A System of Differential Equations on the Torus

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

Charles F. DUNKL C.F. Dunkl

Department of Mathematics, University of Virginia,
PO Box 400137, Charlottesville VA 22904-4137, USA<EMAIL_ADDRESS>http://people.virginia.edu/~cfd5z/

Received December 11, 2016, in final form June 02, 2017; Published online June
08, 2017

For each irreducible module of the symmetric group $\mathcal{S}_{N}$ 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
$x_{i}=x_{j}$, $i<j$, 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.

nonsymmetric Jack polynomials; matrix-valued weight function; symmetric group
modules

33C52; 32W50; 35F35; 20C30; 42B05

###### Contents

  1. 1 Introduction
  2. 2 Modules of the symmetric group
    1. 2.1 Jack polynomials
  3. 3 The differential system
    1. 3.1 The adjoint operation on Laurent polynomials and $L(x)$
  4. 4 Integration by parts
  5. 5 Local power series near the singular set
    1. 5.1 Behavior on boundary
  6. 6 Bounds
  7. 7 Sufficient condition for the inner product property
  8. 8 The orthogonality measure on the torus
    1. 8.1 Maximal singular support
    2. 8.2 Boundary values for the measure
  9. 9 Analytic matrix arguments
    1. Acknowledgement

##  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 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
$L^{2}\big{(}\mathbb{T}^{N},K_{\kappa}(x)\mathrm{d}m(x)\big{)}$, where

$\displaystyle\mathbb{T}^{N}:=\big{\\{}x\in\mathbb{C}^{N}\colon|x_{j}|=1,\,1\leq
j\leq N\big{\\}},$
$\displaystyle\mathrm{d}m(x)=(2\pi)^{-N}\mathrm{d}\theta_{1}\cdots\mathrm{d}\theta_{N},\qquad
x_{j}=\exp(\mathrm{i}\theta_{j}),\qquad-\pi<\theta_{j}\leq\pi,\qquad 1\leq
j\leq N,$ $\displaystyle K_{\kappa}(x)=\prod_{1\leq i<j\leq
N}|x_{i}-x_{j}|^{2\kappa},\qquad\kappa>-\frac{1}{N};$

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 polynomials. 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 $n^{\rm th}$ roots of unity and the product of these entries is a
$(n/p)^{\rm th}$ 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
$\tau$ of $N$, and a parameter $\kappa$ restricted to an interval determined
by the partition, namely $-1/h_{\tau}<\kappa<1/h_{\tau}$ where $h_{\tau}$ is
the maximum hook-length of the partition $\tau$. 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 $\mathbb{Z}^{N}$. By using known properties of the
NSJP’s we produced a positive-definite matrix function on $\mathbb{Z}^{N}$ and
this implied the existence of the desired orthogonality measure. Additionally
we showed that the part of the measure supported by $\mathbb{T}_{\rm
reg}^{N}:=\mathbb{T}^{N}\backslash\bigcup_{i<j}\\{x\colon x_{i}=x_{j}\\}$ is
absolutely continuous with respect to the Haar measure $\mathrm{d}m$ 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 $\kappa>1$ has great difficulty with
the singularities of the measure of the form $|x_{i}-x_{j}|^{-2|\kappa|}$. We
use a Cauchy principal-value argument based on a weak continuity condition
across the faces $\\{x\colon x_{i}=x_{j}\\}$ (as an over-simplified one-
dimensional example consider the integral
$\int_{-1}^{1}\frac{\mathrm{d}}{\mathrm{d}x}f(x)\mathrm{d}x$ with
$f(x)=|2x+x^{2}|^{-1/4}$: the integral is divergent but the principal value
$\lim\limits_{\varepsilon\rightarrow
0_{+}}\big{\\{}\int_{-1}^{-\varepsilon}+\int_{\varepsilon}^{1}\big{\\}}f^{\prime}(x)\mathrm{d}x=f(1)-f(-1)+\lim\limits_{\varepsilon\rightarrow
0_{+}}\\{f(-\varepsilon)-f((\varepsilon))\\}$ and
$f(-\varepsilon)-f(\varepsilon)=O\big{(}\varepsilon^{3/4}\big{)}$ hence the
limit exists).

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 $\\{(ux_{1},\ldots$, $ux_{N})\colon|u|=1\\}$. The
purpose of the latter condition is to allow solutions analytic on connected
components of $\mathbb{T}_{\rm reg}^{N}$. Denote the degree of $\tau$ by
$n_{\tau}$. The solutions of the differential system are locally analytic
$n_{\tau}\times n_{\tau}$ matrix functions with initial condition given by a
constant matrix. That is, the solution space is of dimension $n_{\tau}^{2}$
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 $\kappa$ 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 $\kappa$ in
$-1/h_{\tau}<\kappa<1/h_{\tau}$.

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 $1/r^{2}$ 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 demonstrated to have a unique matrix solution $L(x)$ such that
$L(x)^{\ast}L(x)\mathrm{d}m(x)$ 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 $L(x)^{\ast}L(x)$ 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 $L^{\ast}L\mathrm{d}m$.

  * •

Section 5: deriving power series expansions of $L(x)$ near the singular set
$\bigcup_{i<j}\big{\\{}x\in\mathbb{T}^{N}\colon$ $x_{i}=x_{j}\big{\\}}$, in
particular near the set $\\{x\colon x_{N-1}=x_{N}\\}$; description of
commutation properties of the coefficients with respect to the reflection
$(N-1,N)$; the behavior of $L$ across the mirror $\\{x\colon
x_{N-1}=x_{N}\\}$.

  * •

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 $\kappa$ 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

$\displaystyle\big{\\{}\big{(}e^{\mathrm{i}\theta_{1}},e^{\mathrm{i}\theta_{2}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i}\theta_{N-1}}\big{)}\colon\theta_{1}<\theta_{2}<\cdots<\theta_{N-2}<\theta_{N-1}<\theta_{1}+2\pi\big{\\}};$

in turn this property is shown to imply the validity of the sufficient
condition set up in Section 7.

  * •

Section 9: analyticity properties of the solutions of matrix equations with
analytic coefficients; the results are used to extend the validity of the
Hermitian form to the desired interval $-1/h_{\tau}<\kappa<1/h_{\tau}$ from
the smaller interval found in Section 7.

##  2 Modules of the symmetric group

The symmetric group $\mathcal{S}_{N}$, the set of permutations of
$\\{1,2,\ldots,N\\}$, acts on $\mathbb{C}^{N}$ by permutation of coordinates.
For $\alpha\in\mathbb{Z}^{N}$ the norm is
$|\alpha|:=\sum\limits_{i=1}^{N}|\alpha_{i}|$ and the monomial is
$x^{\alpha}:=\prod\limits_{i=1}^{N}x_{i}^{\alpha_{i}}$. Denote
$\mathbb{N}_{0}:=\\{0,1,2,\ldots\\}$. The space of polynomials
$\mathcal{P}:=\operatorname{span}_{\mathbb{C}}\big{\\{}x^{\alpha}\colon\alpha\in\mathbb{N}_{0}^{N}\big{\\}}$.
Elements of
$\operatorname{span}_{\mathbb{C}}\big{\\{}x^{\alpha}\colon\alpha\in\mathbb{Z}^{N}\big{\\}}$
are called Laurent polynomials. The action of $\mathcal{S}_{N}$ 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, $[w]_{ij}=\delta_{i,w(j)}$, then
$xw=x[w]$). This is a representation of $\mathcal{S}_{N}$, 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
$w_{1},w_{2}\in\mathcal{S}_{N}$.

Furthermore $\mathcal{S}_{N}$ is generated by reflections in the mirrors
$\\{x\colon x_{i}=x_{j}\\}$ for $1\leq i<j\leq N$. These are transpositions,
denoted by $(i,j)$, so that $x(i,j)$ denotes the result of interchanging
$x_{i}$ and $x_{j}$. Define the $\mathcal{S}_{N}$-action on
$\alpha\in\mathbb{Z}^{N}$ so that $(xw)^{\alpha}=x^{w\alpha}$

$\displaystyle(xw)^{\alpha}=\prod_{i=1}^{N}x_{w(i)}^{\alpha_{i}}=\prod_{j=1}^{N}x_{j}^{\alpha_{w^{-1}(j)}},$

that is $(w\alpha)_{i}=\alpha_{w^{-1}(i)}$ (take $\alpha$ as a column vector,
then $w\alpha=[w]\alpha$).

The simple reflections $s_{i}:=(i,i+1)$, $1\leq i\leq N-1$, generate
$\mathcal{S}_{N}$. They are the key devices for applying inductive methods,
and satisfy the braid relations:

$\displaystyle s_{i}s_{j}=s_{j}s_{i},\qquad|i-j|\geq 2;$ $\displaystyle
s_{i}s_{i+1}s_{i}=s_{i+1}s_{i}s_{i+1}.$

We consider the situation where the group $\mathcal{S}_{N}$ acts on the range
as well as on the domain of the polynomials. We use vector spaces, called
$\mathcal{S}_{N}$-modules, on which $\mathcal{S}_{N}$ has an irreducible
unitary (orthogonal) representation: $\tau\colon\mathcal{S}_{N}\rightarrow
O_{m}(\mathbb{R})$
$\big{(}\tau(w)^{-1}=\tau\big{(}w^{-1}\big{)}=\tau(w)^{T}\big{)}$. See James
and Kerber [8] for representation theory, including a modern discussion of
Young’s methods.

Denote the set of partitions

$\displaystyle\mathbb{N}_{0}^{N,+}:=\big{\\{}\lambda\in\mathbb{N}_{0}^{N}\colon\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{N}\big{\\}}.$

We identify $\tau$ with a partition of $N$ given the same label, that is
$\tau\in\mathbb{N}_{0}^{N,+}$ and $|\tau|=N$. The length of $\tau$ is
$\ell(\tau):=\max\\{i\colon\tau_{i}>0\\}$. There is a Ferrers diagram of shape
$\tau$ (also given the same label), with boxes at points $(i,j)$ with $1\leq
i\leq\ell(\tau)$ and $1\leq j\leq\tau_{i}$. A tableau of shape $\tau$ is a
filling of the boxes with numbers, and a reverse standard Young tableau (RSYT)
is a filling with the numbers $\\{1,2,\ldots,N\\}$ 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 $(1,1,\ldots,1)$
partitions (the trivial and determinant representations, respectively). We
need the important quantity $h_{\tau}:=\tau_{1}+\ell(\tau)-1$, the maximum
hook-length of the diagram (the hook-length of the node $(i,j)\in\tau$ is
defined to be $\tau_{i}-j+\\#\\{k\colon
i<k\leq\ell(\tau)\&j\leq\tau_{k}\\}+1$). Denote the set of RSYT’s of shape
$\tau$ by $\mathcal{Y}(\tau)$ and let

$\displaystyle V_{\tau}=\operatorname{span}\\{T\colon
T\in\mathcal{Y}(\tau)\\}$

(the field is $\mathbb{C}(\kappa)$) with orthogonal basis $\mathcal{Y}(\tau)$.
For $1\leq i\leq N$ and $T\in\mathcal{Y}(\tau)$ 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 $T\in\mathcal{Y}(\tau)$ is uniquely determined by its content vector
$[c(i,T)]_{i=1}^{N}$. Let $S_{1}(\tau)=\sum\limits_{i=1}^{N}c(i,T)$ (this sum
depends only on $\tau$) and $\gamma:=S_{1}(\tau)/N$. The
$\mathcal{S}_{N}$-invariant inner product on $V_{\tau}$ is defined by

$\displaystyle\langle
T,T^{\prime}\rangle_{0}:=\delta_{T,T^{\prime}}\times\prod_{\begin{subarray}{c}1\leq
i<j\leq N,\\\ c(i,T)\leq
c(j,T)-2\end{subarray}}\left(1-\frac{1}{(c(i,T)-c(j,T))^{2}}\right),\qquad
T,T^{\prime}\in\mathcal{Y}(\tau).$

It is unique up to multiplication by a constant.

The Jucys–Murphy elements $\sum\limits_{j=i+1}^{N}(i,j)$ satisfy
$\sum\limits_{j=i+1}^{N}\tau((i,j))T=c(i,T)T$ and thus the central element
$\sum\limits_{1\leq i<j\leq N}(i,j)$ satisfies $\sum\limits_{1\leq i<j\leq
N}\tau((i,j))T=S_{1}(\tau)T$ for each $T\in\mathcal{Y}(\tau)$. 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
$\tau((N-1,N))$ being

$\displaystyle\left[\begin{matrix}-I_{m_{\tau}}&O\\\
O&I_{n_{\tau}-m_{\tau}}\end{matrix}\right],$

where $n_{\tau}:=\dim V_{\tau}=\\#\mathcal{Y}(\tau)$ and $m_{\tau}$ is given
by $\operatorname{tr}(\tau((N-1,N)))=n_{\tau}-2m_{\tau}$. From the sum
$\sum\limits_{i<j}\tau((i,j))=S_{1}(\tau)I$ it follows that
$\binom{N}{2}\operatorname{tr}(\tau((N-1,N)))=S_{1}(\tau)n_{\tau}$ and
$m_{\tau}=n_{\tau}\big{(}\frac{1}{2}-\frac{S_{1}(\tau)}{N(N-1)}\big{)}$. (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 $\mathcal{P}_{\tau}:=\mathcal{P}\otimes V_{\tau}$, the space of
$V_{\tau}$-valued polynomials, which is equipped with the $\mathcal{S}_{N}$
action:

$\displaystyle w\left(x^{\alpha}\otimes
T\right)=(xw)^{\alpha}\otimes\tau(w)T,\qquad\alpha\in\mathbb{N}_{0}^{N},\qquad
T\in\mathcal{Y}(\tau),$ $\displaystyle wp(x)=\tau(w)p(xw),\qquad
p\in\mathcal{P}_{\tau},$

extended by linearity. There is a parameter $\kappa$ which may be
generic/transcendental or complex.

######  Definition 2.1.

The Dunkl and Cherednik–Dunkl operators are ($1\leq i\leq N$,
$p\in\mathcal{P}_{\tau}$)

$\displaystyle\mathcal{D}_{i}p(x):=\frac{\partial}{\partial
x_{i}}p(x)+\kappa\sum_{j\neq i}\tau((i,j))\frac{p(x)-p(x(i,j))}{x_{i}-x_{j}},$
$\displaystyle\mathcal{U}_{i}p(x):=\mathcal{D}_{i}(x_{i}p(x))-\kappa\sum_{j=1}^{i-1}\tau((i,j))p(x(i,j)).$

The commutation relations analogous to the scalar case hold:

$\displaystyle\mathcal{D}_{i}\mathcal{D}_{j}=\mathcal{D}_{j}\mathcal{D}_{i},\qquad\mathcal{U}_{i}\mathcal{U}_{j}=\mathcal{U}_{j}\mathcal{U}_{i},\qquad
1\leq i,j\leq N;$ $\displaystyle
w\mathcal{D}_{i}=\mathcal{D}_{w(i)}w,\qquad\forall\,w\in\mathcal{S}_{N};\qquad
s_{j}\mathcal{U}_{i}=\mathcal{U}_{i}s_{j},\qquad j\neq i-1,i;$ $\displaystyle
s_{i}\mathcal{U}_{i}s_{i}=\mathcal{U}_{i+1}+\kappa
s_{i},\qquad\mathcal{U}_{i}s_{i}=s_{i}\mathcal{U}_{i+1}+\kappa,\qquad\mathcal{U}_{i+1}s_{i}=s_{i}\mathcal{U}_{i}-\kappa.$

The simultaneous eigenfunctions of $\\{\mathcal{U}_{i}\\}$ are called (vector-
valued) nonsymmetric Jack polynomials (NSJP). For generic $\kappa$ these
eigenfunctions form a basis of $\mathcal{P}_{\tau}$ (this property fails for
certain rational numbers outside the interval
$-1/h_{\tau}<\kappa<1/h_{\tau}$). There is a partial order on
$\mathbb{N}_{0}^{N}\times\mathcal{Y}(\tau)$ for which the NSJP’s have a
triangular expression with leading term indexed by
$(\alpha,T)\in\mathbb{N}_{0}^{N}\times\mathcal{Y}(\tau)$. The polynomial with
this label is denoted by $\zeta_{\alpha,T}$, homogeneous of degree
$\sum\limits_{i=1}^{N}\alpha_{i}$ and satisfies

$\displaystyle\mathcal{U}_{i}\zeta_{\alpha,T}=\left(\alpha_{i}+1+\kappa
c\left(r_{\alpha}(i),T\right)\right)\zeta_{\alpha,T},\qquad 1\leq i\leq N,$
$\displaystyle
r_{\alpha}(i):=\\#\\{j\colon\alpha_{j}>\alpha_{i}\\}+\\#\\{j\colon 1\leq j\leq
i,\alpha_{j}=\alpha_{i}\\};$

the rank function $r_{\alpha}\in\mathcal{S}_{N}$ and $r_{\alpha}=I$ if and
only if $\alpha$ is a partition. The vector

$\displaystyle[\alpha_{i}+1+\kappa c(r_{\alpha}(i),T)]_{i=1}^{N}$

is called the spectral vector for $(\alpha,T)$. The NSJP structure can be
extended to Laurent polynomials. Let $e_{N}:=\prod\limits_{i=1}^{N}x_{i}$ and
$\boldsymbol{1}:=(1,1,\ldots,1)\in\mathbb{N}_{0}^{N}$, then
$r_{\alpha+m\boldsymbol{1}}=r_{\alpha}$ for any $\alpha\in\mathbb{N}_{0}^{N}$
and $m\in\mathbb{Z}$. The commutation
$\mathcal{U}_{i}(e_{N}^{m}p)=e_{N}^{m}(m+\mathcal{U}_{i})p$ for $1\leq i\leq
N$ and $p\in\mathcal{P}_{\tau}$ imply that $e_{N}^{m}\zeta_{\alpha,T}$ and
$\zeta_{\alpha+m\boldsymbol{1},T}$ have the same spectral vector for any
$m\in\mathbb{N}_{0}$. They also have the same leading term (see [3, Section
2.2]) and hence $e_{N}^{m}\zeta_{\alpha,T}=\zeta_{\alpha+m\boldsymbol{1},T}$
for $\alpha\in\mathbb{N}_{0}^{N}$. This fact allows the definition of
$\zeta_{\alpha,T}$ for any $\alpha\in\mathbb{Z}^{N}$: let
$m=-\min_{i}\alpha_{i}$ then $\alpha+m\boldsymbol{1}\in\mathbb{N}_{0}^{N}$ and
set $\zeta_{\alpha,T}:=e_{N}^{-m}\zeta_{\alpha+m\boldsymbol{1},T}$.

For a complex vector space $V$ a Hermitian form is a mapping
$\langle\cdot,\cdot\rangle\colon V\otimes V\rightarrow\mathbb{C}$ such that
$\langle u,cv\rangle=c\langle u,v\rangle$, $\langle
u,v_{1}+v_{2}\rangle=\langle u,v_{1}\rangle+\langle u,v_{2}\rangle$ and
$\langle u,v\rangle=\overline{\langle v,u\rangle}$ for $u,v_{1},v_{2}\in V$,
$c\in\mathbb{C}$. The form is positive semidefinite if $\langle u,u\rangle\geq
0$ for all $u\in V$. The concern of this paper is with a particular Hermitian
form on $\mathcal{P}_{\tau}$ which has the properties (for all
$f,g\in\mathcal{P}_{\tau},T,T^{\prime}\in\mathcal{Y}(\tau)$,
$w\in\mathcal{S}_{N}$, $1\leq i\leq N$):

$\displaystyle\langle 1\otimes T,1\otimes T^{\prime}\rangle=\langle
T,T^{\prime}\rangle_{0},$ (2.1) $\displaystyle\langle wf,wg\rangle=\langle
f,g\rangle,$ $\displaystyle\langle x_{i}\mathcal{D}_{i}f,g\rangle=\langle
f,x_{i}\mathcal{D}_{i}g\rangle,$ $\displaystyle\langle
x_{i}f,x_{i}g\rangle=\langle f,g\rangle.$

The commutation
$\mathcal{U}_{i}=\mathcal{D}_{i}x_{i}-\kappa\sum\limits_{j<i}(i,j)=x_{i}\mathcal{D}_{i}+1+\kappa\sum\limits_{j>i}(i,j)$
together with $\langle(i,j)f,g\rangle=\langle f,(i,j)g\rangle$ show that
$\langle\mathcal{U}_{i}f,g\rangle=\langle f,\mathcal{U}_{i}g\rangle$ for all
$i$. Thus uniqueness of the spectral vectors (for all but a certain set of
rational $\kappa$ values) implies that
$\langle\zeta_{\alpha,T},\zeta_{\beta,T^{\prime}}\rangle=0$ whenever
$(\alpha,T)\neq(\beta,T^{\prime})$. In particular polynomials homogeneous of
different degrees are mutually orthogonal, by the basis property of
$\\{\zeta_{\alpha,T}\\}$. For this particular Hermitian form, multiplication
by any $x_{i}$ is an isometry for all $1\leq i\leq N$. 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 $\mathbb{C}_{\times}:=\mathbb{C}\backslash\\{0\\}$ and $\mathbb{C}_{\rm
reg}^{N}:=\mathbb{C}_{\times}^{N}\backslash\bigcup\limits_{i<j}\\{x\colon
x_{i}=x_{j}\\}$. The torus is a compact multiplicative abelian group. The
notations for the torus and its Haar measure in terms of polar coordinates are

$\displaystyle\mathbb{T}^{N}:=\big{\\{}x\in\mathbb{C}^{N}\colon|x_{j}|=1,\,1\leq
j\leq N\big{\\}},$
$\displaystyle\mathrm{d}m(x)=(2\pi)^{-N}\mathrm{d}\theta_{1}\cdots\mathrm{d}\theta_{N},\qquad
x_{j}=\exp(\mathrm{i}\theta_{j}),\qquad-\pi<\theta_{j}\leq\pi,\qquad 1\leq
j\leq N.$

Let $\mathbb{T}_{\rm reg}^{N}:=\mathbb{T}^{N}\cap\mathbb{C}_{\rm reg}^{N}$,
then $\mathbb{T}_{\rm reg}^{N}$ has $(N-1)!$ connected components and each
component is homotopic to a circle (if $x$ is in some component then so is
$ux=(ux_{1},\ldots,ux_{N})$ for each $u\in\mathbb{T}$).

######  Definition 2.2.

Let $\omega:=\exp\frac{2\pi\mathrm{i}}{N}$ and
$x_{0}:=\big{(}1,\omega,\ldots,\omega^{N-1}\big{)}$. Denote the connected
component of $\mathbb{T}_{\rm reg}^{N}$ containing $x_{0}$ by
$\mathcal{C}_{0}$.

Thus $\mathcal{C}_{0}$ is the set consisting of
$\big{(}e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N}}\big{)}$ with
$\theta_{1}<\theta_{2}<\cdots<\theta_{N}<\theta_{1}+2\pi$.

