Orthogonality Measure on the Torus for Vector-Valued Jack Polynomials

Orthogonality Measure on the Torus
for Vector-Valued Jack Polynomials\star\star$\star$This paper is a
contribution to the Special Issue on Orthogonal Polynomials, Special Functions
and Applications. The full collection is available at
http://www.emis.de/journals/SIGMA/OPSFA2015.html

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 November 26, 2015, in final form March 23, 2016; Published online
March 27, 2016

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

nonsymmetric Jack polynomials; Fourier–Stieltjes coefficients; matrix-valued
measure; symmetric group modules

33C52; 42B10; 20C30; 46G10; 35F35

##  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

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

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 [12] established orthogonality structures on
the torus for trigonometric polynomials associated with Weyl groups; the
nonsymmetric Jack polynomials form a special case. Details on the derivation
of the norm formulae can be found in the treatise by Xu and the author [6,
Section 10.6.3]. The weight function $K_{\kappa}$ turned out to be the square
of the base state for the Calogero–Sutherland quantum mechanical model of $N$
identical particles located at $x_{1},x_{2},\ldots,x_{N}$ on the circle with a
$1/r^{2}$ potential. This means that the particles repel each other with a
force corresponding to a potential $C|x_{i}-x_{j}|^{-2}$. See Lapointe and
Vinet [10] for the construction of wavefunctions in terms of Jack polynomials
for this model. More recently Griffeth [8] constructed vector-valued Jack
polynomials for the family $G\left(n,p,N\right)$ of complex reflection groups.
These are the groups of permutation matrices (exactly one nonzero entry in
each row and each column) whose nonzero entries are $n^{th}$ roots of unity
and the product of these entries is a $(n/p)^{\rm th}$ 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 [3] together with Luque [5] investigated the symmetric
group case more intensively. The results from these two papers are the
foundation for the present work.

Since the torus structure is such an important aspect of the theory of Jack
polynomials it seemed like an obvious research topic to find the role of the
torus in the vector-valued Jack case. Is there a matrix-valued weight function
on the torus for which the vector-valued Jack polynomials are mutually
orthogonal? Some explorations in the $N=3$ and $N=4$ situation showed that the
theory is much more complicated than the ordinary (scalar) case. For two-
dimensional representations the weight function has hypergeometric function
entries (see [4]); this is quite different from the rather natural product
$\prod\limits_{1\leq i<j\leq N}|x_{i}-x_{j}|^{2\kappa}$, a power of the
discriminant.

In this paper we will produce a matrix-valued measure on the torus for which
the vector-valued nonsymmetric polynomials are mutually orthogonal. The result
applies to arbitrary irreducible representations of the symmetric groups. In
each case there is a permitted range of the parameter. We start with a concise
outline of the definitions and construction of the polynomials using the
Yang–Baxter graph technique in Section 2, based on [3] and [5]. Section 3
contains the construction of the abstract Hermitian form which is designed to
act like an integral over the torus; that is, multiplication by a coordinate
function $x_{j}$ is an isometry. The method is algebraic and based on the
Yang–Baxter graph. In Section 4 we use techniques from Fourier analysis to
produce the desired measure. The Section begins by using the formulae from the
previous sections to define the hypothetical Fourier–Stieltjes coefficients,
defined on $\mathbb{Z}^{N}$ which is the dual group of the torus, a
multiplicative group, and then applying a matrix version of a theorem of
Bochner about positive-definite functions to get the measure. There is an
application of approximate-identity theory using a Cesàro kernel to construct
a sequence of positive matrix-valued Laurent polynomials which converges to
the orthogonality measure.

Section 5 develops a recurrence relation satisfied by the Fourier–Stieltjes
coefficients of the orthogonality measure. The relation allows an inductive
calculation for the coefficients (but actual work, even with symbolic
computation software, may not be feasible unless the dimensions are reasonably
small), and it describes the list of parameter values (certain rational
numbers) for which the construction fails.

The scalar weight function on the torus

$\displaystyle K_{\kappa}(x):=\prod\limits_{1\leq i<j\leq
N}\big{\\{}(x_{i}-x_{j})\big{(}x_{i}^{-1}-x_{j}^{-1}\big{)}\big{\\}}^{\kappa}$

satisfies a first-order differential system,

$\displaystyle x_{i}\frac{\partial}{\partial x_{i}}K_{\kappa}(x)=\kappa
K(x)\sum_{j\neq i}\frac{x_{i}+x_{j}}{x_{i}-x_{j}},\qquad 1\leq i\leq N.$

In Section 6 we show that there is an analogous matrix differential system
which is solved in a distribution sense by the orthogonality measure. We
outline a result asserting that the orthogonality measure restricted to the
complement of $\bigcup\limits_{1\leq i<j\leq N}\\{x\colon x_{i}=x_{j}\\}$ is
equal to an analytic solution of the differential system times the Haar
measure $\mathrm{d}m$. Finally there is Appendix A containing some technical
background results.

