ jk for all j, k, depending only on F (0). Suppose f , g are trigonometric (Laurent) polynomials (equivalent to finite support on Z N ), then .
By definition f , f F ≥ 0 and by the density of the trigonometric polynomials in C T N ; C n it follows that ·, · F is a positive semidefinite (it is possible that f , f F = 0 for some f = 0) Hermitian form.
The next result is used for the approximate identity arguments.
Theorem A.4. Suppose σ ∈ C T N and σ(x) ≥ 0 for all x ∈ T N then each σ * µ ij ∈ C T N and [σ * µ ij (x)] n i,j=1 ∈ M n (C) is positive semidefinite for each x ∈ T N . Proof . Suppose (f i ) n i=1 ∈ C T N ; C n . A convolution formula (similar to that in Lemma 4.13) shows that f (yu) i f (yu) j dµ ij (y)dm(u) ≥ 0, (change-of-variable x = yu) because y → f (yu) ∈ C T N ; C n and the double sum is continuous in u and nonnegative by the above theorem. Now let f (x) = g(x)v where v ∈ C n and g ∈ C T N , g ≥ 0, T N g 2 dm = 1 and g = 0 off an ε-neighborhood of a fixed z ∈ T N (i.e., |x − z| ≥ ε implies g(x) = 0) then Let ε → 0 then the integral tends to n i,j=1 v i (σ * µ ij (z))v j , and this completes the proof.