In [3] we showed that if $-1/h_{\tau}<\kappa<1/h_{\tau}$ then there exists a
positive matrix-valued measure $\mathrm{d}\mu$ on $\mathbb{T}^{N}$ such that
for $f,g\in C^{(1)}\big{(}\mathbb{T}^{N};V_{\tau}\big{)}$,
$w\in\mathcal{S}_{N}$, $1\leq i\leq N$,

$\displaystyle\int_{\mathbb{T}^{N}}f(x)^{\ast}\mathrm{d}\mu(x)g(x)=\int_{\mathbb{T}^{N}}f(xw)^{\ast}\tau(w)^{-1}\mathrm{d}\mu(x)\tau(w)g(xw),$
$\displaystyle\int_{\mathbb{T}^{N}}(x_{i}\mathcal{D}_{i}f(x))^{\ast}\mathrm{d}\mu(x)g(x)=\int_{\mathbb{T}^{N}}f(x)^{\ast}\mathrm{d}\mu(x)x_{i}\mathcal{D}_{i}g(x),$
$\displaystyle\int_{\mathbb{T}^{N}}(1\otimes
T)^{\ast}\mathrm{d}\mu(x)(1\otimes T)=\langle T,T\rangle_{0},\qquad
T\in\mathcal{Y}(\tau).$

We introduced the notation

$\displaystyle
f(x)^{\ast}\mathrm{d}\mu(x)g(x):=\sum\limits_{T,T^{\prime}\in\mathcal{Y}(\tau)}\overline{f(x)_{T}}g(x)_{T^{\prime}}\mathrm{d}\mu_{T,T^{\prime}}(x),$

where $f,g\in\mathcal{P}_{\tau}$ have the components $(f_{T}),(g_{T})$ with
respect to the orthonormal basis

$\displaystyle\big{\\{}\langle T,T\rangle_{0}^{-1/2}T\colon
T\in\mathcal{Y}(\tau)\big{\\}}.$

Thus the Hermitian form $\langle
f,g\rangle=\int_{\mathbb{T}^{N}}f(x)^{\ast}\mathrm{d}\mu(x)g(x)$ satisfies
(2.1). Furthermore we showed that

$\displaystyle\mathrm{d}\mu(x)=\mathrm{d}\mu_{s}(x)+L(x)^{\ast}H(\mathcal{C})L(x)\mathrm{d}m(x),$

where the singular part $\mu_{s}$ is the restriction of $\mu$ to
$\mathbb{T}^{N}\backslash\mathbb{T}_{\rm reg}^{N}$, $H(\mathcal{C})$ is
constant and positive-definite on each connected component $\mathcal{C}$ of
$\mathbb{T}_{\rm reg}^{N}$ 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 $\mu$ has no singular part.

##  3 The differential system

Consider the system (with $\partial_{i}:=\frac{\partial}{\partial x_{i}}$,
$1\leq i\leq N$) for $n_{\tau}\times n_{\tau}$ matrix functions $L(x)$

$\displaystyle\partial_{i}L(x)=\kappa L(x)\left\\{\sum_{j\neq
i}\frac{1}{x_{i}-x_{j}}\tau((i,j))-\frac{\gamma}{x_{i}}I\right\\},\qquad 1\leq
i\leq N,$ (3.1)
$\displaystyle\gamma:=\frac{S_{1}(\tau)}{N}=\frac{1}{2N}\sum_{i=1}^{\ell(\tau)}\tau_{i}(\tau_{i}-2i+1).$

The effect of the term $\frac{\gamma}{x_{i}}I$ is to make $L(x)$ homogeneous
of degree zero, that is, $\sum\limits_{i=1}^{N}x_{i}\partial_{i}L(x)=0$. The
differential system is defined on $\mathbb{C}_{\rm reg}^{N}$, Frobenius
integrable and analytic, thus any local solution can be continued analytically
to any point in $\mathbb{C}_{\rm reg}^{N}$. 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 $\mathbb{C}_{\rm reg}^{N}$ then there is no
change.

Integrability means that $\partial_{i}(\kappa
L(x)A_{j}(x))=\partial_{j}(\kappa L(x)A_{i}(x))$ for $i\neq j$, writing the
system as $\partial_{i}L(x)=\kappa L(x)A_{i}(x)$, $1\leq i\leq N$ (where
$A_{i}(x)$ is defined by equation (3.1)). The condition becomes

$\displaystyle\kappa^{2}L(x)A_{i}(x)A_{j}(x)+\kappa
L(x)\partial_{i}A_{j}(x)=\kappa^{2}L(x)A_{j}(x)A_{i}(x)+\kappa
L(x)\partial_{j}A_{i}(x).$

Since
$\partial_{i}A_{j}(x)=\frac{\tau((i,j))}{(x_{i}-x_{j})^{2}}=\partial_{j}A_{i}(x)$
it suffices to show that $A_{i}(x)^{\prime}:=\sum\limits_{k\neq
i}\frac{\tau((i,k))}{x_{i}-x_{k}}$ and
$A_{j}(x)^{\prime}:=\sum\limits_{\ell\neq
j}\frac{\tau((j,\ell))}{x_{j}-x_{\ell}}$ commute with each other. The product
$A_{i}(x)^{\prime}A_{j}(x)^{\prime}$ is a sum of
$-\frac{1}{(x_{i}-x_{j})^{2}}I$, terms of the form
$\frac{\tau((i,k)(j,\ell))}{(x_{i}-x_{k})(x_{j}-x_{\ell})}+\frac{\tau((i,\ell)(j,k))}{(x_{i}-x_{\ell})(x_{j}-x_{k})}$
for $\\{i,k\\}\cap\\{j,\ell\\}=\varnothing$, and terms involving the
$3$-cycles $(i,j,k)$ and $(j,i,k)$ occurring as

$\displaystyle\frac{\tau((i,j)(j,k))}{(x_{i}-x_{j})(x_{j}-x_{k})}+\frac{\tau((i,k)(j,i))}{(x_{i}-x_{k})(x_{j}-x_{i})}+\frac{\tau((i,k)(j,k))}{(x_{i}-x_{k})(x_{j}-x_{k})}$
$\displaystyle\qquad{}=\frac{\tau((i,j,k))}{(x_{i}-x_{k})(x_{j}-x_{k})}+\frac{\tau((j,i,k))}{(x_{i}-x_{k})(x_{j}-x_{k})},$

(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, $\det L(x)\neq 0$. Recall
Jacobi’s identity:

$\displaystyle\frac{\partial}{\partial t}\det
F(t)=\operatorname{tr}\left(\operatorname{adj}(F(t))\frac{\partial}{\partial
t}F(t)\right),$

where $F(t)$ is a differentiable matrix function and
$\operatorname{adj}(F(t))F(t)=\det F(t)I$, that is,
$\operatorname{adj}(F(t))=\\{\det F(t)\\}F(t)^{-1}$ when $F(t)$ is invertible;
thus

$\displaystyle\partial_{i}\det L(x)=\operatorname{tr}\big{(}\\{\det
L(x)\\}L(x)^{-1}\partial_{i}L(x)\big{)}=\kappa\det
L(x)\operatorname{tr}A_{i}(x).$

This can be solved: from $\sum\limits_{i<j}\tau((i,j))=S_{1}(\tau)I$ it
follows that
$\operatorname{tr}(\tau((i,j)))=\binom{N}{2}^{-1}S_{1}(\tau)n_{\tau}=\frac{2}{N-1}\gamma
n_{\tau}$ (and $n_{\tau}=\\#\mathcal{Y}(\tau)$). We obtain the system

$\displaystyle\partial_{i}\det L(x)=\kappa\gamma
n_{\tau}\left\\{\frac{2}{N-1}\sum_{j\neq
i}\frac{1}{x_{i}-x_{j}}-\frac{1}{x_{i}}\right\\}\det L(x),\qquad 1\leq i\leq
N.$

By direct verification

$\displaystyle\det L(x)=c\prod\limits_{1\leq i<j\leq
N}\left(-\frac{(x_{i}-x_{j})^{2}}{x_{i}x_{j}}\right)^{\kappa\lambda/2},\qquad\lambda:=\frac{\gamma
n_{\tau}}{2(N-1)}=\operatorname{tr}(\tau((1,2))),$

is a local solution for any $c\in\mathbb{C}_{\times}$, if
$x_{k}=e^{\mathrm{i}\theta_{k}}$, $1\leq k\leq N$ then
$-\frac{(x_{i}-x_{j})^{2}}{x_{i}x_{j}}=4\sin^{2}\frac{\theta_{i}-\theta_{j}}{2}$
(with the principal branch of the power function, positive on positive reals).
This implies $\det L(x)\neq 0$ for $x\in\mathbb{C}_{\rm reg}^{N}$ (and of
course $\det L(x)$ is homogeneous of degree zero).

######  Proposition 3.1.

If $L(x)$ is a solution of (3.1) in some connected open subset $U$ of
$\mathbb{C}_{\rm reg}^{N}$ then $L(xw)\tau(w)^{-1}$ is a solution in
$Uw^{-1}$.

###### Proof.

First let $w=(j,k)$ for some fixed $j$, $k$. If $i\neq j,k$ then replace $x$
by $x(j,k)$ in $\partial_{i}L$ to obtain

$\displaystyle\partial_{i}L(x(j,k))\tau((j,k))=\kappa
L(x(j,k))\left\\{\sum_{\ell\neq
i,j,k}\frac{\tau((i,\ell))}{x_{i}-x_{\ell}}+\frac{\tau((i,j))}{x_{i}-x_{k}}+\frac{\tau((i,k))}{x_{i}-x_{j}}-\frac{\gamma}{x_{i}}I\right\\}\tau(j,k)$
$\displaystyle\hphantom{\partial_{i}L(x(j,k))\tau((j,k))}{}=\kappa
L(x(j,k))\tau(j,k)\left\\{\sum_{\ell\neq
i,j,k}\frac{\tau((i,\ell))}{x_{i}-x_{\ell}}+\frac{\tau((i,k))}{x_{i}-x_{k}}+\frac{\tau((i,j))}{x_{i}-x_{j}}-\frac{\gamma}{x_{i}}I\right\\},$

because $(i,j)(j,k)=(j,k)(i,k)$. Next let $w=(i,j)$, then
$\partial_{i}[L(x(i,j))]=(\partial_{j}L)(x(i,j))$ and

$\displaystyle\partial_{i}[L(x(i,j))\tau((i,j))]=\kappa
L(x(i,j))\left\\{\sum_{\ell\neq
i,j}\frac{\tau((j,\ell))}{x_{i}-x_{\ell}}+\frac{\tau((i,j))}{x_{i}-x_{j}}-\frac{\gamma}{x_{i}}I\right\\}\tau((i,j))$
$\displaystyle\hphantom{\partial_{i}[L(x(i,j))\tau((i,j))]}{}=\kappa
L(x(i,j))\tau((i,j))\left\\{\sum_{\ell\neq
i,j}\frac{\tau((i,\ell))}{x_{i}-x_{\ell}}+\frac{\tau((i,j))}{x_{i}-x_{j}}-\frac{\gamma}{x_{i}}I\right\\},$

by use of $(j,\ell)(i,j)=(i,j)(i,\ell)$. Arguing by induction suppose
$L(xw)\tau(w)^{-1}$ is a solution then so is
$L(x(j,k)w)\tau(w)^{-1}\tau((j,k))=L(x(j,k)w)\tau((j,k)w)^{-1}$, for
any$(j,k)$, that is, the statement holds for $w^{\prime}=(j,k)w$. ∎

Let $w_{0}:=(1,2,3,\ldots,N)=(12)(23)\cdots(N-1,N)$ denote the $N$-cycle and
let $\langle w_{0}\rangle$ denote the cyclic group generated by $w_{0}$. There
are two components of $\mathbb{T}_{\rm reg}^{N}$ which are set-wise invariant
under $\langle w_{0}\rangle$ namely $\mathcal{C}_{0}$ and the reverse
$\\{\theta_{N}<\theta_{N-1}<\cdots<\theta_{1}<\theta_{N}+2\pi\\}$. Indeed
$\langle w_{0}\rangle$ is the stabilizer of $\mathcal{C}_{0}$ as a subgroup of
$\mathcal{S}_{N}$.

Henceforth we use $L(x)$ to denote the solution of (3.1) in $\mathcal{C}_{0}$
which satisfies $L(x_{0})=I$.

######  Proposition 3.2.

Suppose $x\in\mathcal{C}_{0}$ and $m\in\mathbb{Z}$ then
$L(xw_{0}^{m})=\tau(w_{0})^{-m}L(x)\tau(w_{0})^{m}$.

###### Proof.

Consider the solution $L(xw_{0})\tau(w_{0})^{-1}$ which agrees with $\Xi L(x)$
for all $x\in\mathcal{C}_{0}$ for some fixed matrix $\Xi$. In particular for
$x=x_{0}$ where $x_{0}w_{0}=\big{(}\omega,\ldots,\omega^{N-1},1\big{)}=\omega
x_{0}$ (recall $(xw)_{i}=x_{w(i)}$) we obtain $\Xi
L(x_{0})=L(x_{0}w_{0})\tau(w_{0})^{-1}=L(\omega
x_{0})\tau(w_{0})^{-1}=L(x_{0})\tau(w_{0})^{-1}$; because $L(x)$ is
homogeneous of degree zero. Thus $\Xi=\tau(w_{0})^{-1}$. Repeated use of the
relation shows $L(xw_{0}^{m})=\tau(w_{0})^{-m}L(x)\tau(w_{0})^{m}$. ∎

Because of its frequent use denote $\upsilon:=\tau(w_{0})$ (the letter
$\upsilon$ occurs in the Greek word cycle).

######  Definition 3.3.

For $w\in\mathcal{S}_{N}$ set $\nu(w):=\upsilon^{1-w(1)}$. For any
$x\in\mathbb{T}_{\rm reg}^{N}$ there is a unique $w_{x}$ such that
$w_{x}(1)=1$ and $xw_{x}^{-1}\in\mathcal{C}_{0}$. Set $M(w,x):=\nu(w_{x}w)$.

As a consequence $\nu(w_{0}^{m}w)=\upsilon^{-m}\nu(w)$ for any
$w\in\mathcal{S}_{N}$ and $m\in\mathbb{Z}$; since
$w_{0}^{m}w(1)-1=(w(1)+m-1)\operatorname{mod}N$. Also $M(I,x)=I$. There is a
1-1 correspondence $w\mapsto\mathcal{C}_{0}w$ between
$\\{w\in\mathcal{S}_{N}\colon w(1)=1\\}$ and the connected components of
$\mathbb{T}_{\rm reg}^{N}$.

######  Proposition 3.4.

For any $w_{1},w_{2}\in\mathcal{S}_{N}$ and $x\in\mathbb{T}_{\rm reg}^{N}$

$\displaystyle M(w_{1}w_{2},x)=M(w_{2},xw_{1})M(w_{1},x).$

###### Proof.

By definition $M(w_{1}w_{2},x)=\nu(w_{x}w_{1}w_{2})$ and
$M(w_{1},x)=\nu(w_{x}w_{1})=\upsilon^{-m}$ where $w_{x}w_{1}(1)=m+1$. Let
$w_{3}=w_{xw_{1}}$, that is, $w_{3}(1)=1$ and
$xw_{1}w_{3}^{-1}\in\mathcal{C}_{0}$. From
$\big{(}xw_{x}^{-1}\big{)}\big{(}w_{x}w_{1}w_{3}^{-1}\big{)}$
$\in\mathcal{C}_{0}$ it follows that $w_{x}w_{1}w_{3}^{-1}\in\langle
w_{0}\rangle$, in particular $w_{x}w_{1}w_{3}^{-1}=w_{0}^{m}$ because
$w_{x}w_{1}w_{3}^{-1}(1)=w_{x}w_{1}(1)=m+1=w_{0}^{m}(1)$. Thus
$M(w_{2},xw_{1})=\nu(w_{3}w_{2})=\nu\big{(}w_{0}^{-m}w_{x}w_{1}w_{2}\big{)}=\upsilon^{m}\nu(w_{x}w_{1}w_{2})$,
and $\upsilon^{m}=M(w_{1},x)^{-1}$. This completes the proof. ∎

######  Corollary 3.5.

Suppose $w\in\mathcal{S}_{N}$ and $x\in\mathbb{T}_{\rm reg}^{N}$ then
$M\big{(}w^{-1},xw\big{)}=M(w,x)^{-1}$.

###### Proof.

Indeed $M\left(w^{-1},xw\right)M(w,x)=M\big{(}ww^{-1},x\big{)}=I$. ∎

We can now extend $L(x)$ to all of $\mathbb{T}_{\rm reg}^{N}$ from its values
on $\mathcal{C}_{0}$.

######  Definition 3.6.

For $x\in\mathbb{T}_{\rm reg}^{N}$ let

$\displaystyle L(x):=L\big{(}xw_{x}^{-1}\big{)}\tau(w_{x}).$

######  Proposition 3.7.

For any $x\in\mathbb{T}_{\rm reg}^{N}$ and $w\in\mathcal{S}_{N}$

$\displaystyle L(xw)=M(w,x)L(x)\tau(w).$

###### Proof.

Let $w_{1}=w_{xw}$, that is, $w_{1}(1)=1$ and
$xww_{1}^{-1}\in\mathcal{C}_{0}$, then by definition
$L(xw)=L\big{(}xww_{1}^{-1}\big{)}\tau(w_{1})$ and
$L\big{(}xw_{x}^{-1}\big{)}=L(x)\tau(w_{x})^{-1}$. Let $m=w_{x}w(1)-1$. Since
$w_{x}ww_{1}^{-1}$ fixes $\mathcal{C}_{0}$ and
$w_{x}ww_{1}^{-1}(1)=w_{x}w(1)=m+1$ it follows that
$w_{x}ww_{1}^{-1}=w_{0}^{m}$. Thus $w_{1}=w_{0}^{-m}w_{x}w$,

$\displaystyle
L\big{(}xww_{1}^{-1}\big{)}\tau(w_{1})=L\big{(}xw_{x}^{-1}w_{0}^{m}\big{)}\tau\big{(}w_{0}^{-m}w_{x}w\big{)}=\upsilon^{-m}L\big{(}xw_{x}^{-1}\big{)}\upsilon^{m}\tau\big{(}w_{0}^{-m}w_{x}w\big{)}$
$\displaystyle\hphantom{L\big{(}xww_{1}^{-1}\big{)}\tau(w_{1})}{}=\upsilon^{-m}L\big{(}xw_{x}^{-1}\big{)}\tau(w_{x}w)=\upsilon^{-m}L(x)\tau(w)$

and $M(w,x)=\nu(w_{x}w)=\upsilon^{-m}$. ∎

###  3.1 The adjoint operation on Laurent polynomials and $\boldsymbol{L(x)}$

The purpose is to define an operation which agrees with taking complex
conjugates of functions and Hermitian adjoints of matrix functions when
restricted to $\mathbb{T}^{N}$, and which preserves analyticity. The parameter
$\kappa$ is treated as real in this context even where it may be complex (to
preserve analyticity in $\kappa$). For $x\in\mathbb{C}_{\times}^{N}$ define
$\phi x:=\big{(}x_{1}^{-1},x_{2}^{-1},\ldots,x_{N}^{-1}\big{)}$, then
$\phi(xw)=(\phi x)w$ for all $w\in\mathcal{S}_{N}$.

######  Definition 3.8.

  1. (1)

If $f(x)=\sum\limits_{\alpha\in\mathbb{Z}^{N}}c_{\alpha}x^{\alpha}$ is a
Laurent polynomial then
$f^{\ast}(x):=\sum\limits_{\alpha\in\mathbb{Z}^{N}}\overline{c_{\alpha}}x^{-\alpha}$.

  2. (2)

If $f(x)=\sum\limits_{\alpha\in\mathbb{Z}^{N}}A_{\alpha}x^{\alpha}$ is a
Laurent polynomial with matrix coefficients then
$f^{\ast}(x):=\sum\limits_{\alpha\in\mathbb{Z}^{N}}A_{\alpha}^{\ast}x^{-\alpha}$.

  3. (3)

if $F(x)$ is a matrix-valued function analytic in an open subset $U$ of
$\mathbb{C}_{\times}^{N}$ then $F^{\ast}(x):=\overline{(F(\overline{\phi
x}))}^{T}\\!$ and $F^{\ast}$ is analytic on $\phi U$, that is, if
$F(x)\\!=\\![F_{ij}(x)]_{i,j=1}^{N}$ then
$F^{\ast}(x)\\!=\\![\overline{F_{ji}(\overline{\phi x})}]_{i,j=1}^{N}\\!$ (for
example if $F_{12}(x)=c_{1}\kappa
x_{1}x_{3}^{-1}+c_{2}x_{2}^{2}x_{3}^{-1}x_{4}^{-1}$ then
$F_{21}^{\ast}(x)=\overline{c_{1}}\kappa
x_{1}^{-1}x_{3}+\overline{c_{2}}x_{2}^{-2}x_{3}x_{4}$).

Loosely speaking $F^{\ast}(x)$ is obtained by replacing $x$ by $\phi x$,
conjugating the complex constants and transposing. The fundamental chamber
$\mathcal{C}_{0}$ is mapped by $\phi$ onto
$\big{\\{}\big{(}e^{\mathrm{i}\theta_{j}}\big{)}_{j=1}^{N}\colon\theta_{1}>\theta_{2}>\cdots$
$>\theta_{N}>\theta_{1}-2\pi\big{\\}}$, again set-wise invariant under
$w_{0}$. Using
$\frac{\mathrm{d}}{\mathrm{d}t}\big{\\{}f\big{(}\frac{1}{t}\big{)}\big{\\}}=-\frac{1}{t^{2}}\big{(}\frac{d}{dt}f\big{)}\big{(}\frac{1}{t}\big{)}$
we obtain the system

$\displaystyle\partial_{i}L(\phi x)=\kappa L(\phi x)\left\\{\sum_{j\neq
i}\frac{x_{j}}{x_{i}}\frac{\tau((i,j))}{x_{i}-x_{j}}+\frac{\gamma}{x_{i}}\right\\},\qquad
1\leq i\leq N.$

Transposing this system leads to (note
$\tau(w)^{T}=\tau(w)^{\ast}=\tau\big{(}w^{-1}\big{)}$)

$\displaystyle\partial_{i}L(\phi x)^{T}=\kappa\left\\{\sum_{j\neq
i}\frac{x_{j}}{x_{i}}\frac{\tau((i,j))}{x_{i}-x_{j}}+\frac{\gamma}{x_{i}}\right\\}L(\phi
x)^{T},\qquad 1\leq i\leq N.$

Now use part (3) of Definition 3.8 and set up the system whose solution of

$\displaystyle\partial_{i}L^{\ast}(x)=\kappa\left\\{\sum_{j\neq
i}\frac{x_{j}}{x_{i}}\frac{\tau((i,j))}{x_{i}-x_{j}}+\frac{\gamma}{x_{i}}\right\\}L^{\ast}(x),\qquad
1\leq i\leq N.$ (3.2)

satisfying $L^{\ast}(x_{0})=I$ is denoted by $L^{\ast}(x)$. 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 $L(x)^{\ast}$
agrees with the Hermitian adjoint of $L(x)$ for $x\in\mathcal{C}_{0}$ (for
real $\kappa$). The goal here is to establish conditions on a constant
Hermitian matrix $H$ so that $K(x):=L^{\ast}(x)HL(x)$ has desirable
properties, such as $K(xw)=\tau(w)^{-1}K(x)\tau(w)$ and $K(x)\geq 0$ (i.e.,
positive definite).

Similarly to the above $\tau((i,j))L^{\ast}(x(i,j))$ is also a solution of
(3.2), implying that $\tau(w)L^{\ast}(xw)$ is a solution for any
$w\in\mathcal{S}_{N}$, the inductive step is

$\displaystyle\tau((i,j))\tau(w)L(x(i,j)w)^{\ast}=\tau((i,j)w)L(x(i,j)w)^{\ast}.$

Also
$L^{\ast}(x_{0}w_{0})=L^{\ast}\big{(}\omega^{-1}x_{0}\big{)}=L^{\ast}(x_{0})=I$
(thus there is a matrix $\widetilde{\Xi}$ such that
$\tau(w_{0})L^{\ast}(xw_{0})=L^{\ast}(\phi x)\widetilde{\Xi}$ for all
$x\in\mathcal{C}_{0}$, and $\widetilde{\Xi}=\tau(w_{0})=\upsilon$. In analogy
to $L$ for $x\in\mathbb{T}_{\rm reg}^{N}$ and the same $w_{x}$ as above let
$L(\phi x_{0})^{T}=I$, $L(\phi x)^{T}:=\tau(w_{x})^{-1}L\big{(}\phi
xw_{x}^{-1}\big{)}^{T}$ (since $\phi xw_{x}^{-1}\in\phi\mathcal{C}_{0})$. Then
$L(\phi xw)^{T}=\tau(w)^{-1}L(\phi x)^{T}M(w,x)^{-1}$.