##  2 Vector-valued Jack polynomials and the Yang–Baxter graph

This is a summary of the definitions and results from [3] and [5]. For
$x=(x_{1},\ldots,x_{N})\in\mathbb{C}^{N}$ the monomial
$x^{\alpha}:=\prod\limits_{i=1}^{N}x_{i}^{\alpha_{i}}$,
$\alpha=(\alpha_{1},\ldots,\alpha_{N})\in\mathbb{N}_{0}^{N}$,
$|\alpha|:=\sum\limits_{i=1}^{N}\alpha_{i}$,
$\mathbb{N}_{0}:=\\{0,1,2,3,\ldots\\}$ and $\alpha$ is called a multi-index or
a composition of $|\alpha|$. We denote two distinguished elements by
$\mathbf{0}=(0,0,\ldots,0)$, and $\mathbf{1}=(1,1,\ldots,1)$. The degree of
$x^{\alpha}$ is $|\alpha|$, and a polynomial is a finite linear combination of
monomials. The linear space of all polynomials is denoted by $\mathcal{P}$,
and $\mathcal{P}_{n}=\operatorname{span}\\{x^{\alpha}\colon|\alpha|=n\\}$ is
the subspace of polynomials homogeneous of degree $n$. The specific
polynomials considered here have coefficients in $\mathbb{Q}(\kappa)$ where
$\kappa$ is transcendental (indeterminate) but which will also take on certain
real values. The multi-indices $\alpha$ have an important partial order: let
$\alpha^{+}$ denote the nonincreasing rearrangement of $\alpha$, for example
if $\alpha=(1,2,1,4)$ then $\alpha^{+}=(4,2,1,1)$. Let $\mathbb{N}_{0}^{N,+}$
denote the set of partition multi-indices, that is,
$\big{\\{}\lambda\in\mathbb{N}_{0}^{N}\colon\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{N}\big{\\}}$.

######  Definition 2.1.

$\displaystyle\alpha\prec\beta~{}\Longleftrightarrow\sum_{j=1}^{i}\alpha_{j}\leq\sum_{j=1}^{i}\beta_{j},\qquad
1\leq i\leq N,\qquad\alpha\neq\beta,$
$\displaystyle\alpha\vartriangleleft\beta\Longleftrightarrow(|\alpha|=|\beta|)\wedge\big{[}(\alpha^{+}\prec\beta^{+})\vee(\alpha^{+}=\beta^{+}\wedge\alpha\prec\beta)\big{]}.$

For example $(3,2,1)\vartriangleleft(0,2,4)\vartriangleleft(4,0,2)$, while
$(4,1,1)$, $(3,3,0)$ are not $\vartriangleleft$-comparable. The symmetric
group $\mathcal{S}_{N}$, the set of permutations of $\\{1,2,\ldots,N\\}$, acts
on $\mathbb{C}^{N}$ by permutation of coordinates. The action is extended to
polynomials by $wp(x)=p(xw)$ where $(xw)_{i}=x_{w(i)}$ (consider $x$ as a row
vector and $w$ as a permutation matrix, $[w]_{ij}=\delta_{i,w(j)}$, 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
$\\{x\colon x_{i}-x_{j}=0\\}$ for $1\leq i<j\leq N$. These are transpositions,
denoted by $(i,j)$, interchanging $x_{i}$ and $x_{j}$. Define the
$\mathcal{S}_{N}$-action on $\alpha\in\mathbb{N}_{0}^{N}$ 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 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: $\tau\colon\mathcal{S}_{N}\rightarrow
O_{m}(\mathbb{R})$ ($\tau(w)^{-1}=\tau(w^{-1})=\tau(w)^{T}$). See James and
Kerber [9] for representation theory, including a modern discussion of Young’s
methods. We will specify an orthogonal basis and the images $\tau(s_{i})$ for
each $i$, which suffices for our purposes. Identify $\tau$ with a partition of
$N$: $(\tau_{1},\tau_{2},\ldots)\in\mathbb{N}_{0}^{N,+}$ such that $|\tau|=N$.
The length of $\tau$ is $\ell(\tau)=\max\\{i\colon\tau_{i}>0\\}$. There is a
Ferrers diagram of shape $\tau$ (this diagram is given the same name), 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, that is, we require $\dim
V_{\tau}\geq 2$. The hook-length of the node $(i,j)\in\tau$ is defined to be

$\displaystyle\operatorname{hook}(\tau;i,j):=\tau_{i}-j+\\#\big{\\{}k\colon
i<k\leq\ell(\tau)\wedge j\leq\tau_{k}\big{\\}}+1.$

We will need the key quantity
$h_{\tau}:=\operatorname{hook}(\tau;1,1)=\tau_{1}+\ell(\tau)-1$, the maximum
hook-length of the diagram.