For any nonsingular constant matrix $C$ the function $CL(x)$ also satisfies
(3.1) and the function $K(x):=L^{\ast}(x)C^{\ast}CL(x)$ satisfies the system

$\displaystyle x_{i}\partial_{i}K(x)=\kappa\sum_{j\neq
i}\left\\{\frac{x_{j}}{x_{i}-x_{j}}\tau((i,j))K(x)+K(x)\tau((i,j))\frac{x_{i}}{x_{i}-x_{j}}\right\\},\qquad
1\leq i\leq N.$ (3.3)

This formulation can be slightly generalized by replacing $C^{\ast}C$ 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
$K(xw)=\tau(w)^{-1}K(x)\tau(w)$, that is,

$\displaystyle K(xw)=\tau(w)^{-1}L^{\ast}(x)M(w,x)^{-1}HM(w,x)L(x)\tau(w)$
$\displaystyle\hphantom{K(xw)}{}=\tau(w)^{-1}L^{\ast}(x)\upsilon^{m}H\upsilon^{-m}L(x)\tau(w)$

(from Proposition 3.7), where $m=w_{x}w(1)-1$. The condition is equivalent to

$\displaystyle\upsilon H=H\upsilon,$

which is now added to the hypotheses, summarized here:

######  Condition 3.9.

$L(x)$ is the solution of (3.1) such that $L(x_{0})=I$ and
$L(x)=L\big{(}xw_{x}^{-1}\big{)}\tau(w_{x})$ for $x\in\mathbb{T}_{\rm
reg}^{N}$ where $w_{x}(1)=1$ and $xw_{x}^{-1}\in\mathcal{C}_{0}$;
$L^{\ast}(x)$ is the solution of (3.2) satisfying $L^{\ast}(x_{0})=I$,
$K(x)=L^{\ast}(x)HL(x)$ is a solution of (3.3) and $H$ satisfies $H^{\ast}=H$,
$\upsilon H=H\upsilon$.

##  4 Integration by parts

In this section we establish the relation between the differential system and
the abstract relation $\langle x_{i}\mathcal{D}_{i}f,g\rangle=\langle
f,x_{i}\mathcal{D}_{i}g\rangle$ holding for $1\leq i\leq N$ and $f,g\in
C^{1}\big{(}\mathbb{T}^{N};V_{\tau}\big{)}$. We demonstrate how close $L$ is
to providing the desired inner product, by performing an integration-by-parts
over an $\mathcal{S}_{N}$-invariant closed set $\subset\mathbb{T}_{\rm
reg}^{N}$. Here $L(x)$ and $H$ satisfy the hypotheses listed in Condition 3.9
above. We use the identity
$x_{i}\partial_{i}f^{\ast}(x)=-(x_{i}\partial_{i}f)^{\ast}(x)$. For $\delta>0$
let

$\displaystyle\Omega_{\delta}:=\Big{\\{}x\in\mathbb{T}^{N}\colon\min\limits_{1\leq
i<j\leq N}|x_{i}-x_{j}|\geq\delta\Big{\\}}.$

This set is invariant under $\mathcal{S}_{N}$ and $K(x)$ is bounded and smooth
on it. Thus the following integrals exist.

######  Proposition 4.1.

Suppose $H$ satisfies Condition 3.9 then for $f,g\in
C^{1}\big{(}\mathbb{T}^{N};V_{\tau}\big{)}$ and $1\leq i\leq N$

$\displaystyle\int_{\Omega_{\delta}}\big{\\{}{-}(x_{i}\mathcal{D}_{i}f(x))^{\ast}K(x)g(x)+f(x)^{\ast}K(x)x_{i}\mathcal{D}_{i}g(x)\big{\\}}\mathrm{d}m(x)$
$\displaystyle\qquad{}=\int_{\Omega_{\delta}}x_{i}\partial_{i}\\{f(x)^{\ast}K(x)g(x)\\}\mathrm{d}m(x).$

###### Proof.

By definition

$\displaystyle
x_{i}\mathcal{D}_{i}g(x)=x_{i}\partial_{i}g(x)+\kappa\sum_{j\neq
i}\frac{x_{i}}{x_{i}-x_{j}}\tau((i,j))(g(x)-g(x(i,j))),$
$\displaystyle(x_{i}\mathcal{D}_{i}f(x))^{\ast}=-x_{i}\partial_{i}f(x)^{\ast}+\kappa\sum_{j\neq
i}\frac{x_{j}}{x_{j}-x_{i}}(f(x)^{\ast}-f(x(i,j))^{\ast})\tau((i,j)).$

Thus

$\displaystyle-(x_{i}\mathcal{D}_{i}f(x))^{\ast}K(x)g(x)+f(x)^{\ast}K(x)x_{i}\mathcal{D}_{i}g(x)$
$\displaystyle\qquad{}=x_{i}\partial_{i}f(x)^{\ast}+x_{i}\partial_{i}g(x)$
$\displaystyle\qquad\quad{}+\kappa f(x)^{\ast}\sum_{j\neq
i}\left\\{\frac{x_{j}}{x_{i}-x_{j}}\tau((i,j))K(x)+K(x)\tau((i,j))\frac{x_{i}}{x_{i}-x_{j}}\right\\}g(x)$
$\displaystyle\qquad\quad{}-\kappa\sum_{j\neq
i}\frac{1}{x_{i}-x_{j}}\big{\\{}x_{j}f(x(i,j))^{\ast}\tau((i,j))K(x)g(x)+x_{i}f(x)^{\ast}K(x)\tau((i,j))g(x(i,j))\big{\\}}$
$\displaystyle\qquad{}=x_{i}\partial_{i}\\{f(x)^{\ast}K(x)g(x)\\}$ (4.1)
$\displaystyle\qquad\quad{}-\kappa\sum_{j\neq
i}\frac{1}{x_{i}-x_{j}}\big{\\{}x_{j}f(x(i,j))^{\ast}\tau((i,j))K(x)g(x)+x_{i}f(x)^{\ast}K(x)\tau((i,j))g(x(i,j))\big{\\}}.$

For each pair $\\{i,j\\}$ the term inside $\\{\cdot\\}$ is invariant under
$x\mapsto x(i,j)$, because $K(x(i,j))=\tau((i,j))K(x)\tau((i,j))$, and
$x_{i}-x_{j}$ changes sign under this transformation. Thus

$\displaystyle\int_{\Omega_{\delta}}\frac{x_{j}f(x(i,j))^{\ast}\tau((i,j))K(x)g(x)+x_{i}f(x)^{\ast}K(x)\tau((i,j))g(x(i,j))}{x_{i}-x_{j}}\mathrm{d}m(x)=0$

for each $j\neq i$ because $\Omega_{\delta}$ and $\mathrm{d}m$ are invariant
under $(i,j)$. ∎

Observe the value of $\kappa$ is not involved in the proof. Since
$x_{j}\partial_{j}=-\mathrm{i}\frac{\partial}{\partial\theta_{j}}$ when
$x_{j}=e^{\mathrm{i}\theta_{j}}$ and
$\mathrm{d}m(x)=(2\pi)^{-N}\mathrm{d}\theta_{1}\cdots\mathrm{d}\theta_{N}$ one
step of integration can be directly evaluated. Consider the case $i=N$ and for
a fixed $(N-1)$-tuple $(\theta_{1},\ldots,\theta_{N-1})$ with
$\theta_{1}<\theta_{2}<\cdots<\theta_{N-1}<\theta_{1}+2\pi$ such that
$\big{|}e^{\mathrm{i}\theta_{j}}-e^{\mathrm{i}\theta_{i}}\big{|}\geq\delta$
the integral over $\theta_{N}$ is over a union of closed intervals. These are
the complement of $\bigcup\limits_{1\leq j\leq
N-1}\\{\theta\colon\theta_{j}-\delta^{\prime}<\theta<\theta_{j}+\delta^{\prime}\\}$
in the circle, where $\sin\frac{\delta^{\prime}}{2}=\frac{\delta}{2}$. This
results in an alternating sum of values of $f^{\ast}Kg$ at the end-points of
the closed intervals. Analyzing the resulting integral (over
$(\theta_{1},\ldots,\theta_{N-1})$ with respect to
$\mathrm{d}\theta_{1}\cdots\mathrm{d}\theta_{N-1}$) 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 $\kappa\notin\mathbb{Z+}\frac{1}{2}$. We consider the
system (3.1) in a neighborhood of the face $\\{x\colon x_{N-1}=x_{N}\\}$ of
$\mathcal{C}_{0}$. We use a coordinate system which treats the singularity in
a simple way. For a more concise notation define

$\displaystyle
x(u,z)=(x_{1},x_{2},\ldots,x_{N-2},u-z,u+z)\in\mathbb{C}_{\times}^{N}$

We consider the system in terms of the variable $x(u,z)$ subject to the
conditions that the points $x_{1},x_{2},\ldots,x_{N-2},u$ are pairwise
distinct and $|z|<\min\limits_{1\leq j\leq N-2}|x_{j}-u|$, also $|z|<|u|$,
$\operatorname{Im}\frac{z}{u}>0$ (these conditions imply
$\arg(u-z)<\arg(u+z)$). This allows power series expansions in $z$.

For $z_{1},z_{2}\in\mathbb{C}_{\times}$ let

$\displaystyle\rho(z_{1},z_{2}):=\left[\begin{matrix}z_{1}I_{m_{\tau}}&O\\\
O&z_{2}I_{n_{\tau}-m_{\tau}}\end{matrix}\right].$

Let $\sigma:=\tau((N-1,N))=\rho(-1,1)$. We analyze the local solution
$L(x(u-z,u+z))$ with an initial condition specified later. We obtain the
differential system (using $\partial_{z}:=\frac{\partial}{\partial z}$,
$\partial_{u}:=\frac{\partial}{\partial u}$)

$\displaystyle\partial_{z}L(x)=\partial_{N}L-\partial_{N-1}L$
$\displaystyle\hphantom{\partial_{z}L(x)}{}=\kappa
L\left\\{\sum_{j=1}^{N-2}\left(\frac{\tau((j,N))}{u-x_{j}+z}-\frac{\tau((j,N-1))}{u-x_{j}-z}\right)+\frac{\tau(N-1,N)}{z}-\frac{\gamma}{u+z}I+\frac{\gamma}{u-z}I\right\\},$
$\displaystyle\partial_{u}L(x)=\partial_{N}L+\partial_{N-1}L$
$\displaystyle\hphantom{\partial_{u}L(x)}{}=\kappa
L\left\\{\sum_{j=1}^{N-2}\left(\frac{\tau((j,N))}{u-x_{j}+z}+\frac{\tau((j,N-1))}{u-x_{j}-z}\right)-\frac{\gamma}{u+z}I-\frac{\gamma}{u-z}I\right\\},$
$\displaystyle\partial_{j}L(x)=\kappa L(x)\left\\{\sum_{i=1,i\neq
j}^{N-2}\frac{\tau((i,j))}{x_{j}-x_{i}}-\frac{\gamma}{x_{j}}I+\frac{\tau((j,N-1))}{x_{j}-u+z}+\frac{\tau((j,N))}{x_{j}-u-z}\right\\},$
$\displaystyle\hphantom{\partial_{j}L(x)=}{}1\leq j\leq N-2.$

Using the expansion
$\frac{1}{t-z}=\sum\limits_{n=0}^{\infty}\frac{z^{n}}{t^{n+1}}$ for $|z|<|t|$
we let

$\displaystyle\beta_{n}(x(u,0)):=\sum_{j=1}^{N-2}\frac{\tau((j,N))}{(u-x_{j})^{n+1}}$

for $n=0,1,2,\ldots$and express the equations as (since
$\sigma\tau((j,N))\sigma=\tau((j,N-1))$)

$\displaystyle\partial_{z}L(x)=\kappa
L(x)\left\\{\sum_{n=0}^{\infty}\big{\\{}(-1)^{n}\beta_{n}(x(u,0))-\sigma\beta_{n}(x(u,0))\sigma\big{\\}}z^{n}+\frac{\sigma}{z}-\frac{\gamma}{u+z}I+\frac{\gamma}{u-z}I\right\\},$
$\displaystyle\partial_{u}L(x)=\kappa
L(x)\left\\{\sum_{n=0}^{\infty}\big{\\{}(-1)^{n}\beta_{n}(x(u,0))+\sigma\beta_{n}(x(u,0))\sigma\big{\\}}z^{n}-\frac{\gamma}{u+z}I-\frac{\gamma}{u-z}I\right\\},$
$\displaystyle\partial_{j}L(x)=\kappa L(x)\left\\{\sum_{i=1,i\neq
j}^{N-2}\frac{\tau((i,j))}{x_{j}-x_{i}}-\frac{\gamma}{x_{j}}I-\sum_{n=0}^{\infty}\frac{\tau((j,N-1))+(-1)^{n}\tau((j,N))}{(u-x_{j})^{n+1}}z^{n}\right\\},$
$\displaystyle\hphantom{\partial_{j}L(x)=}{}1\leq j\leq N-2.$

Set

$\displaystyle
B_{n}(x(u,0))=(-1)^{n}\beta_{n}(x(u,0))-\sigma\beta_{n}(x(u,0))\sigma,\qquad
n=0,1,2,\ldots.$

Note $\sigma B_{n}x(u,0)\sigma=(-1)^{n+1}B_{n}(x(u,0))$. Suggested by the
relation

$\displaystyle\frac{\partial}{\partial
z}\rho\big{(}z^{-\kappa},z^{\kappa}\big{)}=\frac{\kappa}{z}\rho\big{(}{}-z^{-\kappa},z^{\kappa}\big{)}=\frac{\kappa}{z}\rho\big{(}z^{-\kappa},z^{\kappa}\big{)}\sigma$

we look for a solution of the form

$\displaystyle
L(x)=\left(\big{(}u^{2}-z^{2}\big{)}\prod_{j=1}^{N-2}x_{j}\right)^{-\gamma\kappa}\rho\big{(}z^{-\kappa},z^{\kappa}\big{)}\sum_{n=0}^{\infty}\alpha_{n}(x(u,0))z^{n},$
(5.1)

where each $a_{n}(x(u,0))$ is matrix-valued and analytic in $x(u,0)$, and the
initial condition is $\alpha_{0}\big{(}x^{(0)}\big{)}=I$, where $x^{(0)}$ is a
base point, chosen as
$\big{(}1,\omega,\omega^{2},\ldots,\omega^{N-3},\omega^{-3/2},\omega^{-3/2}\big{)}$
(that is, $u=\omega^{-3/2}$, $z=0$), where $\omega:=e^{2\pi\mathrm{i}/N}$.
Implicitly restrict $(x_{1},\ldots,x_{N-1},u)$ to a simply connected open
subset of $\mathbb{C}_{\rm reg}^{N-1}$ containing
$\big{(}1,\omega,\omega^{2},\ldots,\omega^{N-3},\omega^{-3/2}\big{)}$.
Substitute (5.1) in the $\partial_{z}$ equation (suppressing the $x(u,0)$
argument in the $\alpha_{n}$’s)

$\displaystyle\partial_{z}L=\kappa\gamma\left(\frac{1}{u+z}-\frac{1}{u-z}\right)\left(\big{(}u^{2}-z^{2}\big{)}\prod_{j=1}^{N-2}x_{j}\right)^{-\gamma\kappa}\rho\big{(}z^{-\kappa},z^{\kappa}\big{)}\sum_{n=0}^{\infty}\alpha_{n}z^{n}$
$\displaystyle\hphantom{\partial_{z}L=}{}+\left(\big{(}u^{2}-z^{2}\big{)}\prod_{j=1}^{N-2}x_{j}\right)^{-\gamma\kappa}\frac{\kappa}{z}\rho\big{(}z^{-\kappa},z^{\kappa}\big{)}\sigma\sum_{n=0}^{\infty}\alpha_{n}z^{n}$
$\displaystyle\hphantom{\partial_{z}L=}{}+\left(\big{(}u^{2}-z^{2}\big{)}\prod_{j=1}^{N-2}x_{j}\right)^{-\gamma\kappa}\rho\big{(}z^{-\kappa},z^{\kappa}\big{)}\sum_{n=1}^{\infty}n\alpha_{n}z^{n-1}$
$\displaystyle\hphantom{\partial_{z}L}{}=\kappa\left(\big{(}u^{2}-z^{2}\big{)}\prod_{j=1}^{N-2}x_{j}\right)^{-\gamma\kappa}\rho\big{(}z^{-\kappa},z^{\kappa}\big{)}$
$\displaystyle\hphantom{\partial_{z}L=}{}\times\sum_{n=0}^{\infty}\alpha_{n}z^{n}\left\\{\sum_{m=0}^{\infty}B_{m}(u)z^{m}+\frac{\sigma}{z}-\gamma\left(\frac{1}{u+z}-\frac{1}{u-z}\right)\right\\},$

which simplifies to

$\displaystyle\frac{\kappa}{z}\sum_{n=0}^{\infty}(\sigma\alpha_{n}-\alpha_{n}\sigma)z^{n}+\sum_{n=1}^{\infty}n\alpha_{n}z^{n-1}=\kappa\sum_{n=0}^{\infty}\alpha_{n}z^{n}\sum_{m=0}^{\infty}B_{m}(x(u,0))z^{m}.$
(5.2)

The equations for $\partial_{u}$ and $\partial_{j}$ simplify to

$\displaystyle\left(\big{(}u^{2}-z^{2}\big{)}\prod_{j=1}^{N-2}x_{j}\right)^{-\gamma\kappa}\left\\{\sum_{n=0}^{\infty}\partial_{u}\alpha_{n}z^{n}-\kappa\gamma\left(\frac{1}{u+v}+\frac{1}{u-v}\right)\sum_{n=0}^{\infty}\alpha_{n}z^{n}\right\\}$
$\displaystyle\qquad{}=\kappa\left(\big{(}u^{2}-z^{2}\big{)}\prod_{j=1}^{N-2}x_{j}\right)^{-\gamma\kappa}\sum_{n=0}^{\infty}\alpha_{n}z^{n}$
$\displaystyle\qquad\quad{}\times\left\\{\sum_{m=0}^{\infty}\big{\\{}(-1)^{m}\beta_{m}(x(u,0))+\sigma\beta_{m}(x(u,0))\sigma\big{\\}}z^{m}-\frac{\gamma}{u+v}I-\frac{\gamma}{u-v}I\right\\},$

leading to (with $1\leq j\leq N-2$)

$\displaystyle\sum_{n=0}^{\infty}\partial_{u}\alpha_{n}(x(u,0))z^{n}=\kappa\sum_{n=0}^{\infty}\alpha_{n}(x(u,0))z^{n}\sum_{m=0}^{\infty}\big{\\{}(-1)^{m}\beta_{m}(x(u,0))+\sigma\beta_{m}(x(u,0))\sigma\big{\\}}z^{m},$
$\displaystyle\sum_{n=0}^{\infty}\partial_{j}\alpha_{n}(x(u,0))z^{n}=\kappa\sum_{n=0}^{\infty}\alpha_{n}(x(u,0))z^{n}$
$\displaystyle\hphantom{\sum_{n=0}^{\infty}\partial_{j}\alpha_{n}(x(u,0))z^{n}=}{}\times\left\\{\sum_{i=1,i\notin
j}^{N-2}\frac{\tau((i,j))}{x_{j}-x_{i}}-\sum_{m=0}^{\infty}\frac{\tau((j,N-1))+(-1)^{m}\tau((j,N))}{(u-x_{j})^{m+1}}z^{m}\right\\}.$

We only need the equations for $\alpha_{0}(x(u,0))$ (that is, the coefficient
of $z^{0}$) to initialize the $\partial_{z}$ equation (this is valid because
the system is Frobenius integrable):

$\displaystyle\partial_{u}\alpha_{0}(x(u,0))=\kappa\alpha_{0}(x(u,0))\big{\\{}\beta_{0}(x(u,0))+\sigma\beta_{0}(x(u,0))\sigma\big{\\}},$
(5.3)
$\displaystyle\partial_{j}\alpha_{0}(x(u,0))=\kappa\alpha_{0}(x(u,0))\left\\{\sum_{i=1,i\notin
j}^{N-2}\frac{\tau((i,j))}{x_{j}-x_{i}}-\frac{\tau((j,N-1))+\tau((j,N))}{(u-x_{j})}\right\\},$
$\displaystyle\hphantom{\partial_{j}\alpha_{0}(x(u,0))=}{}2\leq j\leq N-2.$

######  Lemma 5.1.

$\sigma\alpha_{0}(x(u,0))\sigma=\alpha_{0}(x(u,0))$ and $\alpha_{0}(x(u,0))$
is invertible.

###### Proof.

By hypothesis $\alpha_{0}\big{(}x^{(0)}\big{)}=I$. The right hand sides of the
system are invariant under the transformation $Q\mapsto\sigma Q\sigma$ thus
$\alpha_{0}(x(u,0))$ and $\sigma\alpha_{0}(x(u,0))\sigma$ 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
$\lambda:=\operatorname{tr}(\sigma)=n_{\tau}-2m_{\tau}$)

$\displaystyle\partial_{u}\det\alpha_{0}(x(u,0))=\kappa\det\alpha_{0}(x(u,0))\operatorname{tr}\\{\beta_{0}(x(u,0))+\sigma\beta_{0}(x(u,0))\sigma\\}$
$\displaystyle\hphantom{\partial_{u}\det\alpha_{0}(x(u,0))}{}=2\kappa\det\alpha_{0}(x(u,0))\mathrm{\lambda}\sum_{j=1}^{N-2}\frac{1}{(u-x_{j})},$
$\displaystyle\partial_{j}\det\alpha_{0}(x(u,0))=\kappa\lambda\det\alpha_{0}(x(u,0))\left\\{\sum_{i=1,i\notin
j}^{N-2}\frac{1}{x_{j}-x_{i}}-\frac{2}{u-x_{j}}\right\\},\qquad 1\leq j\leq
N-2,$ $\displaystyle\det\alpha_{0}(x(u,0))=\prod_{1\leq i<j\leq
N-2}\left(\frac{x_{i}-x_{j}}{x_{i}^{(0)}-x_{j}^{(0)}}\right)^{\lambda\kappa}\prod_{i=1}^{N-2}\left(\frac{x_{i}-u}{x_{i}^{(0)}-x_{N-1}^{(0)}}\right)^{2\lambda\kappa},$

the multiplicative constant follows from $\alpha_{0}\big{(}x^{(0)}\big{)}=I$.
Thus $\alpha_{0}(x(u,0))$ is nonsingular in its domain. ∎

We turn to the inductive definition of $\\{\alpha_{n}(x(u,0))\colon n\geq
1\\}$.

In terms of the block decomposition
$(m_{\tau}+(n_{\tau}-m_{\tau}))\times(m_{\tau}+(n_{\tau}-m_{\tau}))$
(henceforth called the $\sigma$-_block decomposition_) of a matrix

$\displaystyle\alpha=\left[\begin{matrix}\alpha_{11}&\alpha_{12}\\\
\alpha_{21}&\alpha_{22}\end{matrix}\right]$

$\sigma\alpha\sigma=\alpha$ if and only if $\alpha_{12}=O=\alpha_{21}$ and
$\sigma\alpha\sigma=-\alpha$ if and only if $\alpha_{11}=O=\alpha_{22}$. Write
the $\sigma$-block decomposition of $\alpha_{n}(u)$ as

$\displaystyle\alpha_{n}=\left[\begin{matrix}\alpha_{n,11}&\alpha_{n,12}\\\
\alpha_{n,21}&\alpha_{n,22}\end{matrix}\right]$

then the coefficient of $z^{n-1}$ on the left side of equation (5.2) is

$\displaystyle\kappa(\sigma\alpha_{n}-\alpha_{n}\sigma)+n\alpha_{n}=\left[\begin{matrix}n\alpha_{n,11}&(n-2\kappa)\alpha_{n,12}\\\
(n+2\kappa)\alpha_{n,21}&n\alpha_{n,22}\end{matrix}\right],$

and on the right side it is

$\displaystyle\kappa
S_{n}(x(u,0)):=\kappa\sum_{i=0}^{n-1}\alpha_{n-1-i}B_{i}(x(u,0)),$

for $n\geq 1$. Arguing inductively suppose
$\sigma\alpha_{m}\sigma=(-1)^{m}\alpha_{m}$ for $0\leq m\leq n$, then $\sigma
S_{n}\sigma=\sum\limits_{i=0}^{n-1}(\sigma\alpha_{n-1-i}\sigma)(\sigma
B_{i}\sigma)=\sum\limits_{i=0}^{n-1}(-1)^{n-1-i+i-1}\alpha_{n-1-i}B_{i}$ and
thus $\sigma S_{n}(u)\sigma=(-1)^{n}S_{n}(u)$. In terms of the $\sigma$-block
decomposition $\left[\begin{matrix}S_{n,11}&S_{n,12}\\\
S_{n,21}&S_{n,22}\end{matrix}\right]$ of $S_{n}(x(u,0))$ this condition
implies $S_{n,12}=O=S_{n,21}$ when $n$ is even, and $S_{n,11}=O=S_{n,22}$ when
$n$ is odd. This implies (for $n=1,2,3,\ldots$)

$\displaystyle\alpha_{2n}(x(u,0))=\frac{\kappa}{2n}S_{2n}(x(u,0)),$ (5.4)
$\displaystyle\alpha_{2n-1}(x(u,0))=\rho\left(\frac{\kappa}{2n-1-2\kappa},\frac{\kappa}{2n-1+2\kappa}\right)S_{2n-1}(x(u,0)),$

and thus $\sigma\alpha_{n}(x(u,0))\sigma=(-1)^{n}\alpha_{n}(x(u,0))$. In
particular

$\displaystyle\alpha_{1}(x(u,0))=\rho\left(\frac{\kappa}{1-2\kappa},\frac{\kappa}{1+2\kappa}\right)\alpha_{0}(x(u,0))B_{0}(x(u,0)),$

and all the coefficients are determined; by hypothesis
$\kappa\notin\mathbb{Z+}\frac{1}{2}$ and the denominators are of the form
$2m+1\pm 2\kappa$.

Henceforth denote the series (5.1), solving (3.1) with the normalization
$\alpha_{0}\big{(}x^{(0)}\big{)}=I$ by $L_{1}(x)$. It is defined for all
$x(u,z)\in\mathcal{C}_{0}$ subject to $|z|<\min\limits_{1\leq j\leq
N-2}|x_{j}-u|$, also $|z|<|u|$, $\operatorname{Im}\frac{z}{u}>0$. 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 $\mathbb{T}_{\rm reg}^{N}$ and
$L(x_{0})=I$). In terms of $x(u,z)$ the point $x_{0}$ corresponds to
$u=\frac{1}{2}\big{(}\omega^{-1}+\omega^{-2}\big{)}$,
$z=\frac{1}{2}\big{(}\omega^{-1}-\omega^{-2}\big{)}$,
$x(u,z)=\big{(}1,\omega,\ldots,\omega^{N-3},u-z,u+z\big{)}$, then
$\min\limits_{1\leq j\leq
N-2}|u-x_{j}|=\sin\frac{\pi}{N}\big{(}5+4\cos\frac{2\pi}{N}\big{)}$ and
$|z|=\sin\frac{\pi}{N}$ (also $\frac{z}{u}=\mathrm{i}\tan\frac{\pi}{N}$) and
$x_{0}$ is in the domain of convergence of the series $L_{1}(x)$. Thus the
relation $L_{1}(x)=L_{1}(x_{0})L(x)$ holds in the domain of $L_{1}$ in
$\mathcal{C}_{0}$. This implies the important fact that $L_{1}(x_{0})$ is an
analytic function of $\kappa$, to be exploited in Section 9.

###  5.1 Behavior on boundary

The term $\rho(z^{-\kappa},z^{\kappa})$ implies that $L_{1}(x)$ is not
continuous at $z=0$, that is, on the boundary $\\{x\colon x_{N-1}=x_{N}\\}$.
However there may be a weak type of continuity, specifically

$\displaystyle\lim\limits_{x_{N-1}-x_{N}\rightarrow 0}(K(x)-K(x(N-1,N)))=0.$

With the aim of expressing the desired $K(x)$ in the form
$L(x)^{\ast}C^{\ast}CL(x)$ (and $C$ is unknown at this stage) we consider
$CL(x)$ in series form, that is $CL_{1}(x_{0})^{-1}L_{1}(x)$ (recall $\det
L(x)\neq 0$ in $\mathcal{C}_{0}$). We analyze the effect of $C$ on the weak
continuity condition. Denote $C^{\prime}:=CL_{1}(x_{0})^{-1}$.

From Proposition 3.7 $L(x(N-1,N))=\nu((N-1,N))L(x)\tau((N-1,N))=L(x)\sigma$,
because $w(1)=1$ for $w=(N-1,N)$, [for the special case $N=3$, $\tau=(2,1)$,
$\mathbb{T}_{\rm reg}^{3}$ has two components and we define
$L(x)=L(x(2,3))\sigma$ for the component $\neq\mathcal{C}_{0}$]. By use of
$x(u,z)(N-1,N)=x(u,-z)$ it follows that

$\displaystyle
CL(x(u,z)(N-1,N))=CL(x(u,z))\sigma=C^{\prime}\left(x_{N}x_{N-1}\right)^{-\gamma\kappa}\rho\big{(}z^{-\kappa},z^{\kappa}\big{)}\sum_{n=0}^{\infty}\alpha_{n}(u)z^{n}\sigma$
$\displaystyle\hphantom{CL(x(u,z)(N-1,N))}{}=C^{\prime}\sigma(x_{N}x_{N-1})^{-\gamma\kappa}\rho\big{(}z^{-\kappa},z^{\kappa}\big{)}\sum_{n=0}^{\infty}\alpha_{n}(u)(-1)^{n}z^{n},$

because $\sigma\alpha_{n}(u)\sigma=(-1)^{n}\alpha_{n}(u)$ and
$\sigma=\rho(-1,1)$.

Recall $L^{\ast}(x)$ is defined as $L(\phi x)^{T}$ with complex constants
replaced by their conjugates. Then $\phi
x(u,z)=\big{(}x_{1}^{-1},x_{2}^{-1},\ldots,x_{N-2}^{-1},\frac{1}{u-z},\frac{1}{u+z}\big{)}$.
To compute $L_{1}(\phi x(u,z))$ replace $u$ by
$u^{\prime}=\frac{u}{(u+z)(u-z)}$ and replace $z$ by
$z^{\prime}=-\frac{z}{(u+z)(u-z)}$. When restricted to the torus
$u^{\prime}=\frac{1}{2}\big{(}\frac{1}{x_{N-1}}+\frac{1}{x_{N}}\big{)}=\overline{u}$
and
$z^{\prime}=\frac{1}{2}\big{(}\frac{1}{x_{N-1}}-\frac{1}{x_{N}}\big{)}=\overline{z}$.
The terms
$\beta_{n}(u):=\sum\limits_{j=1}^{N-2}\frac{\tau((j,N))}{(u-x_{j})^{n+1}}$ in
the intermediate formulae for $L_{1}$ are replaced by their complex conjugates
when $x(u,z)\in\mathbb{T}^{N}$. Similarly
$\widetilde{\beta}_{k}:=\sum\limits_{m=0}^{\infty}\sum\limits_{j=1}^{N-2}\frac{\tau((j,N))}{(u_{0}-x_{j})^{k+1}}$
transforms to $\overline{(\widetilde{\beta}_{k})}$ because the constant
$u_{0}$ is conjugated. Thus for $x(u,z)\in\mathbb{T}_{\rm reg}^{N}$

$\displaystyle
L_{1}(x(u,z))^{\ast}=\sum_{m=0}^{\infty}\alpha_{m}(u)^{\ast}\overline{z}^{m}\rho\big{(}\overline{z}^{-\kappa},\overline{z}^{\kappa}\big{)}C^{{}^{\prime}\ast}(\overline{x_{N}x_{N-1}})^{-\gamma\kappa};$

$\alpha_{m}(u)^{\ast}$ denotes the adjoint of the matrix $\alpha_{m}(u)$. Then

$\displaystyle
L_{1}(x(u,z)(N-1,N))^{\ast}=\sum_{m=0}^{\infty}(-1)^{m}\alpha_{m}(u)^{\ast}\overline{z}^{m}\rho(\overline{z}^{-\kappa},\overline{z}^{\kappa})\sigma
C^{{}^{\prime}\ast}(\overline{x_{N}x_{N-1}})^{-\gamma\kappa}.$

Furthermore (recall $K(x(N-1,N))=\sigma K(x)\sigma$ by definition)

$\displaystyle
K(x(u,z))=\sum_{m,n=0}^{\infty}\overline{z}^{m}z^{n}\alpha_{m}(u)^{\ast}\rho\big{(}\overline{z}^{-\kappa},\overline{z}^{\kappa}\big{)}C^{\prime\ast}C^{\prime}\rho\big{(}z^{-\kappa},z^{\kappa}\big{)}\alpha_{n}(u),$
$\displaystyle
K(x(u,-z))=\sum_{m,n=0}^{\infty}\overline{z}^{m}z^{n}(-1)^{m+n}\alpha_{m}(u)^{\ast}\rho\big{(}\overline{z}^{-\kappa},\overline{z}^{\kappa}\big{)}\sigma
C^{\prime\ast}C^{\prime}\sigma\rho\big{(}z^{-\kappa},z^{\kappa}\big{)}\alpha_{n}(u).$

The term of lowest order in $z$ in $K(x(u,z))-K(x(u,-z))$ is

$\displaystyle\alpha_{0}(u)^{\ast}\rho\big{(}\overline{z}^{-\kappa},\overline{z}^{\kappa}\big{)}\big{\\{}C^{\prime\ast}C^{\prime}-\sigma
C^{\prime\ast}C^{\prime}\sigma\big{\\}}\rho\big{(}z^{-\kappa},z^{\kappa}\big{)}\alpha_{0}(u).$

In terms of the $\sigma$-block decomposition, with

$\displaystyle C^{\prime\ast}C^{\prime}=\begin{bmatrix}c_{11}&c_{12}\\\
c_{12}^{\ast}&c_{22}\end{bmatrix},\qquad\alpha_{0}(u)=\begin{bmatrix}a_{11}(u)&O\\\
O&a_{22}(u)\end{bmatrix}$

the expression equals

$\displaystyle
2\begin{bmatrix}O&\left(\dfrac{z}{\overline{z}}\right)^{\kappa}a_{11}(u)^{\ast}c_{12}a_{22}(u)\\\
\left(\dfrac{\overline{z}}{z}\right)^{\kappa}a_{22}(u)^{\ast}c_{12}^{\ast}a_{11}(u)&O\end{bmatrix},$

which tends to zero as $z\rightarrow 0$ if and only if $c_{12}=0$, that is,
$\sigma C^{\ast}C\sigma=C^{\ast}C$.

######  Proposition 5.2.

Suppose $C^{\prime\ast}C^{\prime}$ commutes with $\sigma$ then

$\displaystyle K(x(u,z))-K(x(u,z)(N-1,N))=O\big{(}|z|^{1-2|\kappa|}\big{)}.$

###### Proof.

The hypothesis implies $C^{\prime\ast}C^{\prime}$ commutes with
$\rho(z^{-\kappa},z^{\kappa})$, thus

$\displaystyle K(x(u,z))-K(x(u,z)(N-1,N))$
$\displaystyle\qquad{}=\sum_{m,n=0}^{\infty}\overline{z}^{m}z^{n}\big{(}1-(-1)^{m+n}\big{)}\alpha_{m}(u)^{\ast}\rho\big{(}|z|^{-2\kappa},|z|^{2\kappa}\big{)}C^{\prime\ast}C^{\prime}\alpha_{n}(u)$
$\displaystyle\qquad{}=2z\alpha_{0}(u)^{\ast}\rho\big{(}|z|^{-2\kappa},|z|^{2\kappa}\big{)}C^{\prime\ast}C^{\prime}\alpha_{1}(u)+2\overline{z}\alpha_{1}(u)^{\ast}\rho\big{(}|z|^{-2\kappa},|z|^{2\kappa}\big{)}C^{{}^{\prime}\ast}C^{\prime}\alpha_{0}(u)$
$\displaystyle\qquad\quad{}+\sum_{m+n\geq
2}^{\infty}\overline{z}^{m}z^{n}\big{(}1-(-1)^{m+n}\big{)}\alpha_{m}(u)^{\ast}\rho\big{(}|z|^{-2\kappa},|z|^{2\kappa}\big{)}C^{\prime\ast}C^{\prime}\alpha_{n}(u).$

The dominant terms come from $m=0,n=1$ and $m=1,n=0$; both of order
$O\big{(}|z|^{1-2|\kappa|}\big{)}$. ∎

We will see later for purpose of integration by parts, that the change in $K$
between the points
$\big{(}x_{1},\ldots,x_{N-2},e^{\mathrm{i}\theta},e^{\mathrm{i}(\theta-\varepsilon)}\big{)}$
and
$\big{(}x_{1},\ldots,x_{N-2},e^{\mathrm{i}\theta},e^{\mathrm{i}(\theta+\varepsilon)}\big{)}$
is a key part of the analysis; this uses the relation
$K\big{(}\big{(}x_{1},\ldots,x_{N-2},e^{\mathrm{i}\theta},e^{\mathrm{i}(\theta-\varepsilon)}\big{)}\big{)}=\sigma
K\big{(}\big{(}x_{1},\ldots,x_{N-2},e^{\mathrm{i}(\theta-\varepsilon)},e^{\mathrm{i}\theta}\big{)}\big{)}\sigma$.

##  6 Bounds

In this section we derive bounds on $L(x)$ of global and local type.
Throughout we adopt the normalization $L(x_{0})=I$. The operator norm on
$n_{\tau}\times n_{\tau}$ complex matrices is defined by
$\|M\|=\sup\\{|Mv|\colon|v|=1\\}$.

######  Theorem 6.1.

There is a constant $c$ depending on $\kappa$ such that $\|L(x)\|\leq
c\prod\limits_{1\leq i<j\leq N}|x_{i}-x_{j}|^{-|\kappa|}$ for each
$x\in\mathbb{T}_{\rm reg}^{N}$.

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 $M(0)=I$, $\frac{d}{dt}M(t)=M(t)F(t)$ and $\|F(t)\|\leq f(t)$ for
$0\leq t\leq 1$ then $\|M(t)-I\|\leq\exp\int_{0}^{t}f(s)\mathrm{d}s-1$ and
$\|M(1)\|\leq\exp\int_{0}^{1}f(s)\mathrm{d}s$.

###### Proof from [11, Theorem 7.1.11].

Let $\ell(t):=\|M(t)-I\|$ then the equation
$M(t)-I=\int_{0}^{t}M(s)F(s)\mathrm{d}s$ and the inequalities
$\|M(t)\|\leq\|M(t)-I\|+\|I\|$ (and $\|I\|=1$) imply that
$\ell(t)\leq\int_{0}^{t}(\ell(s)+1)f(s)\mathrm{d}s$. Define differentiable
functions $b(t)$ and $h(t)$ by

$\displaystyle h(t):=\exp\int_{0}^{t}f(s)\mathrm{d}s,$ $\displaystyle
b(t)h(t)=\int_{0}^{t}(\ell(s)+1)f(s)\mathrm{d}s+1.$

Apply $\frac{d}{dt}$to the latter equation:

$\displaystyle b^{\prime}(t)h(t)+b(t)f(t)h(t)=(\ell(t)+1)f(t),$ $\displaystyle
b^{\prime}(t)h(t)=f(t)\left\\{\ell(t)+1-\int_{0}^{t}(\ell(s)+1)f(s)\mathrm{d}s-1\right\\}$
$\displaystyle\hphantom{b^{\prime}(t)h(t)}{}=f(t)\left\\{\ell(t)-\int_{0}^{t}(\ell(s)+1)f(s)\mathrm{d}s\right\\}\leq
0.$

Hence $b^{\prime}(t)\leq 0$ and $b(t)\leq b(0)=1$ which implies

$\displaystyle\ell(t)\leq\int_{0}^{t}(\ell(s)+1)f(s)\mathrm{d}s=b(t)h(t)-1\leq
h(t)-1.$

Finally $\|M(1)\|\leq$ $\|M(1)-I\|+1\leq\exp\int_{0}^{1}f(s)\mathrm{d}s$. ∎

Next we set up a differentiable path $p(t)=(p_{1}(t),\ldots,p_{N}(t))$ in
$\mathbb{C}_{\rm reg}^{N}$ starting at $x_{0}$ and obtain the equation

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}L(p(t))=\kappa
L(p(t))\sum_{i=1}^{N}\left\\{\sum_{j\neq
i}\frac{p_{i}^{\prime}(t)}{p_{i}(t)-p_{j}(t)}\tau((i,j))-\gamma\frac{p_{i}^{\prime}(t)}{p_{i}(t)}I\right\\}$
$\displaystyle\hphantom{\frac{\mathrm{d}}{\mathrm{d}t}L(p(t))}{}=\kappa
L(p(t))\left\\{\sum_{1\leq i<j\leq
N}\frac{p_{j}^{\prime}(t)-p_{i}^{\prime}(t)}{p_{j}(t)-p_{i}(t)}\tau((i,j))-\gamma\sum_{i=1}^{N}\frac{p_{i}^{\prime}(t)}{p_{i}(t)}I\right\\}.$

Suppose
$x=\big{(}e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N}}\big{)}\in\mathcal{C}_{0}$
and $\theta_{1}<\theta_{2}<\cdots<\theta_{N}<\theta_{1}+2\pi$. Define the path
$p(t)=\big{(}e^{\mathrm{i}g_{1}(t)},\ldots,e^{\mathrm{i}g_{N}(t)}\big{)}$
where $g_{j}(t)=(1-t)\frac{2(j-1)\pi}{N}+t\theta_{j}$ for $1\leq j\leq N$.
Then $p(t)\in\mathcal{C}_{0}$ for $0\leq t\leq 1$ because
$g_{i+1}(t)-g_{i}(t)=(1-t)\frac{2\pi}{N}+t(\theta_{i+1}-\theta_{i})>0$ for
$1\leq i<N$ and
$2\pi+g_{1}(t)-g_{N}(t)=2\pi+t\theta_{1}-(1-t)\frac{2(N-1)\pi}{N}-t\theta_{N}=(1-t)\frac{2\pi}{N}+t(2\pi+\theta_{1}-\theta_{N})>0$.
The factor of $\tau((i,j))$ in the equation is

$\displaystyle\mathrm{i}\frac{g_{j}^{\prime}(t)e^{\mathrm{i}g_{j}(t)}-g_{i}^{\prime}(t)e^{\mathrm{i}g_{i}(t)}}{e^{\mathrm{i}g_{j}(t)}-e^{\mathrm{i}g_{i}(t)}}=\frac{1}{2}\left\\{(g_{j}^{\prime}(t)-g_{i}^{\prime}(t))\frac{\cos\big{(}\frac{1}{2}(g_{j}(t)-g_{i}(t))\big{)}}{\sin\big{(}\frac{1}{2}(g_{j}(t)-g_{i}(t))\big{)}}+\mathrm{i}(g_{j}^{\prime}(t)+g_{i}^{\prime}(t))\right\\}.$

We will apply Lemma 6.2 to
$\widetilde{L}(x)=\prod\limits_{j=1}^{N}x_{j}^{\gamma\kappa}L(x)$; this only
changes the phase and removes the
$\sum\limits_{i=1}^{N}\frac{p_{i}^{\prime}(t)}{p_{i}(t)}$ term. In the
notation of Lemma 6.2

$\displaystyle
f(t)=|\kappa|\sum\limits_{i<j}\left|\mathrm{i}\frac{g_{j}^{\prime}(t)e^{\mathrm{i}g_{j}(t)}-g_{i}^{\prime}(t)e^{\mathrm{i}g_{i}(t)}}{e^{\mathrm{i}g_{j}(t)}-e^{\mathrm{i}g_{i}(t)}}\right|$

(since $\|\tau((i,j))\|=1$). To set up the integral
$\int_{0}^{1}f(t)\mathrm{d}t$ let

$\displaystyle\phi_{ij}(t)=\frac{1}{2}(g_{j}(t)-g_{i}(t))=\frac{1}{2}\left\\{(1-t)\frac{2(j-i)\pi}{N}+t(\theta_{j}-\theta_{i})\right\\}$

so that
$\phi_{ij}^{{}^{\prime}}(t)=\frac{1}{2}\big{(}\theta_{j}-\theta_{i}+\frac{2(j-i)\pi}{N}\big{)}$
and $0<\phi_{ij}(t)<\pi$ for $i<j$ and $0\leq t\leq 1$. The terms
$|\mathrm{i}(g_{j}^{\prime}(t)+g_{i}^{\prime}(t))|\leq 4\pi$ provide a simple
bound (no singularities off $\mathbb{T}_{\rm reg}^{N}$). The dominant terms
come from
$\int_{0}^{1}\frac{|\phi_{ij}^{\prime}\cos\phi_{ij}(t)|}{\sin\phi_{ij}(t)}\mathrm{d}t$.
There are two cases. Let $\phi_{0},\phi_{1}$ satisfy $0<\phi_{0},\phi_{1}<\pi$
and let $\phi(t)=(1-t)\phi_{0}+t\phi_{1}$. The antiderivative
$\int\frac{\phi^{\prime}\cos\phi(t)}{\sin\phi(t)}\mathrm{d}t=\log\sin\phi(t)$.
The first case applies when either $0<\phi_{0},\phi_{1}\leq\frac{\pi}{2}$ or
$\frac{\pi}{2}\leq\phi_{0},\phi_{1}<\pi$ (assign
$\phi_{0}=\frac{\pi}{2}=\phi_{1}$ to the first interval); then
$\phi^{\prime}\cos\phi(t)\geq 0$ if $0<\phi_{0}\leq\phi_{1}\leq\frac{\pi}{2}$
or $\frac{\pi}{2}\leq\phi_{1}\leq\phi_{0}<\pi$ and
$\phi^{\prime}\cos\phi(t)<0$ otherwise. These imply

$\displaystyle\int_{0}^{1}\frac{|\phi^{\prime}\cos\phi(t)|}{\sin\phi(t)}\mathrm{d}t=\left|\log\frac{\sin\phi_{1}}{\sin\phi_{0}}\right|.$

The second case applies when either $0<\phi_{0}<\frac{\pi}{2}<\phi_{1}<\pi$
(thus $\phi^{\prime}>0$) or $0<\phi_{1}<\frac{\pi}{2}<\phi_{0}<\pi$. Let
$\phi(t_{0})=\frac{\pi}{2}$ (that is,
$t_{0}=\frac{\pi/2-\phi_{0}}{\phi_{1}-\phi_{0}})$. In the first situation

$\displaystyle\int_{0}^{1}\frac{|\phi^{\prime}\cos\phi(t)|}{\sin\phi(t)}\mathrm{d}t=\int_{0}^{t_{0}}\frac{\phi^{\prime}\cos\phi(t)}{\sin\phi(t)}\mathrm{d}t-\int_{t_{0}}^{1}\frac{\phi^{\prime}\cos\phi(t)}{\sin\phi(t)}\mathrm{d}t=-\log\sin\phi_{0}-\log\sin\phi_{1},$

since $\log\sin\frac{\pi}{2}=0$; and the same value holds for the second
situation. We obtain

$\displaystyle\int_{0}^{1}f(t)\mathrm{d}t\leq|\kappa|\sum_{1\leq i<j\leq
N}\left\\{-\log\sin\frac{\theta_{j}-\theta_{i}}{2}-\log\sin\frac{(j-i)\pi}{N}+4\pi\right\\}.$

Taking exponentials and using the lemma (recall
$\big{|}e^{\mathrm{i}\phi_{1}}-e^{\mathrm{i}\phi_{2}}\big{|}=2\sin\big{|}\frac{\phi_{1}-\phi_{2}}{2}\big{|}$)
we obtain

$\displaystyle\|L(x)\|\leq c\prod\limits_{1\leq i<j\leq
N}|x_{i}-x_{j}|^{-|\kappa|}.$

This bound applies to all of $\mathbb{T}_{\rm reg}^{N}$ when $L(x_{0})$
commutes with $\upsilon$ and $L$ is extended to $\mathbb{T}_{\rm reg}^{N}$ as
in Definition 3.6. This completes the proof of Theorem 6.1.

Next we find bounds on the series expansion from (5.1)

$\displaystyle
L_{1}(x)=\left(\big{(}u^{2}-z^{2}\big{)}\prod_{j=1}^{N-2}x_{j}\right)^{-\gamma\kappa}\rho\big{(}z^{-\kappa},z^{\kappa}\big{)}\sum_{n=0}^{\infty}\alpha_{n}(x(u,0))z^{n},$

where $|z|<\delta_{0}:=\min\limits_{1\leq j\leq N-2}\left|u-x_{j}\right|$ and
$\operatorname{Im}\frac{z}{u}>0$. Recall the recurrence (5.4)

$\displaystyle
S_{n}:=\sum_{i=0}^{n-1}\alpha_{n-1-i}\big{\\{}(-1)^{i}\beta_{i}-\sigma\beta_{i}\sigma\big{\\}},\qquad\beta_{i}:=\sum_{j=0}^{N-2}\frac{\tau((j,N))}{(u-x_{j})^{i+1}},$
$\displaystyle\alpha_{2n}(x(u,0))=\frac{\kappa}{2n}S_{2n},$
$\displaystyle\alpha_{2n+1}(x(u,0))=\rho\left(\frac{\kappa}{2n+1-2\kappa},\frac{\kappa}{2n+1+2\kappa}\right)S_{2n+1}.$

######  Proposition 6.3.

Suppose $|\kappa|\leq\kappa_{0}<\frac{1}{2}$ and $\lambda:=(N-2)\kappa_{0}$
then for $n\geq 0$

$\displaystyle\|\alpha_{2n}(x(u,0))\|\leq\|\alpha_{0}(x(u,0))\|\frac{(\lambda)_{n}\big{(}\lambda+\frac{1}{2}-\kappa_{0}\big{)}_{n}}{n!\big{(}\frac{1}{2}-\kappa_{0}\big{)}_{n}}\delta_{0}^{-2n},$
(6.1)
$\displaystyle\|\alpha_{2n+1}(x(u,0))\|\leq\|\alpha_{0}(x(u,0))\|\frac{(\lambda)_{n+1}\big{(}\lambda+\frac{1}{2}-\kappa_{0}\big{)}_{n}}{n!\big{(}\frac{1}{2}-\kappa_{0}\big{)}_{n+1}}\delta_{0}^{-2n-1}.$

###### Proof.

Suppose $n\geq 1$ then
$\|S_{n}\|\leq\sum\limits_{i=0}^{n-1-i}\|\alpha_{n-1-i}\|(2N-4)\delta_{0}^{-i-1}$
(since $\|\tau((j,N-1))\|=1$). Furthermore, since $\big{|}\frac{\kappa}{n\pm
2\kappa}\big{|}\leq\frac{\kappa_{0}}{n-2\kappa_{0}}$ for $n\geq 2$, we find

$\displaystyle\|\alpha_{2n+1}\|\leq\frac{2\lambda}{2n+1-2\kappa_{0}}\sum\limits_{i=0}^{2n}\|\alpha_{2n-i}\|\delta_{0}^{-i-1},$
$\displaystyle\|\alpha_{2n}\|\leq\frac{\lambda}{n}\sum\limits_{i=0}^{2n-1}\|\alpha_{2n-1-i}\|\delta_{0}^{-i-1}.$

To set up an inductive argument let $t_{n}$ denote the hypothetical bound on
$\|\alpha_{n}(x(u,0))\|$ and set
$v_{n}=\sum\limits_{i=0}^{n-1}t_{n-1-i}\delta_{0}^{-i-1}$; then
$v_{n}=\delta_{0}^{-1}(t_{n-1}+v_{n-1})$ for $n\geq 2$. Setting
$t_{2n}=\frac{\lambda}{n}v_{2n}$ and
$t_{2n+1}=\frac{2\lambda}{2n+1-2\kappa_{0}}v_{2n+1}$ the recurrence relations
become

$\displaystyle
t_{2n}=\frac{\lambda}{n}\left(t_{2n-1}+\frac{2n-1-2\kappa_{0}}{2\lambda}t_{2n-1}\right)\delta_{0}^{-1}=\frac{2\lambda+2n-1-2\kappa_{0}}{2n}t_{2n-1}\delta_{0}^{-1},$
$\displaystyle
t_{2n+1}=\frac{2\lambda}{2n+1-2\kappa_{0}}\left(t_{2n}+\frac{n}{\lambda}t_{2n}\right)\delta_{0}^{-1}=\frac{2\lambda+2n}{2n+1-2\kappa_{0}}t_{2n}\delta_{0}^{-1}.$

Starting with
$\|\alpha_{1}\|\leq\frac{2\lambda}{1-2\kappa_{0}}\|\alpha_{0}\|\delta_{0}^{-1}=t_{1}$
the stated bounds are proved inductively. ∎

By use of Stirling’s formula for $\frac{\Gamma(n+a)}{\Gamma(n+b)}\sim n^{a-b}$
we see that $t_{n}$ behaves like (a multiple of) $n^{2\lambda-1}$ for large
$n$. Also there is a constant $c^{\prime}$ depending on $N$ and $\kappa_{0}$
such that

$\displaystyle\sum_{n=2}^{\infty}\|\alpha_{n}(x(u,0))\||z|^{n}\leq
c^{\prime}\|\alpha_{0}(x(u,0))\|\left(\frac{|z|}{\delta_{0}}\right)^{2}\left(1-\frac{|z|}{\delta_{0}}\right)^{-2\lambda-2}.$
(6.2)

We also need to analyze the effect of small changes in $u$. Fix a point
$x(\widetilde{u},0)$ and consider series expansions of
$\alpha_{n}(x(\widetilde{u},0))$ in powers of $(u-\widetilde{u})$. Let
$\delta_{1}:=\min\limits_{1\leq j\leq N-2}|\widetilde{u}-x_{j}|$. Recall
equation (5.3)

$\displaystyle\partial_{u}\alpha_{0}(x(u,0))=\kappa\alpha_{0}(x(u,0))\\{\beta_{0}(x(u,0))+\sigma\beta_{0}(x(u,0))\sigma\\},$

and solve this in the form

$\displaystyle\alpha_{0}(x(u,0))=\sum_{n=0}^{\infty}\alpha_{0,n}(x(\widetilde{u},0))(u-\widetilde{u})^{n}.$

This leads to the recurrence (suppressing the arguments $x(\widetilde{u},0)$)

$\displaystyle\sum_{n=1}^{\infty}n\alpha_{0,n}(u-\widetilde{u})^{n-1}$
$\displaystyle\qquad{}=\kappa\sum_{n=0}^{\infty}\alpha_{0,n}(u-\widetilde{u})^{n}\sum_{m=0}^{\infty}(-1)^{m}(u-\widetilde{u})^{m}\sum_{j=0}^{N-2}\frac{\tau((j,N-1))+\tau((j,N))}{(\widetilde{u}-x_{j})^{m+1}},$
$\displaystyle(n+1)\alpha_{0,n+1}=\kappa\sum_{m=0}^{n}\alpha_{0,n-m}\widetilde{\beta}_{m}(x(\widetilde{u},0)),$
$\displaystyle\widetilde{\beta}_{m}(x(\widetilde{u},0)):=(-1)^{m}\sum_{j=0}^{N-2}\frac{\tau((j,N-1))+\tau((j,N))}{(\widetilde{u}-x_{j})^{m+1}}.$

Thus $\|\widetilde{\beta}_{m}\|\leq\frac{2(N-2)}{\delta_{1}^{m+1}}$ and by a
similar method as above we find

$\displaystyle\|\alpha_{0,n}(x(\widetilde{u},0))\|\leq\frac{(2\lambda)_{n}}{n!}\delta_{1}^{-n}\|\alpha_{0}(x(\widetilde{u},0))\|,$
(6.3)

where $\lambda=(N-2)\kappa_{0}$. From

$\displaystyle\alpha_{1}(x(u,0))=\rho\left(\frac{\kappa}{1-2\kappa},\frac{\kappa}{1+2\kappa}\right)\alpha_{0}(x(u,0))\sum_{j=0}^{N-2}\frac{\tau((j,N-1))-\tau((j,N))}{(u-x_{j})}$
$\displaystyle\hphantom{\alpha_{1}(x(u,0))}{}=\rho\left(\frac{\kappa}{1-2\kappa},\frac{\kappa}{1+2\kappa}\right)\sum_{n=0}^{\infty}\alpha_{0,n}(u-\widetilde{u})^{n}$
$\displaystyle\hphantom{\alpha_{1}(x(u,0))=}{}\times\sum_{m=0}^{\infty}(-1)^{m}(u-\widetilde{u})^{m}\sum_{j=0}^{N-2}\frac{\tau((j,N-1))-\tau((j,N))}{(\widetilde{u}-x_{j})^{m+1}},$

we can derive a recurrence for the coefficients in
$\alpha_{1}(x(u,0))=\sum\limits_{n=0}^{\infty}\alpha_{1,n}(x(\widetilde{u},0))(u-\widetilde{u})^{n}$.
Also

$\displaystyle\|\alpha_{1,n}(x(\widetilde{u},0))\|\leq\frac{2\lambda}{1-2\kappa_{0}}\|\alpha_{0}(x(\widetilde{u},0))\|\sum_{j=0}^{n}\frac{(2\lambda)_{n-j}}{(n-j)!}\delta_{1}^{j-n}\delta_{1}^{-j-1}$
$\displaystyle\hphantom{\|\alpha_{1,n}(x(\widetilde{u},0))\|}{}=\frac{2\lambda(2\lambda+1)_{n}}{(1-2\kappa_{0})n!}\delta_{1}^{-n-1}\|\alpha_{0}(x(\widetilde{u},0))\|;$

note $2\lambda(2\lambda+1)_{n}=(2\lambda)_{n+1}$.

Essentially we are setting up bounds on behavior of $L(x(u,z))$ for points
near $x(\widetilde{u},0)$ in terms of $\|\alpha_{0}(x(\widetilde{u},0))\|$
which is handled by the global bound.

In the series

$\displaystyle\sum_{n=0}^{\infty}\alpha_{n}(x(u,0))z^{n}=\sum_{m,n=0}^{\infty}\alpha_{n,m}(x(\widetilde{u},0))z^{n}(u-\widetilde{u})^{m}$

the first order terms are

$\displaystyle\alpha_{00}(x(\widetilde{u},0))+\alpha_{0,1}(x(\widetilde{u},0))(u-\widetilde{u})+\alpha_{1,0}(x(\widetilde{u},0))z,$

and the bounds (6.1) for the omitted terms

$\displaystyle\sum_{n=2}^{\infty}\|\alpha_{n}(x(u,0))\||z|^{n}\leq
c^{\prime}\|\alpha_{0}(x(u,0))\|\left(\frac{|z|}{\delta_{0}}\right)^{2}\left(1-\frac{|z|}{\delta_{0}}\right)^{-2\lambda-2},$
$\displaystyle\sum\limits_{n=2}^{\infty}\|\alpha_{0,n}(x(\widetilde{u},0))\||u-\widetilde{u}|^{n}\leq(2\lambda)_{2}\left(\frac{|u-\widetilde{u}|}{\delta_{1}}\right)^{2}\left(1-\frac{|u-\widetilde{u}|}{\delta_{1}}\right)^{-2\lambda-2}\|\alpha_{0}(x(\widetilde{u},0))\|,$
(6.4)
$\displaystyle|z|\sum_{n=1}^{\infty}\|\alpha_{1,n}(x(\widetilde{u},0))\||u-\widetilde{u}|^{n}\leq\frac{(2\lambda)_{2}}{1-2\kappa_{0}}\left(\frac{|z(u-\widetilde{u})|}{\delta_{1}^{2}}\right)\left(1-\frac{|u-\widetilde{u}|}{\delta_{1}}\right)^{-2\lambda-2}\|\alpha_{0}(x(\widetilde{u},0))\|.$

Note there is a difference between $\delta_{0}$ and $\delta_{1}$:
$\delta_{0},\delta_{1}$ are the distances from the nearest $x_{j}$ ($1\leq
j\leq N-2)$ to $u,\widetilde{u}$ respectively; thus the double series
converges in $|z|+|u-\widetilde{u}|<\delta_{1}$ because this implies
$|z|<\delta_{1}-|u-\widetilde{u}|\leq\delta_{0}$, by the triangle inequality:
$\delta_{1}\leq|u-\widetilde{u}|+\delta_{0}$ .

##  7 Sufficient condition for the inner product property

In this section we will use the series

$\displaystyle
L_{1}(y,u-z,u+z)=\left(\prod\limits_{j=1}^{N}x_{j}\right)^{-\gamma\kappa}\rho\big{(}z^{-\kappa},z^{\kappa}\big{)}\sum_{n=0}^{\infty}\alpha_{n}(x(u,0))z^{n},$

normalized by $\alpha_{0}\big{(}x^{(0)}\big{)}=I$ where
$x^{(0)}=\big{(}1,\omega,\omega^{2},\ldots,\omega^{N-3},\omega^{-3/2},\omega^{-3/2}\big{)}$,
$\omega=e^{2\pi\mathrm{i}/N}$. The hypothesis is that there exists a Hermitian
matrix $H$ such that $\upsilon H=H\upsilon$ (recall $\upsilon:=\tau(w_{0})$)
and the matrix $H_{1}$ defined by

$\displaystyle L_{1}(x)^{\ast}H_{1}L_{1}(x)=L(x)^{\ast}HL(x)$ (7.1)

commutes with $\sigma=\tau(N-1,N)$ (recall $L(x_{0})=I$). Setting $x=x_{0}$ we
find that $H=L_{1}(x_{0})^{\ast}H_{1}L_{1}(x_{0})$. The analogous condition
has to hold for each face of $\mathcal{C}_{0}$ and any such face can be
obtained from $\\{x_{N-1}=x_{N}\\}$ by applying $x\mapsto xw_{0}^{m}$ with
suitable $m$. For notational simplicity we will work out only the
$\\{x_{N-2}=x_{N-1}\\}$ case. From the general relation
$w(i,j)w^{-1}=(w(i),w(j))$ we obtain $w_{0}^{-1}(N-1,N)w_{0}=(N-2,N-1)$. A
matrix $M$ commutes with $\tau(N-2,N-1)$ if and only if $\upsilon
M\upsilon^{-1}$ commutes with $\sigma$. Let

$\displaystyle
x^{\prime}=\big{(}x_{1}^{\prime},\ldots,x_{N-3}^{\prime},u-z,u+z,x_{N}^{\prime}\big{)},$
$\displaystyle
x^{\prime}w_{0}^{-1}=\big{(}x_{N}^{\prime},x_{1}^{\prime},\ldots,x_{N-3}^{\prime},u-z,u+z\big{)}=x,$
$\displaystyle
L_{2}(x^{\prime}):=\upsilon^{-1}L_{1}\big{(}x^{\prime}w_{0}^{-1}\big{)}\upsilon=\left(\prod\limits_{j=1}^{N}x_{j}\right)^{-\gamma\kappa}\upsilon^{-1}\rho\big{(}z^{-\kappa},z^{\kappa}\big{)}\sum_{n=0}^{\infty}\alpha_{n}(y,u)\upsilon
z^{n}.$

This is a solution of (3.1) by Proposition 3.1. This has the analogous
behavior to $L_{1}$; writing

$\displaystyle\upsilon^{-1}\rho\big{(}z^{-\kappa},z^{\kappa}\big{)}\alpha_{n}(y,u)\upsilon=\big{\\{}\upsilon^{-1}\rho\big{(}z^{-\kappa},z^{\kappa}\big{)}\upsilon\big{\\}}\big{\\{}\upsilon^{-1}\alpha_{n}(y,u)\upsilon\big{\\}}$

implies the relations

$\displaystyle\tau(N-2,N-1)\big{\\{}\upsilon^{-1}\rho\big{(}z^{-\kappa},z^{\kappa}\big{)}\upsilon\big{\\}}=\big{\\{}\upsilon^{-1}\rho\big{(}z^{-\kappa},z^{\kappa}\big{)}\upsilon\big{\\}}\tau(N-2,N-1),$
$\displaystyle\tau(N-2,N-1)\big{\\{}\upsilon^{-1}\alpha_{n}(y,u)\upsilon\big{\\}}\tau(N-2,N-1)=(-1)^{n}\big{\\{}\upsilon^{-1}\alpha_{n}(y,u)\upsilon\big{\\}},\qquad
n\geq 0.$

We claim that the Hermitian matrix $H_{2}$ defined by

$\displaystyle L_{2}(x)^{\ast}H_{2}L_{2}(x)=L(x)^{\ast}HL(x)$ (7.2)

commutes with $\tau(N-2,N-1)$. There is a subtle change: the base point
$x^{(0)}=\big{(}1,\omega,\ldots,\omega^{N-2}$,
$\omega^{-3/2},\omega^{-3/2}\big{)}$ is replaced by
$\big{(}\omega,\ldots,\omega^{N-2},\omega^{-3/2},\omega^{-3/2},1\big{)}$ and
now $\omega x_{0}=\big{(}\omega,\ldots,\omega^{N-1},1\big{)}$ is in the domain
of convergence of $L_{2}$. Set $x=\omega x_{0}$ in (7.2) to obtain

$\displaystyle L_{2}(\omega x_{0})=\upsilon^{-1}L_{1}\big{(}\omega
x_{0}w_{0}^{-1}\big{)}\upsilon=\upsilon^{-1}L_{1}(x_{0})\upsilon,$
$\displaystyle H_{2}=(L_{2}(\omega x_{0})^{\ast})^{-1}HL_{2}(\omega
x_{0})^{-1}=\upsilon^{-1}(L_{1}(x_{0})^{\ast})^{-1}\upsilon
H\upsilon^{-1}L_{1}(x_{0})^{-1}\upsilon$
$\displaystyle\hphantom{H_{2}}{}=\upsilon^{-1}(L_{1}(x_{0})^{\ast})^{-1}HL_{1}(x_{0})^{-1}\upsilon=\upsilon^{-1}H_{1}\upsilon,$

because $H$ commutes with $\upsilon$ (and $L(\omega x_{0})=L(x_{0})=I$ by the
homogeneity). Thus $H_{2}$ commutes with $\tau(N-2,N-1)$.

From Theorem 6.1 we have the bound

$\displaystyle\|L(x)^{\ast}HL(x)\|\leq
c\prod\limits_{i<j}|x_{i}-x_{j}|^{-2|\kappa|}.$

Denote $K(x)=L(x)^{\ast}HL(x)$. We will show that there is an interval
$-\kappa_{1}<\kappa<\kappa_{1}$ where $\kappa_{1}$ depends on $N$ such that

$\displaystyle\int_{\mathbb{T}^{N}}\big{\\{}(x_{N}\mathcal{D}_{N}f)^{\ast}(x)K(x)g(x)-f^{\ast}(xt)K(x)x_{N}\mathcal{D}_{N}g(x)\big{\\}}\mathrm{d}m(x)=0,$

for each $f,g\in C^{1}\big{(}\mathbb{T}^{N};V_{\tau}\big{)}$. Consider the
Haar measure of
$\big{\\{}x\colon\min\limits_{i<j}|x_{i}-x_{j}|<\varepsilon\big{\\}}$; let
$\sin\frac{\varepsilon^{\prime}}{2}=\frac{\varepsilon}{2}$ and $i<j$ then
$m\big{\\{}x\colon|x_{i}-x_{j}|\leq\varepsilon\big{\\}}=\frac{1}{\pi}\varepsilon^{\prime}$,
thus
$m\big{\\{}x\colon\min\limits_{i<j}|x_{i}-x_{j}|<\varepsilon\big{\\}}\leq\binom{N}{2}\frac{\varepsilon^{\prime}}{\pi}$.
The integral is broken up into three pieces. The aim is to let
$\delta\rightarrow 0$; where $\delta$ satisfies an upper bound
$\delta<\min\big{(}\big{(}2\sin\frac{\pi}{N}\big{)}^{2},\frac{1}{9}\big{)}$;
the first term comes from the maximum spacing of $N$ points on $\mathbb{T}$
and the second is equivalent to $3\delta<\delta^{1/2}$. Also
$\delta^{\prime}:=2\arcsin\frac{\delta}{2}$.

  1. 1.

$\min\limits_{i<j}|x_{i}-x_{j}|<\delta$, done with the integrability of
$\prod\limits_{i<j}|x_{i}-x_{j}|^{-2|\kappa|}$ for $|\kappa|<\frac{1}{N}$
(from the Selberg integral
$\int_{\mathbb{T}^{N}}\prod\limits_{i<j}|x_{i}-x_{j}|^{-2|\kappa|}\mathrm{d}m(x)=\frac{\Gamma
1-N|\kappa|)}{\Gamma(1-|\kappa|)^{N}}$), and the measure of the set is
$O(\delta)$. The limit as $\delta\rightarrow 0$ is zero by the dominated
convergence theorem.

  2. 2.

$\delta\leq\min\limits_{1\leq i<j\leq N-1}|x_{i}-x_{j}|<\delta^{1/2}$ and
$\delta\leq\min\limits_{1\leq i\leq N-1}|x_{i}-x_{N}|$; this case uses the
same bound on $K$ and the $\mathbb{T}^{N-1}$-Haar measure of
$\big{\\{}x\in\mathbb{T}^{N-1}\colon\min\limits_{1\leq i<j\leq
N-1}|x_{i}-x_{j}|<\delta^{1/2}\big{\\}};$

  3. 3.

$\min\limits_{1\leq i<j\leq N-1}|x_{i}-x_{j}|\geq\delta^{1/2}$ and
$\min\limits_{1\leq i\leq N-1}|x_{i}-x_{N}|\geq\delta$. 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 $\Omega_{\delta}$,
equals

$\displaystyle\int_{\Omega_{\delta}}x_{N}\partial_{N}\\{f(x)^{\ast}K(x)g(x)\\}\mathrm{d}m(x).$

We use the coordinates $x_{j}=e^{\mathrm{i}\theta_{j}}$, $1\leq j\leq N$; thus
$x_{N}\partial_{N}=-\mathrm{i}\frac{\partial}{\partial\theta_{N}}$. For fixed
$(\theta_{1},\ldots,\theta_{N-1})$ the condition $x\in\Omega_{\delta}$ implies
that the set of $\theta_{N}$-values is a union of disjoint closed intervals
(it is possible there is only one, in the extreme case
$\theta_{j}=j\delta^{\prime}$ for $1\leq j\leq N-1$ the interval is
$N\delta^{\prime}\leq\theta_{N}\leq 2\pi$). In case (2) the
$\theta_{N}$-integration results in a sum of terms
$(f^{\ast}Kg)\big{(}e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i}\phi}\big{)}$
with coefficients $\pm 1$ where $\min\limits_{1\leq i\leq
N-1}|\phi-\theta_{i}|=\delta$. Each such sum is bounded by
$2(N-1)c\|f\|_{\infty}\|g\|_{\infty}\delta^{-N(N-1)|\kappa|}$, because
$\prod\limits_{i<j}|x_{i}-x_{j}|\geq\delta^{N(N-1)/2}$ on $\Omega_{\delta}$.
Thus the integral for part (2) is bounded by

$\displaystyle
2(N-1)\|f\|_{\infty}\|g\|_{\infty}\delta^{-N(N-1)|\kappa|}\binom{N-1}{2}\left(2\arcsin\frac{\delta^{1/2}}{2}\right)\leq
c^{\prime}\delta^{1/2-N(N-1)|\kappa|},$

for some finite constant $c^{\prime}$ (depending on $f$, $g$). This term tends
to zero as $\delta\rightarrow 0$ if $|\kappa|<\frac{1}{2N(N-1)}$.

In part (3) the intervals
$[\theta_{i}-\delta^{\prime},\theta_{i}+\delta^{\prime}]$ are pairwise
disjoint because $|\theta_{i}-\theta_{j}|\geq 3\delta^{\prime}$ for $i\neq j$
(recall $\sqrt{\delta}>3\delta$). To simplify the notation assume
$\theta_{1}<\theta_{2}<\cdots$ (the other cases follow from the group
invariance of the setup). Then the $\theta_{N}$-integration yields

$\displaystyle(2\pi)^{1-N}\sum_{j=1}^{N-1}\int_{R_{\delta}}\left\\{\begin{matrix}(f^{\ast}Kg)\big{(}e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{j}},\ldots,e^{\mathrm{i}(\theta_{j}-\delta^{\prime})}\big{)}\\\
-(f^{\ast}Kg)\big{(}e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{j}},\ldots,e^{\mathrm{i}(\theta_{j}+\delta^{\prime})}\big{)}\end{matrix}\right\\}\mathrm{d}\theta_{1}\cdots\mathrm{d}\theta_{N-1},$

where
$R_{\delta}:=\big{\\{}(\theta_{1},\ldots,\theta_{N-1}):\theta_{1}<\theta_{2}<\cdots<\theta_{N-1}<\theta_{1}+2\pi,\min\big{|}e^{\mathrm{i}\theta_{j}}-e^{\mathrm{i}\theta_{k}}\big{|}\geq\sqrt{\delta}\big{\\}}$.
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
$(f^{\ast}Kg)\big{(}e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i}(\theta_{N-1}-\delta^{\prime})}\big{)}-(f^{\ast}Kg)\big{(}e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i}(\theta_{N-1}+\delta^{\prime})}\big{)}$
tends to zero with $\delta$ but this is not enough to control the integral.
The idea is to show that

$\displaystyle\big{|}(f^{\ast}Kg)\big{(}e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i}(\theta_{N-1}-\delta^{\prime})}\big{)}-(f^{\ast}Kg)\big{(}e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i}(\theta_{N-1}+\delta^{\prime})}\big{)}\big{|}$
$\displaystyle\qquad{}\leq
c^{\prime\prime}\delta^{1/2-2|\kappa|}\big{|}(f^{\ast}Kg)\big{(}e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i}(\theta_{N-1}+\delta^{\prime})}\big{)}\big{|}$

for some constant $c^{\prime\prime}$. This can then be bounded using the
$\|K\|$ bound for sufficiently small $|\kappa|$. Fix
$y=\big{(}e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-2}}\big{)}$
and let $x(y,u-v,u+v)$ denote
$\big{(}e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-2}},u-z,u+z\big{)}$.
We will use the form $K=L_{1}^{\ast}H_{1}L_{1}$ from (7.1) with two pairs of
values along with $\widetilde{u}=e^{\mathrm{i}\theta_{N-1}}$, and set
$\zeta=e^{\mathrm{i\delta}^{\prime}}$

  1. 1)

$\eta^{(1)}=x(y,u_{1}-z_{1},u_{1}+z_{1})=x\big{(}y,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i}(\theta_{N-1}+\delta^{\prime})}\big{)}$,
then $u_{1}=\frac{1}{2}\widetilde{u}(1+\zeta)$,
$z_{1}=\frac{1}{2}\widetilde{u}(\zeta-1)$, $u_{1}-\widetilde{u}=z_{1}$,
$|z_{1}|=\delta$,

  2. 2)

$\eta^{(2)}=x(y,u_{2}-z_{2},u_{2}+z_{2})=x\big{(}y,e^{\mathrm{i}(\theta_{N-1}-\delta^{\prime})},e^{\mathrm{i}\theta_{N-1}}\big{)}$,
then $u_{2}=\frac{1}{2}\widetilde{u}\big{(}1+\zeta^{-1}\big{)}$,
$z_{2}=\frac{1}{2}\widetilde{u}\big{(}1-\zeta^{-1}\big{)}=\zeta^{-1}z_{1}$,
$u_{2}-\widetilde{u}=-z_{2}$, $|z_{2}|=\delta$.

Let
$\eta^{(3)}\\!=\\!x\big{(}y,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i}(\theta_{N-1}-\delta^{\prime})}\big{)}\\!=\\!\eta^{(2)}(N-1,N)$,
then by construction $K\big{(}\eta^{(3)}\big{)}\\!=\\!\sigma
K\big{(}\eta^{(2)}\big{)}\sigma$. We start by disposing of the $f$ and $g$
factors: by uniform continuous differentiability there is a constant
$c^{\prime\prime\prime}$ such that
$\big{\|}f\big{(}\eta^{(1)}\big{)}-f\big{(}\eta^{(3)}\big{)}\big{\|}\leq
c^{\prime\prime\prime}\delta^{\prime}$ and
$\big{\|}g\big{(}\eta^{(1)}\big{)}-g\big{(}\eta^{(3)}\big{)}\big{\|}\leq
c^{\prime\prime\prime}\delta^{\prime}$ (same constant for all of
$\mathbb{T}^{N}$). So the error made by assuming $f$ and $g$ are constant is
bounded by
$c^{\prime\prime\prime}\delta^{\prime}\big{(}\big{\|}K\big{(}\eta^{(1)}\big{)}\big{\|}+\big{\|}K\big{(}\eta^{(2)}\big{)}\big{\|}\big{)}$.
The problem is reduced to bounding $K\big{(}\eta^{(1)}\big{)}-\sigma
K\big{(}\eta^{(2)}\big{)}\sigma$. To add more detail about the effect of the
$\ast$-operation on $u$ and $z$ we compute

$\displaystyle
z^{\ast}=\frac{1}{2}\left(\frac{1}{u+z}-\frac{1}{u-z}\right)=-\frac{z}{u^{2}-z^{2}},\qquad
u^{\ast}=\frac{1}{2}\left(\frac{1}{u+z}+\frac{1}{u-z}\right)=\frac{u}{u^{2}-z^{2}}$

and if $u-z=e^{i\theta_{N-1}}$, $u+z=e^{i\theta_{N}}$ then

$\displaystyle
z=\frac{1}{2}e^{\mathrm{i}(\theta_{N-1}+\theta_{N})/2}\big{(}e^{\mathrm{i}(\theta_{N}-\theta_{N-1})/2}-e^{\mathrm{i}(\theta_{N-1}-\theta_{N})/2}\big{)}=\mathrm{i}e^{\mathrm{i}(\theta_{N-1}+\theta_{N})/2}\sin\frac{\theta_{N}-\theta_{N-1}}{2},$
$\displaystyle
z^{\ast}=\mathrm{i}e^{-\mathrm{i}(\theta_{N-1}+\theta_{N})/2}\sin\frac{\theta_{N-1}-\theta_{N}}{2}=\overline{z},$
$\displaystyle
u=\frac{1}{2}e^{\mathrm{i}(\theta_{N-1}+\theta_{N})/2}\big{(}e^{\mathrm{i}(\theta_{N}-\theta_{N-1})/2}+e^{\mathrm{i}(\theta_{N-1}-\theta_{N})/2}\big{)}=e^{\mathrm{i}(\theta_{N-1}+\theta_{N})/2}\cos\frac{\theta_{N}-\theta_{N-1}}{2},$
$\displaystyle
u^{\ast}=e^{-\mathrm{i}(\theta_{N-1}+\theta_{N})/2}\cos\frac{\theta_{N-1}-\theta_{N}}{2}=\overline{u};$

the $\ast$-operation agrees with complex conjugate on the torus and
$\rho(z^{-\kappa},z^{\kappa})^{\ast}=\rho(\overline{z}^{-\kappa},\overline{z}^{\kappa})$.
The reason for this is to emphasize that $L(x)^{\ast}$ is an analytic function
agreeing with the (Hermitian) adjoint of $L(x)$. Thus

$\displaystyle
K\big{(}\eta^{(1)}\big{)}=\sum_{n,m=0}^{\infty}\alpha_{n}(x(u_{1},0))^{\ast}\rho\big{(}z_{1}^{-\kappa},z_{1}^{\kappa}\big{)}^{\ast}H_{1}\rho\big{(}z_{1}^{-\kappa},z_{1}^{\kappa}\big{)}\alpha_{m}(x(u_{1},0))(z_{1}^{\ast})^{m}z_{1}^{n}$
$\displaystyle\hphantom{K\big{(}\eta^{(1)}\big{)}}{}=\sum_{n,m=0}^{\infty}\alpha_{n}(x(u_{1},0))^{\ast}H_{1}\rho\big{(}|z_{1}|^{-2\kappa},|z_{1}|^{2\kappa}\big{)}\alpha_{m}(x(u_{1},0))\overline{z_{1}}^{m}z_{1}^{n},$
(7.3)

because $H_{1}$ commutes with $\sigma$ and hence with
$\rho(z_{1}^{-\kappa},z_{1}^{\kappa})$, and

$\displaystyle K\big{(}\eta^{(3)}\big{)}=\sigma
K\big{(}\eta^{(2)}\big{)}\sigma$
$\displaystyle\hphantom{K\big{(}\eta^{(3)}\big{)}}{}=\sum_{n,m=0}^{\infty}(-1)^{m+n}\alpha_{n}(x(u_{2},0))^{\ast}H_{1}\rho\big{(}|z_{2}|^{-2\kappa},|z_{2}|^{2\kappa}\big{)}\alpha_{m}(x(u_{2},0))\overline{z_{2}}^{m}z_{2}^{n},$
(7.4)

because $\sigma\alpha_{n}(x(u_{2},0))\sigma=(-1)^{n}\alpha_{n}(x(u_{2},0))$
for $n\geq 0$. Now we use the expansion in powers of $(u-\widetilde{u})^{n}$
to evaluate $K\big{(}\eta^{(1)}\big{)}-K\big{(}\eta^{(3)}\big{)}$. From the
inequality (6.2)

$\displaystyle\sum_{n=2}^{\infty}\|\alpha_{n}(x(u,0))\||z|^{n}\leq
c^{\prime}\|\alpha_{0}(x(u,0))\|\left(\frac{|z|}{\delta_{0}}\right)^{2}\left(1-\frac{|z|}{\delta_{0}}\right)^{-2\lambda-2}$
$\displaystyle\hphantom{\sum_{n=2}^{\infty}\|\alpha_{n}(x(u,0))\||z|^{n}}{}=c^{\prime}\delta\|\alpha_{0}(x(u,0))\|\big{(}1-\delta^{1/2}\big{)}^{-2\lambda-2}$
(7.5)

with $\delta_{0}=\min\limits_{1\leq j\leq N-2}|u-x_{j}|=\delta^{1/2}$ we can
restrict the problem to $0\leq n,m\leq 1$. The omitted terms in
$K\big{(}\eta^{(1)}\big{)}-K\big{(}\eta^{(3)}\big{)}$ are bounded by
$c^{\prime\prime}\delta^{1-2|\kappa|}\|B_{1}\|\|\alpha_{0}(x(u_{2},x))\|^{2}$,
for some constant $c^{\prime\prime}$. Then

$\displaystyle
L_{1}\big{(}\eta^{(1)}\big{)}=\left(\prod\limits_{j=1}^{N}x_{j}^{(1)}\right)^{-\gamma\kappa}\rho\big{(}z_{1}^{-\kappa},z_{1}^{\kappa}\big{)}\left\\{\begin{matrix}\alpha_{00}(x(\widetilde{u},0))+\alpha_{0,1}(x(\widetilde{u},0))(u_{1}-\widetilde{u})\\\
+\alpha_{1,0}(x(\widetilde{u},0))z_{1}+O(\delta)\end{matrix}\right\\},$
$\displaystyle\sigma
L_{1}\big{(}\eta^{(2)}\big{)}\sigma=\left(\prod\limits_{j=1}^{N}x_{j}^{(2)}\right)^{-\gamma\kappa}\rho\big{(}z_{2}^{-\kappa},z_{2}^{\kappa}\big{)}\left\\{\begin{matrix}\alpha_{00}(x(\widetilde{u},0))+\alpha_{0,1}(x(\widetilde{u},0))(u_{2}-\widetilde{u})\\\
-\alpha_{1,0}(x(\widetilde{u},0))z_{2}+O(\delta)\end{matrix}\right\\},$

because $\sigma\alpha_{1}(x(u,0))\sigma=(-1)^{n}\alpha_{1}(x(u,0))$. The terms
$O(\delta)$ correspond to the bound in (7.5). Drop the argument
$x(\widetilde{u},0)$ for brevity. Combining these with (7.3) and (7.4) we
obtain

$\displaystyle
K\big{(}\eta^{(1)}\big{)}-K\big{(}\eta^{(3)}\big{)}=\\{\alpha_{0,1}(u_{1}-\widetilde{u})+\alpha_{1,0}z_{1}\\}^{\ast}H_{1}\rho\big{(}|z_{1}|^{-2\kappa},|z_{1}|^{2\kappa}\big{)}\alpha_{00}$
$\displaystyle\hphantom{K\big{(}\eta^{(1)}\big{)}-K\big{(}\eta^{(3)}\big{)}=}{}+\alpha_{00}^{\ast}H_{1}\rho\big{(}|z_{1}|^{-2\kappa},|z_{1}|^{2\kappa}\big{)}\\{\alpha_{0,1}(u_{1}-\widetilde{u})+\alpha_{1,0}z_{1}\\}$
$\displaystyle\hphantom{K\big{(}\eta^{(1)}\big{)}-K\big{(}\eta^{(3)}\big{)}=}{}+\\{\alpha_{0,1}(u_{1}-\widetilde{u})+\alpha_{1,0}z_{1}\\}^{\ast}H_{1}\rho\big{(}|z_{1}|^{-2\kappa},|z_{1}|^{2\kappa}\big{)}$
$\displaystyle\hphantom{K\big{(}\eta^{(1)}\big{)}-K\big{(}\eta^{(3)}\big{)}=}{}\times\\{\alpha_{0,1}(u_{1}-\widetilde{u})+\alpha_{1,0}z_{1}\\}$
$\displaystyle\hphantom{K\big{(}\eta^{(1)}\big{)}-K\big{(}\eta^{(3)}\big{)}=}{}-\\{\alpha_{0,1}(u_{2}-\widetilde{u})-\alpha_{1,0}z_{2}\\}^{\ast}H_{1}\rho\big{(}|z_{2}|^{-2\kappa},|z_{2}|^{2\kappa}\big{)}\alpha_{00}$
$\displaystyle\hphantom{K\big{(}\eta^{(1)}\big{)}-K\big{(}\eta^{(3)}\big{)}=}{}-\alpha_{00}^{\ast}H_{1}\rho\big{(}|z_{2}|^{-2\kappa},|z_{2}|^{2\kappa}\big{)}\\{\alpha_{0,1}(u_{1}-\widetilde{u})-\alpha_{1,0}z_{1}\\}$
$\displaystyle\hphantom{K\big{(}\eta^{(1)}\big{)}-K\big{(}\eta^{(3)}\big{)}=}{}-\\{\alpha_{0,1}(u_{2}-\widetilde{u})+\alpha_{1,0}z_{2}\\}^{\ast}H_{1}\rho\big{(}|z_{2}|^{-2\kappa},|z_{2}|^{2\kappa}\big{)}$
$\displaystyle\hphantom{K\big{(}\eta^{(1)}\big{)}-K\big{(}\eta^{(3)}\big{)}=}{}\times\\{\alpha_{0,1}(u_{2}-\widetilde{u})+\alpha_{1,0}z_{2}\\}+O(\delta).$

The key fact is that the
$\alpha_{00}^{\ast}H_{1}\rho\big{(}|z_{1}|^{-2\kappa},|z_{1}|^{2\kappa}\big{)}\alpha_{00}$
terms cancel out ($|z_{1}|=|z_{2}|$).

From
$\|\alpha_{1,0}(x(\widetilde{u},0))\|\leq\frac{(2\lambda)_{2}}{(1-2\kappa_{0})}\delta_{1}^{-1}\|\alpha_{0}(x(\widetilde{u},0))\|$
and $\|\alpha_{0,1}(x(\widetilde{u},0))\|\leq
2\lambda\delta_{1}^{-1}\|\alpha_{0}(x(\widetilde{u},0))\|$ (from (6.3))
$\delta_{1}=\delta_{0}-|u|=\delta^{1/2}-\delta=\delta^{1/2}(1-\delta^{1/2})$.
Thus the sum of the first order terms in
$K\big{(}\eta^{(1)}\big{)}-K\big{(}\eta^{(3)}\big{)}$ is bounded by
$c^{\prime\prime\prime}\|\alpha_{0}(x(\widetilde{u},0))\|^{2}\delta^{1/2-2|\kappa|}\big{(}1-\delta^{1/2}\big{)}^{-1}\|H_{1}\|$,
where the constant $c^{\prime\prime\prime}$ is independent of
$x(\widetilde{u},0)$ (but is dependent on $\kappa_{0}$ and $N$). Note
$|u_{1}-\widetilde{u}|=|u_{2}-\widetilde{u}|=|z_{1}|=|z_{2}|=\delta$. The
second last step is to relate $\|\alpha_{0}(x(\widetilde{u},0))\|$ to
$\|L_{1}\big{(}\eta^{(1)}\big{)}\|$; indeed

$\displaystyle
L_{1}\big{(}\eta^{(1)}\big{)}=\left(\prod\limits_{j=1}^{N}x_{j}^{(1)}\right)^{-\gamma\kappa}\rho\big{(}z_{1}^{-\kappa},z_{1}^{\kappa}\big{)}\left\\{\alpha_{0}(x(\widetilde{u},0))+\sum_{n=1}^{\infty}\alpha_{n}(x(\widetilde{u},0))z_{1}^{n}\right\\}.$

Similarly to (7.5)

$\displaystyle\sum_{n=1}^{\infty}\|\alpha_{n}(x(\widetilde{u},0))\||z|^{n}\leq
c^{\prime}\|\alpha_{0}(x(\widetilde{u},0))\|\left(\frac{|z|}{\delta_{0}}\right)\left(1-\frac{|z|}{\delta_{0}}\right)^{-2\lambda-1}$
$\displaystyle\hphantom{\sum_{n=1}^{\infty}\|\alpha_{n}(x(\widetilde{u},0))\||z|^{n}}{}=c^{\prime}\|\alpha_{0}(x(\widetilde{u},0))\|\delta^{1/2}\big{(}1-\delta^{1/2}\big{)}^{-2\lambda-1}\leq
c^{\prime\prime}\|\alpha_{0}(x(\widetilde{u},0))\|\delta^{1/2},$

(if $\delta<\frac{1}{9}$ then $1-\delta^{1/2}>\frac{2}{3}$); thus

$\displaystyle\|\alpha_{0}(x(\widetilde{u},0))\|\big{(}1-c^{\prime\prime}\delta^{1/2}\big{)}\leq\delta^{-|\kappa|}\big{\|}L_{1}\big{(}\eta^{(1)}\big{)}\big{\|}.$

By Theorem 6.1

$\displaystyle\big{\|}L\big{(}\eta^{(1)}\big{)}\big{\|}\leq
c\prod\limits_{1\leq i<j\leq
N-1}|x_{i}-x_{j}|^{-|\kappa|}\prod\limits_{j=1}^{N-2}\big{|}e^{\mathrm{i}\theta_{j}}-e^{\mathrm{i}(\theta_{N-1}+\delta^{\prime})}\big{|}^{-|\kappa|}\big{|}e^{\mathrm{i}\theta_{N-1}}-e^{\mathrm{i}(\theta_{N-1}+\delta^{\prime})}\big{|}^{-|\kappa|}$
$\displaystyle\hphantom{\big{\|}L\big{(}\eta^{(1)}\big{)}\big{\|}}{}\leq
c\delta^{-|\kappa|\\{(N+1)(N-2)/2+1\\}},$

because the first two groups of terms satisfy the bound
$|x_{i}-x_{j}|\geq\delta^{1/2}$. Combining everything we obtain the bound

$\displaystyle\big{\|}K\big{(}\eta^{(1)}\big{)}-K\big{(}\eta^{(3)}\big{)}\big{\|}\leq
c^{\prime\prime\prime}\|\alpha_{0}(x(\widetilde{u},0))\|^{2}\delta^{1/2-2|\kappa|}\big{(}1-\delta^{1/2}\big{)}^{-1}\|B_{1}\|$
$\displaystyle\hphantom{\big{\|}K\big{(}\eta^{(1)}\big{)}-K\big{(}\eta^{(3)}\big{)}\big{\|}}{}\leq
c^{\prime\prime}\|H_{1}\|\delta^{1/2-2|\kappa|-|\kappa|\\{(N+1)(N-2)+2\\}}.$

The constant is independent of $\eta^{(1)}$ and the exponent on $\delta$ is
$\frac{1}{2}-|\kappa|\big{(}N^{2}-N+2\big{)}$. Thus the integral of part (3)
goes to zero as $\delta\rightarrow 0$ if
$|\kappa|<\big{(}2\big{(}N^{2}-N+2\big{)}\big{)}^{-1}$. This is a crude bound,
considering that we know everything works for $-1/h_{\tau}<\kappa<1/h_{\tau}$,
but as we will see, an open interval of $\kappa$ values suffices.

######  Theorem 7.1.

If there exists a Hermitian matrix $H$ such that

$\displaystyle\upsilon
H=H\upsilon\qquad\text{and}\qquad(L_{1}(x_{0})^{\ast})^{-1}HL_{1}(x_{0})^{-1}$

commutes with $\sigma$, and
$-\big{(}2\big{(}N^{2}-N+2\big{)}\big{)}^{-1}<\kappa<\big{(}2\big{(}N^{2}-N+2\big{)}\big{)}^{-1}$
then

$\displaystyle\int_{\mathbb{T}^{N}}\\{(x_{i}\mathcal{D}_{i}f(x))^{\ast}L(x)^{\ast}HL(x)g(x)-f(x)^{\ast}L(x)^{\ast}HL(x)x_{i}\mathcal{D}_{i}g(x)\\}\mathrm{d}m(x)=0$

for $f,g\in C^{(1)}\big{(}\mathbb{T}^{N};V_{\tau}\big{)}$ and $1\leq i\leq N$.

It is important that we can derive uniqueness of $H$ from the relation,
because the conditions $\langle wf,wg\rangle=\langle f,g\rangle$, $\langle
x_{i}f,x_{i}g\rangle=\langle f,g\rangle$, and $\langle
x_{i}\mathcal{D}_{i}f,g\rangle=\langle f,x_{i}\mathcal{D}_{i}g\rangle$ for
$w\in\mathcal{S}_{N}$ and $1\leq i\leq N$ determine the Hermitian form
uniquely up to multiplication by a constant. Thus the measure
$K(x)\mathrm{d}m(x)$ 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 $L^{\ast}HL\mathrm{d}m$, specifically if $H$ commutes
with $\upsilon$, $(L_{1}(x_{0})^{\ast})^{-1}HL_{1}(x_{0})^{-1}$ commutes with
$\sigma$, and $|\kappa|<(2(N^{2}-N+2))^{-1}$. 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
$\mathrm{d}\mu=\mathrm{d}\mu_{S}+L^{\ast}HL\mathrm{d}m$ where
$\operatorname{spt}\mu_{S}\subset\mathbb{T}^{N}\backslash\mathbb{T}_{\rm
reg}^{N}$, $H$ commutes with $\upsilon$, and $-1/h_{\tau}<\kappa<1/h_{\tau}$
(the support of a Baire measure $\nu$, denoted by $\operatorname{spt}\nu$, is
the smallest compact set whose complement has $\nu$-measure zero). In the next
sections we will show that $(L_{1}(x_{0})^{\ast})^{-1}HL_{1}(x_{0})^{-1}$
commutes with $\sigma$ and that $H$ is an analytic function of $\kappa$ 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,
$\mu_{S}=0$. The proof involves the formal differential equation satisfied by
the Fourier–Stieltjes series of $\mu$, which is used to show $\mu_{S}=0$ on
$\big{\\{}x\in\mathbb{T}^{N}\colon\\#\\{x_{j}\\}_{j=1}^{N}=N-1\big{\\}}$ (that
is, $x$ has at least $N-1$ distinct components). In turn this implies
$(L_{1}^{\ast}(x_{0}))^{-1}HL_{1}(x_{0})^{-1}$ commutes with $\sigma$. The
proofs unfortunately are not short. In the sequel $H$ refers to the Hermitian
matrix in the formula for $\mathrm{d}\mu$ and $K$ denotes $L^{\ast}HL$. Also
$H$ is positive-definite since the measure $\mu$ is positive (else there
exists a vector $v$ with $Hv=0$ and then the $C^{(1)}\big{(}\mathbb{T}_{\rm
reg}^{N};V_{\tau}\big{)}$ function given by $f(x):=L(x)^{-1}vg(x)$ where $g$
is a smooth scalar nonnegative function with support in a sufficiently small
neighborhood of $x_{0}$, has norm $\langle f,f\rangle=0$, a contradiction).
Thus $H$ has a positive-definite square root $C$ which commutes with
$\upsilon$. Now extend $CL(x)$ from $\mathcal{C}_{0}$ to all of
$\mathbb{T}_{\rm reg}^{N}$ by Definition 3.6 and so
$K(x)=L^{\ast}(x)C^{\ast}CL(x)$ for all $x\in\mathbb{T}_{\rm reg}^{N}$ (this
follows from $K(xw)=\tau(w)^{-1}K(x)\tau(w)$).

Furthermore $\int_{\mathbb{T}^{N}}\|K(x)\|\mathrm{d}m(x)<\infty$ because
$K\mathrm{d}m$ is the absolutely continuous part of the finite Baire measure
$\mu$.

We will show that $(L_{1}^{\ast}(x_{0}))^{-1}C^{\ast}CL_{1}(x_{0})^{-1}$
commutes with $\sigma$. The proof begins by establishing a recurrence relation
for the Fourier coefficients of $K(x)$, which comes from equation (3.3). For
$F(x)$ integrable on $\mathbb{T}^{N}$, possibly matrix-valued, and
$\alpha\in\mathbb{Z}^{N}$ let
$\widehat{F}_{\alpha}=\int_{\mathbb{T}^{N}}F(x)x^{-\alpha}\mathrm{d}m(x)$.
Clearly
$\int_{\mathbb{T}^{N}}x^{\beta}F(x)x^{-\alpha}\mathrm{d}m(x)=\widehat{F}_{\alpha-\beta}$;
and if $\partial_{i}F(x)$ is also integrable then (integration-by-parts)

$\displaystyle\int_{\mathbb{T}^{N}}x_{i}\partial_{i}F(x)x^{-\alpha}\mathrm{d}m(x)=\alpha_{i}\int_{\mathbb{T}^{N}}F(x)x^{-\alpha}\mathrm{d}m(x).$
(8.1)

For a subset $J\subset\\{1,2,\ldots,N\\}$ let
$\varepsilon_{J}\in\mathbb{N}_{0}^{N}$ be defined by $(\varepsilon_{J})_{i}=1$
if $i\in J$ and $=0$ otherwise; also $\varepsilon_{i}:=\varepsilon_{\\{i\\}}$.
For $1\leq i\leq N$ let

$\displaystyle E_{i}:=\\{1,2,\ldots,N\\}\backslash\\{i\\},\qquad
E_{ij}:=E_{i}\backslash\\{j\\},$ $\displaystyle p_{i}(x):=\prod\limits_{j\neq
i}(x_{i}-x_{j})=\sum\limits_{\ell=0}^{N-1}(-1)^{\ell}x_{i}^{N-1-\ell}\sum\limits_{J\subset
E_{i},\\#J=\ell}x^{\varepsilon_{J}}.$

Equation (3.3) can be rewritten as

$\displaystyle p_{i}(x)x_{i}\partial_{i}K(x)=\kappa\sum_{j\neq
i}\prod\limits_{\ell\neq
i,j}(x_{i}-x_{\ell})\\{x_{j}\tau((i,j))K(x)+K(x)\tau((i,j))x_{i}\\};$ (8.2)

this is a polynomial relation which shows that $p_{i}(x)x_{i}\partial_{i}K(x)$
is integrable and which has implications for the Fourier coefficients of $K$.

######  Proposition 8.1.

For $1\leq i\leq N$ and $\alpha\in\mathbb{Z}^{N}$ the Fourier coefficients
$\widehat{K}$ satisfy

$\displaystyle\sum_{\ell=0}^{N-1}(-1)^{\ell}(\alpha_{i}+\ell)\sum\limits_{J\subset
E_{i},\,\\#J=\ell}\widehat{K}_{\alpha+\ell\varepsilon_{i}-\varepsilon_{J}}$
$\displaystyle\qquad{}=\kappa\sum_{j\neq
i}\sum_{\ell=0}^{N-2}(-1)^{\ell}\sum_{J\subset
E_{ij},\,\\#J=\ell}\big{\\{}\tau((i,j))\widehat{K}_{\alpha+\ell\varepsilon_{i}-\varepsilon_{j}-\varepsilon_{J}}+\widehat{K}_{\alpha+(l+1)\varepsilon_{i}-\varepsilon_{J}}\tau((i,j))\big{\\}}.$
(8.3)

###### Proof.

Multiply both sides of (8.2) by $x_{i}^{1-N}$; this makes the terms
homogeneous of degree zero. Suppose $j\neq i$ then

$\displaystyle x_{i}^{1-N}\prod\limits_{\ell\neq
i,j}(x_{i}-x_{\ell})=\prod\limits_{\ell\neq
i,j}\left(1-\frac{x_{\ell}}{x_{i}}\right)=\sum_{\ell=0}^{N-2}(-1)^{\ell}x_{i}^{-\ell}\sum_{J\subset
E_{ij},\,\\#J=\ell}x^{\varepsilon_{J}}.$

Multiply the right side by $x^{-\alpha}\mathrm{d}m(x)$ and integrate over
$\mathbb{T}^{N}$ to obtain

$\displaystyle\kappa\sum_{j\neq i}\sum_{\ell=0}^{N-2}(-1)^{\ell}\sum_{J\subset
E_{ij},\,\\#J=\ell}\big{\\{}\tau((i,j))\widehat{K}_{\alpha+\ell\varepsilon_{i}-\varepsilon_{j}-\varepsilon_{J}}+\widehat{K}_{\alpha+(l+1)\varepsilon_{i}-\varepsilon_{J}}\tau((i,j))\big{\\}}.$

The sum is zero unless $\alpha\in\boldsymbol{Z}_{N}$ where
$\boldsymbol{Z}_{N}:=\Big{\\{}\alpha\in\mathbb{Z}^{N}\colon\sum\limits_{j=1}^{N}\alpha_{j}=0\Big{\\}}$,
by the homogeneity. For the left side start with (8.1) applied to
$x_{i}^{1-N}p_{i}(x)x_{i}\partial_{i}K(x)$

$\displaystyle(\alpha_{i}+N-1)\int_{\mathbb{T}^{N}}p_{i}(x)K(x)x_{i}^{1-N}x^{-\alpha}\mathrm{d}m(x)$
$\displaystyle\qquad{}=\int_{\mathbb{T}^{N}}\big{\\{}(x_{i}\partial_{i}p_{i}(x))K(x)+p_{i}(x)(x_{i}\partial_{i}K(x))\big{\\}}x_{i}^{1-N}x^{-\alpha}\mathrm{d}m(x),$
$\displaystyle\int_{\mathbb{T}^{N}}p_{i}(x)(x_{i}\partial_{i}K(x))x_{i}^{1-N}x^{-\alpha}\mathrm{d}m(x)$
$\displaystyle\qquad{}=\int_{\mathbb{T}^{N}}((\alpha_{i}+N-1)p_{i}(x)-x_{i}\partial_{i}p_{i}(x))K(x)x_{i}^{1-N}x^{-\alpha}\mathrm{d}m(x)$
$\displaystyle\qquad{}=\int_{\mathbb{T}^{N}}\sum_{\ell=0}^{N-1}(-1)^{\ell}(\alpha_{i}+\ell)x_{i}^{-\ell}\sum\limits_{J\subset
E_{i},\\#J=\ell}x^{\varepsilon_{J}}K(x)x^{-\alpha}\mathrm{d}m(x)$
$\displaystyle\qquad{}=\sum_{\ell=0}^{N-1}(-1)^{\ell}(\alpha_{i}+\ell)\sum\limits_{J\subset
E_{i},\\#J=\ell}\widehat{K}_{\alpha+\ell\varepsilon_{i}-\varepsilon_{J}}.$

Combining the two sides finishes the proof. If
$\alpha\notin\boldsymbol{Z}_{N}$ then both sides are trivially zero. ∎

This system of recurrences has the easy (and quite undesirable) solution
$\widehat{K}_{\alpha}=I$ for all $\alpha\in\boldsymbol{Z}_{N}$ and $0$
otherwise. The right side becomes $2\kappa\sum\limits_{j\neq
i}\tau((i,j))\sum\limits_{\ell=0}^{N-2}(-1)^{\ell}\binom{N-2}{\ell}=0$ (for
$N\geq 3$, an underlying assumption), and the left side is
$\sum\limits_{\ell=0}^{N-1}(-1)^{\ell}(\alpha_{i}+\ell)\binom{N-1}{\ell}I=0$.
This $\widehat{K}$ corresponds to the measure $\frac{1}{2\pi}\mathrm{d}\theta$
on the circle
$\big{\\{}e^{\mathrm{i}\theta}(1,\ldots,1)\colon-\pi<\theta\leq\pi\big{\\}}$.
Next we show that
$\widehat{\mu}_{\alpha}:=\int_{\mathbb{T}^{N}}x^{-\alpha}\mathrm{d}\mu(x)$
satisfies the same recurrences. Proposition 5.2 of [3] asserts that if
$\alpha,\beta\in\mathbb{N}_{0}^{N}$ and
$\sum\limits_{j=1}^{N}(\alpha_{j}-\beta_{j})=0$ then

$\displaystyle(\alpha_{i}-\beta_{i})\widehat{\mu}_{\alpha-\beta}=\kappa\sum_{\alpha_{j}>\alpha_{i}}\sum_{\ell=1}^{\alpha_{j}-\alpha_{i}}\tau((i,j))\widehat{\mu}_{\alpha+\ell(\varepsilon_{i}-\varepsilon_{j})-\beta}$
$\displaystyle\hphantom{(\alpha_{i}-\beta_{i})\widehat{\mu}_{\alpha-\beta}}{}-\kappa\sum_{\alpha_{i}>\alpha_{j}}\sum_{\ell=0}^{\alpha_{i}-\alpha_{j}-1}\tau((i,j))\widehat{\mu}_{\alpha+\ell(\varepsilon_{j}-\varepsilon_{i})-\beta}-\kappa\sum_{\beta_{j}>\beta_{i}}\sum_{\ell=1}^{\beta_{j}-\beta_{i}}\widehat{\mu}_{\alpha-\ell(\varepsilon_{i}-\varepsilon_{j})-\beta}\tau((i,j))$
$\displaystyle\hphantom{(\alpha_{i}-\beta_{i})\widehat{\mu}_{\alpha-\beta}}{}+\kappa\sum_{\beta_{i}>\beta_{j}}\sum_{\ell=0}^{\beta_{i}-\beta_{j}-1}\widehat{\mu}_{\alpha-\ell(\varepsilon_{j}-\varepsilon_{i})-\beta}\tau((i,j)).$
(8.4)

The relation
$\tau(w)^{\ast}\widehat{\mu}_{w\alpha}\tau(w)=\widehat{\mu}_{\alpha}$ is shown
in [3, Theorem 4.4]. Introduce Laurent series
$\sum\limits_{\alpha\in\boldsymbol{Z}_{N}}B_{\alpha}^{(i,j)}x^{\alpha}$
($i\neq j$) satisfying

$\displaystyle
B_{\alpha}^{(i,j)}-B_{\alpha+\varepsilon_{i}-\varepsilon_{j}}^{(i,j)}=\widehat{\mu}_{\alpha},\qquad
B_{\alpha-\alpha_{j}(\varepsilon_{j}-\varepsilon_{i})}^{(i,j)}=0,$

note

$\displaystyle\alpha-\alpha_{j}(\varepsilon_{j}-\varepsilon_{i})=\big{(}\ldots,\overset{i}{\alpha_{i}+\alpha_{j}},\ldots,\overset{j}{0},\ldots,\overset{\ell}{\alpha_{\ell}},\ldots\big{)},\qquad\ell\neq
i,j.$

The purpose of the definition is to produce a formal Laurent series satisfying

$\displaystyle\left(1-\frac{x_{j}}{x_{i}}\right)\sum_{\alpha}B_{\alpha}^{(i,j)}x^{\alpha}=\sum_{\alpha}\widehat{\mu}_{\alpha}x^{\alpha}.$

The ambiguity in the solution is removed by the second condition (note that
$\sum_{\alpha}\big{(}B_{\alpha}^{(i,j)}-cI\big{)}x^{\alpha}$ also solves the
first equation for any constant $c$).

######  Proposition 8.2.

Suppose $i\neq j$ and $\alpha\in\boldsymbol{Z}_{N}$ then
$B_{\alpha}^{(i,j)}\tau((i,j))=\tau((i,j))B_{(i,j)\alpha}^{(j,i)}$.

###### Proof.

Start with
$\widehat{\mu}_{\alpha}\tau((i,j))=\tau((i,j))\widehat{\mu}_{(i,j)\alpha}$ and
the defining relations

$\displaystyle
B_{\alpha}^{(i,j)}\tau((i,j))-B_{\alpha+\varepsilon_{i}-\varepsilon_{j}}^{(i,j)}\tau((i,j))=\widehat{\mu}_{\alpha}\tau((i,j)),$
$\displaystyle\tau((i,j))B_{(i,j)\alpha}^{(j,i)}-\tau((i,j))B_{(i,j)\alpha+\varepsilon_{j}-\varepsilon_{i}}^{(j,i)}=\tau((i,j))\widehat{\mu}_{(i,j)\alpha};$

subtract the second equation from the first:

$\displaystyle
B_{\alpha}^{(i,j)}\tau((i,j))-\tau((i,j))B_{(i,j)\alpha}^{(j,i)}=B_{\alpha+\varepsilon_{i}-\varepsilon_{j}}^{(i,j)}\tau((i,j))-\tau((i,j))B_{(i,j)\alpha+\varepsilon_{j}-\varepsilon_{i}}^{(j,i)}.$

By two-sided induction

$\displaystyle
B_{\alpha}^{(i,j)}\tau((i,j))-\tau((i,j))B_{(i,j)\alpha}^{(j,i)}=B_{\alpha+s(\varepsilon_{i}-\varepsilon_{j})}^{(i,j)}\tau((i,j))-\tau((i,j))B_{(i,j)\alpha+s(\varepsilon_{j}-\varepsilon_{i})}^{(j,i)}$

for all $s\in\mathbb{Z}$, in particular for $s=\alpha_{j}$ where the right
hand side vanishes by definition. ∎

######  Theorem 8.3.

For $\gamma\in\boldsymbol{Z}_{N}$ and $1\leq i\leq N$

$\displaystyle\gamma_{i}\widehat{\mu}_{\gamma}=\kappa\sum_{j\neq
i}\big{\\{}{-}\tau((i,j))B_{\gamma}^{(j,i)}+B_{\gamma}^{(i,j)}\tau((i,j))\big{\\}}.$
(8.5)

###### Proof.

The proof involves a number of cases (for each $(i,j)$ whether $\gamma_{i}\geq
0$ or $\gamma_{i}<0$, $\gamma_{j}\geq 0$ or $\gamma_{j}<0$). Consider equation
(8.4), in the terms on the first line (with $\tau((i,j))$ acting on the left)
use the substitution
$\widehat{\mu}_{\delta}=B_{\delta}^{(j,i)}-B_{\delta-\varepsilon_{i}+\varepsilon_{j}}^{(j,i)}$,
and for the terms on the second line (with $\tau((i,j))$ acting on the right)
use the substitution
$\widehat{\mu}_{\delta}=B_{\delta}^{(i,j)}-B_{\delta+\varepsilon_{i}-\varepsilon_{j}}^{(i,j)}$.
Set $\alpha_{\ell}=\max(\gamma_{\ell},0)$ and
$\beta_{\ell}=\max(0,-\gamma_{\ell})$ for $1\leq\ell\leq N$, thus
$\gamma=\alpha-\beta$. The left hand side is
$(\alpha_{i}-\beta_{i})\widehat{\mu}_{\alpha-\beta}=\gamma_{i}\widehat{\mu}_{\gamma}$.
We consider two possibilities separately: (i) $\alpha_{i}\geq 0$,
$\beta_{i}=0$; (ii) $\alpha_{i}=0$, $\beta_{i}>0$; and describe the typical
$\tau((i,j))$ terms. The sums over $\ell$ telescope. In the following any term
of the form $\tau((i,j))\widehat{\mu}_{\cdot}$ or
$\widehat{\mu}_{\cdot}\tau((i,j))$ not mentioned explicitly is zero.
Proposition 8.2 is used in each case. For case (i) and $\alpha_{j}>\alpha_{i}$

$\displaystyle\tau((i,j))\sum_{\ell=1}^{\alpha_{j}-\alpha_{i}}\widehat{\mu}_{\alpha+\ell(\varepsilon_{i}-\varepsilon_{j})-\beta}=\tau((i,j))\sum_{\ell=1}^{\alpha_{j}-\alpha_{i}}\big{(}B_{\gamma+\ell(\varepsilon_{i}-\varepsilon_{j})}^{(j,i)}-B_{\gamma+\ell(\varepsilon_{i}-\varepsilon_{j})-\varepsilon_{i}+\varepsilon_{j}}^{(j,i)}\big{)}$
$\displaystyle\hphantom{\tau((i,j))\sum_{\ell=1}^{\alpha_{j}-\alpha_{i}}\widehat{\mu}_{\alpha+\ell(\varepsilon_{i}-\varepsilon_{j})-\beta}}{}=\tau((i,j))\sum_{\ell=1}^{\alpha_{j}-\alpha_{i}}\big{(}B_{\gamma+\ell(\varepsilon_{i}-\varepsilon_{j})}^{(j,i)}-B_{\gamma+(\ell-1)(\varepsilon_{i}-\varepsilon_{j})}^{(j,i)}\big{)}$
$\displaystyle\hphantom{\tau((i,j))\sum_{\ell=1}^{\alpha_{j}-\alpha_{i}}\widehat{\mu}_{\alpha+\ell(\varepsilon_{i}-\varepsilon_{j})-\beta}}{}=\tau((i,j))\big{(}B_{\gamma+(\alpha_{j}-\alpha_{i})(\varepsilon_{i}-\varepsilon_{j})}^{(j,i)}-B_{\gamma}^{(j,i)}\big{)}$
$\displaystyle\hphantom{\tau((i,j))\sum_{\ell=1}^{\alpha_{j}-\alpha_{i}}\widehat{\mu}_{\alpha+\ell(\varepsilon_{i}-\varepsilon_{j})-\beta}}{}=\tau((i,j))\big{(}B_{(i,j)\gamma}^{(j,i)}-B_{\gamma}^{(j,i)}\big{)}=-\tau((i,j))B_{\gamma}^{(j,i)}+B_{\gamma}^{(i,j)}\tau((i,j))$

For case (i) and $\alpha_{i}>\alpha_{j}\geq 0=\beta_{j}$

$\displaystyle-\tau((i,j))\sum_{\ell=0}^{\alpha_{i}-\alpha_{j}-1}\widehat{\mu}_{\alpha+\ell(\varepsilon_{j}-\varepsilon_{i})-\beta}=-\tau((i,j))\sum_{\ell=0}^{\alpha_{i}-\alpha_{j}-1}\big{(}B_{\gamma+\ell(\varepsilon_{j}-\varepsilon_{i})}^{(j,i)}-B_{\gamma+(\ell+1)(\varepsilon_{i}-\varepsilon_{j})}^{(j,i)}\big{)}$
$\displaystyle\hphantom{-\tau((i,j))\sum_{\ell=0}^{\alpha_{i}-\alpha_{j}-1}\widehat{\mu}_{\alpha+\ell(\varepsilon_{j}-\varepsilon_{i})-\beta}}{}=-\tau((i,j))\big{(}B_{\gamma}^{(j,i)}-B_{(i,j)\gamma}^{(j,i)}\big{)}$
$\displaystyle\hphantom{-\tau((i,j))\sum_{\ell=0}^{\alpha_{i}-\alpha_{j}-1}\widehat{\mu}_{\alpha+\ell(\varepsilon_{j}-\varepsilon_{i})-\beta}}{}=-\tau((i,j))B_{\gamma}^{(j,i)}+B_{\gamma}^{(i,j)}\tau((i,j)),$

note
$\gamma+(\alpha_{i}-\alpha_{j})(\varepsilon_{j}-\varepsilon_{i})=(i,j)\gamma$.
For case (i) and $\alpha_{i}>\alpha_{j}=0>-\beta_{j}$

$\displaystyle-\tau((i,j))\sum_{\ell=0}^{\alpha_{i}-1}\widehat{\mu}_{\alpha+\ell(\varepsilon_{j}-\varepsilon_{i})-\beta}=-\tau((i,j))\sum_{\ell=0}^{\alpha_{i}-1}\big{(}B_{\gamma+\ell(\varepsilon_{j}-\varepsilon_{i})}^{(j,i)}-B_{\gamma+(\ell+1)(\varepsilon_{j}-\varepsilon_{i})}^{(j,i)}\big{)}$
$\displaystyle\hphantom{-\tau((i,j))\sum_{\ell=0}^{\alpha_{i}-1}\widehat{\mu}_{\alpha+\ell(\varepsilon_{j}-\varepsilon_{i})-\beta}}{}=-\tau((i,j))\big{(}B_{\gamma}^{(j,i)}-B_{\gamma+\gamma_{i}(\varepsilon_{j}-\varepsilon_{i})}^{(j,i)}\big{)},$
$\displaystyle-\sum_{\ell=1}^{\beta_{j}}\widehat{\mu}_{\alpha-\ell(\varepsilon_{i}-\varepsilon_{j})-\beta}\tau((i,j))=-\sum_{\ell=1}^{\beta_{j}}\big{(}B_{\gamma-\ell(\varepsilon_{i}-\varepsilon_{j})}^{(i,j)}-B_{\gamma-\ell(\varepsilon_{i}-\varepsilon_{j})+\varepsilon_{i}-\varepsilon_{j}}^{(i,j)}\big{)}\tau((i,j))$
$\displaystyle\hphantom{-\sum_{\ell=1}^{\beta_{j}}\widehat{\mu}_{\alpha-\ell(\varepsilon_{i}-\varepsilon_{j})-\beta}\tau((i,j))}{}=-\sum_{\ell=1}^{\beta_{j}}\big{(}B_{\gamma-\ell(\varepsilon_{i}-\varepsilon_{j})}^{(i,j)}-B_{\gamma-(\ell-1)(\varepsilon_{i}-\varepsilon_{j})}^{(i,j)}\big{)}\tau((i,j))$
$\displaystyle\hphantom{-\sum_{\ell=1}^{\beta_{j}}\widehat{\mu}_{\alpha-\ell(\varepsilon_{i}-\varepsilon_{j})-\beta}\tau((i,j))}{}=\big{(}B_{\gamma}^{(i,j)}-B_{\gamma+\gamma_{j}(\varepsilon_{i}-\varepsilon_{j})}^{(i,j)}\big{)}\tau((i,j))$

let $\delta=\gamma+\gamma_{i}(\varepsilon_{j}-\varepsilon_{i})$ then
$\delta_{k}=\gamma_{k}$ for $k\neq i,j$, $\delta_{i}=0$, and
$\delta_{j}=\gamma_{i}+\gamma_{j}$; also
$(i,j)\delta=\gamma+\gamma_{j}(\varepsilon_{i}-\varepsilon_{j})$. Thus the sum
of the terms for this case is

$\displaystyle-\tau((i,j))B_{\gamma}^{(j,i)}+B_{\gamma}^{(i,j)}\tau((i,j))+\tau((i,j))B_{\delta}^{(j,i)}+B_{(i,j)\delta}^{(i,j)}\tau((i,j))$
$\displaystyle\qquad{}=-\tau((i,j))B_{\gamma}^{(j,i)}+B_{\gamma}^{(i,j)}\tau((i,j)).$

For case (ii) and $\beta_{j}=-\gamma_{j}>\beta_{i}=-\gamma_{i}>0$

$\displaystyle-\sum_{\ell=1}^{\beta_{j}-\beta_{i}}\widehat{\mu}_{\alpha-\ell(\varepsilon_{i}-\varepsilon_{j})-\beta}\tau((i,j))=-\sum_{\ell=1}^{\beta_{j}-\beta_{i}}\big{(}B_{\gamma-\ell(\varepsilon_{i}-\varepsilon_{j})}^{(i,j)}-B_{\gamma-(\ell-1)(\varepsilon_{i}-\varepsilon_{j})}^{(i,j)}\big{)}\tau((i,j))$
$\displaystyle\hphantom{-\sum_{\ell=1}^{\beta_{j}-\beta_{i}}\widehat{\mu}_{\alpha-\ell(\varepsilon_{i}-\varepsilon_{j})-\beta}\tau((i,j))}{}=\big{(}{-}B_{\gamma-(\gamma_{i}-\gamma_{j})(\varepsilon_{i}-\varepsilon_{j})}^{(i,j)}+B_{\gamma}^{(i,j)}\big{)}\tau((i,j))$
$\displaystyle\hphantom{-\sum_{\ell=1}^{\beta_{j}-\beta_{i}}\widehat{\mu}_{\alpha-\ell(\varepsilon_{i}-\varepsilon_{j})-\beta}\tau((i,j))}{}=\big{(}{-}B_{(i,j)\gamma}^{(i,j)}+B_{\gamma}^{(i,j)}\big{)}\tau((i,j))$
$\displaystyle\hphantom{-\sum_{\ell=1}^{\beta_{j}-\beta_{i}}\widehat{\mu}_{\alpha-\ell(\varepsilon_{i}-\varepsilon_{j})-\beta}\tau((i,j))}{}=-\tau((i,j))B_{\gamma}^{(j,i)}+B_{\gamma}^{(i,j)}\tau((i,j)).$

For case (ii) and $\beta_{i}>\beta_{j}=-\gamma_{j}\geq 0$ (and $\alpha_{j}=0$)

$\displaystyle\sum_{\ell=0}^{\beta_{i}-\beta_{j}-1}\widehat{\mu}_{\alpha-\ell(\varepsilon_{j}-\varepsilon_{i})-\beta}\tau((i,j))=\sum_{\ell=0}^{\beta_{i}-\beta_{j}-1}\big{(}B_{\gamma-\ell(\varepsilon_{j}-\varepsilon_{i})}^{(i,j)}-B_{\gamma-(\ell+1)(\varepsilon_{j}-\varepsilon_{i})}^{(i,j)}\big{)}\tau((i,j))$
$\displaystyle\hphantom{\sum_{\ell=0}^{\beta_{i}-\beta_{j}-1}\widehat{\mu}_{\alpha-\ell(\varepsilon_{j}-\varepsilon_{i})-\beta}\tau((i,j))}{}=\big{(}B_{\gamma}^{(i,j)}-B_{\gamma-(\gamma_{j}-\gamma_{i})(\varepsilon_{j}-\varepsilon_{i})}^{(i,j)}\big{)}\tau((i,j))$
$\displaystyle\hphantom{\sum_{\ell=0}^{\beta_{i}-\beta_{j}-1}\widehat{\mu}_{\alpha-\ell(\varepsilon_{j}-\varepsilon_{i})-\beta}\tau((i,j))}{}=\big{(}B_{\gamma}^{(i,j)}-B_{(i,j)\gamma}^{(i,j)}\big{)}\tau((i,j))$
$\displaystyle\hphantom{\sum_{\ell=0}^{\beta_{i}-\beta_{j}-1}\widehat{\mu}_{\alpha-\ell(\varepsilon_{j}-\varepsilon_{i})-\beta}\tau((i,j))}{}=-\tau((i,j))B_{\gamma}^{(j,i)}+B_{\gamma}^{(i,j)}\tau((i,j)).$

For case (ii) and $-\beta_{i}=\gamma_{i}<0<\gamma_{j}=\alpha_{j}$ (and
$\beta_{j}=0$)

$\displaystyle\tau((i,j))\sum_{\ell=1}^{\alpha_{j}}\widehat{\mu}_{\alpha+\ell(\varepsilon_{i}-\varepsilon_{j})-\beta}+\sum_{\ell=0}^{\beta_{i}-1}\widehat{\mu}_{\alpha-\ell(\varepsilon_{j}-\varepsilon_{i})-\beta}\tau((i,j))$
$\displaystyle\qquad{}=\tau((i,j))\sum_{\ell=1}^{\alpha_{j}}\big{(}B_{\gamma+\ell(\varepsilon_{i}-\varepsilon_{j})}^{(j,i)}-B_{\gamma+(\ell-1)(\varepsilon_{i}-\varepsilon_{j})}^{(j,i)}\big{)}$
$\displaystyle\qquad\quad{}+\sum_{\ell=0}^{\beta_{i}-1}\big{(}B_{\gamma-\ell(\varepsilon_{j}-\varepsilon_{i})}^{(i,j)}-B_{\gamma-(\ell+1)(\varepsilon_{j}-\varepsilon_{i})}^{(i,j)}\big{)}\tau((i,j))$
$\displaystyle\qquad{}=\tau((i,j))\big{(}B_{\gamma+\gamma_{j}(\varepsilon_{i}-\varepsilon_{j})}^{(j,i)}-B_{\gamma}^{(j,i)}\big{)}+\big{(}B_{\gamma}^{(i,j)}-B_{\gamma+\gamma_{i}(\varepsilon_{j}-\varepsilon_{i})}^{(i,j)}\big{)}\tau((i,j))$
$\displaystyle\qquad{}=-\tau((i,j))B_{\gamma}^{(j,i)}+B_{\gamma}^{(i,j)}\tau((i,j)),$

because
$(i,j)(\gamma+\gamma_{j}(\varepsilon_{i}-\varepsilon_{j}))=\gamma+\gamma_{i}(\varepsilon_{j}-\varepsilon_{i})$.
In the trivial case $\gamma_{i}=\gamma_{j}$ so that $(i,j)\gamma=\gamma$ where
are no nonzero $\tau((i,j))$ terms the equation
$-\tau((i,j))B_{\gamma}^{(j,i)}-B_{\gamma}^{(i,j)}\tau((i,j))=0$ applies. Thus
in each case and for each $j\neq i$ the right hand side contains the
expression
$-\kappa\big{(}\tau((i,j))B_{\gamma}^{(j,i)}-B_{\gamma}^{(i,j)}\tau((i,j))\big{)}$.
∎

In the following there is no implied claim about convergence, because any term
$x^{\alpha}$ appears only a finite number of times in the equation.

######  Theorem 8.4.

For $1\leq i\leq N$ the formal Laurent series
$F(x):=\sum\limits_{\alpha\in\boldsymbol{Z}_{N}}\widehat{\mu}_{\alpha}x^{\alpha}$
satisfies the equation

$\displaystyle p_{i}(x)x_{i}\partial_{i}F(x)=\kappa\sum_{j\neq
i}\prod\limits_{\ell\neq
i,j}(x_{i}-x_{\ell})\\{x_{j}\tau((i,j))F(x)+F(x)\tau((i,j))x_{i}\\}.$ (8.6)

###### Proof.

Start with multiplying equation (8.5) by $x_{i}^{1-N}p_{i}(x)x^{\gamma}$ and
sum over $\gamma\in\boldsymbol{Z}_{N}$ to obtain

$\displaystyle\prod\limits_{j=1,\,j\neq
i}^{N}\left(1-\frac{x_{j}}{x_{i}}\right)\sum_{\gamma\in\boldsymbol{Z}_{N}}\gamma_{i}\widehat{\mu}_{\gamma}x^{\gamma}=\kappa\sum_{j\neq
i}\prod\limits_{k\neq
i,j}\left(1-\frac{x_{k}}{x_{i}}\right)\left(1-\frac{x_{j}}{x_{i}}\right)$
$\displaystyle\hphantom{\prod\limits_{j=1,j\neq
i}^{N}\left(1-\frac{x_{j}}{x_{i}}\right)\sum_{\gamma\in\boldsymbol{Z}_{N}}\gamma_{i}\widehat{\mu}_{\gamma}x^{\gamma}}{}\times\left\\{-\tau((i,j))\sum_{\gamma\in\boldsymbol{Z}_{N}}B_{\gamma}^{(j,i)}x^{\gamma}+\sum_{\gamma\in\boldsymbol{Z}_{N}}B_{\gamma}^{(i,j)}x^{\gamma}\tau((i,j))\right\\}.$

By construction

$\displaystyle\left(1-\frac{x_{j}}{x_{i}}\right)\sum_{\gamma\in\boldsymbol{Z}_{N}}B_{\gamma}^{(i,j)}x^{\gamma}=\sum_{\gamma\in\boldsymbol{Z}_{N}}\big{(}B_{\gamma}^{(i,j)}-B_{\gamma+\varepsilon_{i}-\varepsilon_{j}}^{(i,j)}\big{)}x^{\gamma}=\sum_{\gamma\in\boldsymbol{Z}_{N}}\widehat{\mu}_{\gamma}x^{\gamma}$

and

$\displaystyle\left(1-\frac{x_{j}}{x_{i}}\right)\sum_{\gamma\in\boldsymbol{Z}_{N}}B_{\gamma}^{(j,i)}x^{\gamma}=-\frac{x_{j}}{x_{i}}\left(1-\frac{x_{i}}{x_{j}}\right)\sum_{\gamma\in\boldsymbol{Z}_{N}}B_{\gamma}^{(j,i)}x^{\gamma}=-\frac{x_{j}}{x_{i}}\sum_{\gamma\in\boldsymbol{Z}_{N}}\widehat{\mu}_{\gamma}x^{\gamma}.$

Thus the equation becomes

$\displaystyle\prod\limits_{j=1,\,j\neq
i}^{N}\left(1-\frac{x_{j}}{x_{i}}\right)\sum_{\gamma\in\boldsymbol{Z}_{N}}\gamma_{i}\widehat{\mu}_{\gamma}x^{\gamma}$
$\displaystyle\qquad{}=\kappa\sum_{j\neq i}\prod\limits_{k\neq
i,j}\left(1-\frac{x_{k}}{x_{i}}\right)\left\\{\frac{x_{j}}{x_{i}}\tau((i,j))\sum_{\gamma\in\boldsymbol{Z}_{N}}\widehat{\mu}_{\gamma}x^{\gamma}+\sum_{\gamma\in\boldsymbol{Z}_{N}}\widehat{\mu}_{\gamma}x^{\gamma}\tau((i,j))\right\\}.$

This completes the proof. ∎

######  Corollary 8.5.

The coefficients $\\{\widehat{\mu}_{\alpha}\\}$ satisfy the same recurrences
as $\big{\\{}\widehat{K}_{\alpha}\big{\\}}$ in (8.3).

###  8.1 Maximal singular support

Above we showed that $\mu$ and $K$ satisfy the same Laurent series
differential systems (8.2) and (8.6), thus the singular part $\mu_{S}$ also
satisfies this relation. The singular part $\mu_{S}$ is the restriction of
$\mu$ to $\bigcup\limits_{i<j}\big{\\{}x\in\mathbb{T}^{N}\colon
x_{i}=x_{j}\big{\\}}$, a closed set. For each pair $\\{k,\ell\\}$ let
$E_{k\ell}=\big{\\{}x\in\mathbb{T}^{N}\colon x_{k}\neq x_{\ell}\big{\\}}$, an
open subset of $\mathbb{T}^{N}$. For $i\neq j$ let

$\displaystyle T_{i,j}=\big{\\{}x\in\mathbb{T}^{N}\colon
x_{i}=x_{j}\big{\\}}\cap\bigcap\limits_{\\{k,\ell\\}\cap\\{i,j\\}=\varnothing}\\{E_{k\ell}\cap
E_{ik}\cap E_{jk}\\};$

this is an intersection of a closed set and an open set, hence $T_{i,j}$ is a
Baire set and the restriction $\mu_{i,j}$ of $\mu$ to $T_{i,j}$ is a Baire
measure. Informally $T_{i,j}=\big{\\{}x\in\mathbb{T}^{N}\colon
x_{i}=x_{j},\\#\\{x_{k}\\}=N-1\big{\\}}$. We will prove that $\mu_{i,j}=0$ for
all $i\neq j$. That is, $\mu_{S}$ is supported by
$\big{\\{}x\in\mathbb{T}^{N}\colon\\#\\{x_{k}\\}\leq N-2\big{\\}}$ (the number
of distinct coordinate values is $\leq N-2$). In [3, Corollary 4.15] there is
an approximate identity

$\displaystyle\sigma_{n}^{N-1}(x):=\sum_{k=0}^{n}\frac{(-n)_{k}}{(1-n-N)_{k}}\sum_{\alpha\in\boldsymbol{Z}_{N},\,|\alpha|=2k}x^{\alpha},$

which satisfies $\sigma_{n}^{N-1}(x)\geq 0$ and
$\sigma_{n}^{N-1}\ast\nu\rightarrow\nu$ as $n\rightarrow\infty$, in the
weak-$\ast$ sense for any finite Baire measure $\nu$ on
$\mathbb{T}^{N}/\mathbb{D}$ (referring to functions and measures on
$\mathbb{T}^{N}$ homogeneous of degree zero as Laurent series). The set
$T_{i,j}$ is pointwise invariant under $(i,j)$ thus
$\mathrm{d}\mu_{i,j}(x)=\mathrm{d}\mu_{i,j}(x(i,j))=\tau((i,j))\mathrm{d}\mu_{i,j}(x)\tau((i,j))$.

######  Remark 8.6.

The density of Laurent polynomials in $C^{(1)}\big{(}\mathbb{T}^{N}\big{)}$
can be shown by using an approximate identity, for example:
$u_{n}(x)=\Big{\\{}\frac{1}{n+1}\sum\limits_{j=-n}^{n}(n-|j|+1)x_{1}^{j}\Big{\\}}\sigma_{n}^{N-1}(x)$;
for any $\alpha\in\mathbb{Z}^{N}$ the coefficient of $x^{\alpha}$ in
$u_{n}(x)$ tends to $1$ as $n\rightarrow\infty$ (express
$\alpha=(\alpha_{1}-m)\varepsilon_{1}+(-m,\alpha_{2},\ldots,\alpha_{N})$ where
$m=\sum\limits_{j=2}^{N}\alpha_{j}$). Then $f\ast u_{n}\rightarrow f$ in the
$C^{(1)}\big{(}\mathbb{T}^{N}\big{)}$ norm.

Let $K_{n}^{s}=\sigma_{n}^{N-1}\ast\mu_{S}$ (convolution), a Laurent
polynomial, fix $\ell$ in $1\leq\ell\leq N$, and consider the functionals
$F_{\ell,n}$, $G_{\ell,n}$ on scalar functions $p\in
C^{(1)}\big{(}\mathbb{T}^{N}\big{)}$

$\displaystyle
F_{\ell,n}(p):=\int_{\mathbb{T}^{N}}p(x)\prod\limits_{j\neq\ell}\left(1-\frac{x_{j}}{x_{\ell}}\right)x_{\ell}\partial_{\ell}K_{n}^{s}(x)\mathrm{d}m(x),$
$\displaystyle
G_{\ell,n}(p):=\kappa\sum_{i\neq\ell}\int_{\mathbb{T}^{N}}p(x)\prod\limits_{j\neq\ell,i}\left(1-\frac{x_{j}}{x_{\ell}}\right)\left\\{\frac{x_{i}}{x_{\ell}}\tau\left(\ell,i\right)K_{n}^{s}(x)+K_{n}^{s}(x)\tau(\ell,i)\right\\}\mathrm{d}m(x).$

By construction the functionals annihilate $x^{\alpha}$ for
$\alpha\notin\mathbf{Z}_{N}$. For a fixed $\alpha\in\mathbf{Z}_{N}$ the value
$F_{\ell,n}(x^{-\alpha})-G_{\ell,n}(x^{-\alpha})$ is

$\displaystyle\alpha_{\ell}A_{\alpha}b_{n}(\alpha)+\sum_{i=1}^{N-1}(-1)^{i}\sum_{J\subset
E_{\ell},\,\\#J=i}(\alpha_{\ell}+i)A_{\alpha+i\varepsilon_{\ell}-\varepsilon_{J}}b_{n}(\alpha+i\varepsilon_{\ell}-\varepsilon_{J})$
$\displaystyle-\kappa\sum_{j=1,\,j\neq\ell}^{N}\sum_{i=0}^{N-2}(-1)^{\ell}\sum_{J\subset
E_{\ell,j},\,\\#J=i}\left\\{\begin{matrix}\tau(\ell,j)A_{\alpha+(i+1)\varepsilon_{\ell}-\varepsilon_{j}-\varepsilon_{J}}b_{n}(\alpha+(i+1)\varepsilon_{\ell}-\varepsilon_{j}-\varepsilon_{J})\\\
{}+A_{\alpha+i\varepsilon_{\ell}-\varepsilon_{J}}\tau(\ell,j)b_{n}(\alpha+i\varepsilon_{\ell}-\varepsilon_{J})\end{matrix}\right\\},$

where $b_{n}(\gamma):=\frac{(-n)_{|\gamma|/2}}{(1-N-n)_{|\gamma|/2}}$ (from
the Laurent series of $\sigma_{n}^{N-1})$, and
$A_{\gamma}:=\int_{\mathbb{T}^{N}}x^{-\gamma}\mathrm{d}\mu_{S}$ . Thus for
fixed $\alpha$ the coefficients $b_{n}(\cdot)\rightarrow 1$ as
$n\rightarrow\infty$ and the expression tends to the differential system 8.2
and

$\displaystyle\lim_{n\rightarrow\infty}\big{(}F_{\ell,n}(x^{-\alpha})-G_{\ell,n}(x^{-\alpha})\big{)}=0.$

This result extends to any Laurent polynomial by linearity. From the
approximate identity property

$\displaystyle\lim_{n\rightarrow\infty}G_{\ell,n}(p)=\kappa\sum_{i\neq\ell}\int_{\mathbb{T}^{N}}p(x)\prod\limits_{j\neq\ell,i}\left(1-\frac{x_{j}}{x_{\ell}}\right)\left\\{\frac{x_{i}}{x_{\ell}}\tau(\ell,i)\mathrm{d}\mu_{S}(x)+\mathrm{d}\mu_{S}(x)\tau(\ell,i)\right\\},$

and

$\displaystyle\|G_{\ell,n}(p)\|\leq
M\sup_{x,\,i}\left|p(x)\prod\limits_{j\neq\ell,i}\left(1-\frac{x_{j}}{x_{\ell}}\right)\right|,$

where $M$ depends on $\mu_{S}$. Also

$\displaystyle
F_{\ell,n}(p)=-\int_{\mathbb{T}^{N}}x_{\ell}\partial_{\ell}\left\\{p(x)\prod\limits_{j\neq\ell}\left(1-\frac{x_{j}}{x_{\ell}}\right)\right\\}K_{n}(x)\mathrm{d}m(x),$

and

$\displaystyle\lim_{n\rightarrow\infty}F_{\ell,n}(p)=-\int_{\mathbb{T}^{N}}x_{\ell}\partial_{\ell}\left\\{p(x)\prod\limits_{j\neq\ell}\left(1-\frac{x_{j}}{x_{\ell}}\right)\right\\}\mathrm{d}\mu_{S}(x)$

for Laurent polynomials $p$. By density of Laurent polynomials in
$C^{(1)}\big{(}\mathbb{T}^{N}\\!/\mathbb{D}\big{)}$
($\mathbb{D}\\!=\\!\\{(u,u,\ldots,u)\colon\\!$ $|u|=1\\}$ thus functions
homogeneous of degree zero on $\mathbb{T}^{N}$ can be considered as functions
on the quotient group $\mathbb{T}^{N}/\mathbb{D}$) we obtain

$\displaystyle-\int_{\mathbb{T}^{N}}x_{\ell}\partial_{\ell}\left\\{p(x)\prod\limits_{j\neq\ell}\left(1-\frac{x_{j}}{x_{\ell}}\right)\right\\}\mathrm{d}\mu_{S}(x)$
$\displaystyle\qquad{}=\kappa\sum_{i\neq\ell}\int_{\mathbb{T}^{N}}p(x)\prod\limits_{j\neq\ell,i}\left(1-\frac{x_{j}}{x_{\ell}}\right)\left\\{\frac{x_{i}}{x_{\ell}}\tau\left(\ell,i\right)\mathrm{d}\mu_{S}(x)+\mathrm{d}\mu_{S}(x)\tau(\ell,i)\right\\},$
(8.7)

for all $p\in C^{(1)}\big{(}\mathbb{T}^{N}/\mathbb{D}\big{)}$.

######  Theorem 8.7.

For $1\leq i<j\leq N$ the restriction $\mu_{S}|T_{i,j}=0$.

###### Proof.

It suffices to take $i=1,j=2$. Let $E$ be an open neighborhood of a point in
$T_{1,2}$ such that if $x\in\overline{E}$ (the closure) and $x_{i}=x_{j}$ for
some pair $i<j$ then $i=1$ and $j=2$. Let $f(x)\in
C^{(1)}\big{(}\mathbb{T}^{N}/\mathbb{D}\big{)}$ have support $\subset E$. Thus
$f(x)=0=\partial_{1}f(x)$ at each point $x$ such that $x_{i}=x_{j}$ for some
pair $\\{i,j\\}\neq\\{1,2\\}$ ($f=0$ on a neighborhood of
$\bigcup\limits_{i<j}\\{x\colon x_{i}=x_{j}\\}\backslash T_{1,2}$). Then in
formula (8.7) (with $\ell=1$) applied to $f$ the measure $\mu_{S}$ can be
replaced with $\mu_{1,2}$. Evaluate the derivative

$\displaystyle x_{1}\partial_{1}\left\\{f(x)\prod\limits_{j\neq