######  Example 2.2.

Here are the Ferrers diagram, a (column-strict) tableau, and an RSYT, all of
shape $(5,3,2)$

$\displaystyle\begin{matrix}\square&\square&\square&\square&\square\\\
\square&\square&\square&&\\\
\square&\square&&&\end{matrix},\qquad\begin{bmatrix}0&0&1&2&3\\\ 1&2&2&&\\\
2&4&&&\end{bmatrix},\qquad\begin{bmatrix}10&7&4&2&1\\\ 9&6&3&&\\\
8&5&&&\end{bmatrix}.$

Denote the set of RSYT’s of shape $\tau$ by $\mathcal{Y}(\tau)$ and let
$V_{\tau}:=\operatorname{span}\\{T\colon T\in\mathcal{Y}(\tau)\\}$ (the field
is $\mathbb{C}(\kappa)$) with orthogonal basis $\mathcal{Y}(\tau)$.
Furthermore $\dim
V_{\tau}=\\#\mathcal{Y}(\tau)=N!/\prod\limits_{(i,j)\in\tau}\operatorname{hook}(\tau;i,j)$.
For $1\leq i\leq N$ and $T\in\mathcal{Y}(\tau)$ the entry $i$ is at
coordinates $(\operatorname{rw}(i,T),\operatorname{cm}(i,T))$ and the content
is $c(i,T):=\operatorname{cm}(i,T)-\operatorname{rw}(i,T)$. Each
$T\in\mathcal{Y}(\tau)$ is uniquely determined by its content vector
$[c(i,T)]_{i=1}^{N}$. For the example $\tau=(3,1)$

$\displaystyle\begin{bmatrix}4&2&1\\\
3&&\end{bmatrix},\qquad\begin{bmatrix}4&3&1\\\
2&&\end{bmatrix},\qquad\begin{bmatrix}4&3&2\\\ 1&&\end{bmatrix}$

the list of content vectors is $[2,1,-1,0]$, $[2,-1,1,0]$, $[-1,2,1,0]$. To
recover $T$ from its content vector fill in the entries starting with $N$,
then $N-1$ ($c(N-1,T)=\pm 1$) has two possibilities and so on.

######  Example 2.3.

The list of $\mathcal{Y}(\tau)$ for $\tau=(3,1,1)$, $N=5$

$\displaystyle\begin{bmatrix}5&2&1\\\ 4&&\\\
3&&\end{bmatrix},\quad\begin{bmatrix}5&3&1\\\ 4&&\\\
2&&\end{bmatrix},\quad\begin{bmatrix}5&3&2\\\ 4&&\\\
1&&\end{bmatrix},\quad\begin{bmatrix}5&4&1\\\ 3&&\\\
2&&\end{bmatrix},\quad\begin{bmatrix}5&4&2\\\ 3&&\\\
1&&\end{bmatrix},\quad\begin{bmatrix}5&4&3\\\ 2&&\\\ 1&&\end{bmatrix}.$

The corresponding list of content vectors is $[2,1,-2,-1,0]$, $[2,-2,1,-1,0]$,
$[-2,2,1,-1,0]$, $[2,-2,-1,1,0]$, $[-2,2,-1,1,0]$, $[-2,-1,2,1,0]$.

The representation theory can be developed using the content vectors in place
of tableaux; this is due to Okounkov and Vershik [14].

###  2.1 Description of the representation $\boldsymbol{\tau}$

The formulae for the action of $\tau(s_{i})$ on the basis $\mathcal{Y}(\tau)$
are from Murphy [11, Theorem 3.12]. Define $b_{i}(T):=1/(c(i,T)-c(i+1,T))$.
Note that $c(i,T)-c(i+1,T)=0$ is impossible for RSYT’s. If
$|c(i,T)-c(i+1,T)|\geq 2$ let $T^{(i)}\in\mathcal{Y}(\tau)$ denote $T$ with
$i$, $i+1$ interchanged. The following describes the action of $\tau(s_{i})$
(in each case there is an informal subrectangle description of the relative
positions of $i$ and $i+1$ in $T$; in cases (3) and (4) $i$ and $i+1$ are not
necessarily in adjacent rows or columns)

  1. 1.

If $\operatorname{rw}(i,T)=\operatorname{rw}(i+1,T)$ then $\tau(s_{i})T=T$;
position is $[i+1,i]$, $b_{i}(T)=1$.

  2. 2.

If $\operatorname{cm}(i,T)=\operatorname{cm}(i+1,T)$ then $\tau(s_{i})T=-T$;
position is $\begin{bmatrix}i+1\\\ i\end{bmatrix}$, $b_{i}(T)=-1$.

  3. 3.

if $\operatorname{rw}(i,T)<\operatorname{rw}(i+1,T)$ (then
$\operatorname{cm}(i,T)>\operatorname{cm}(i+1,T)$), position
$\begin{bmatrix}\ast&i\\\ i+1&\ast\end{bmatrix}$,
$c(i,T)\geq(\operatorname{cm}(i+1,T)+1)-(\operatorname{rw}(i+1,T)-1)\geq
c(i+1,T)+2$, $0<b_{i}(T)\leq\frac{1}{2}$ then

$\displaystyle\tau(s_{i})T=T^{(i)}+b_{i}(T)T,$
$\displaystyle\tau(s_{i})T^{(i)}=\big{(}1-b_{i}(T)^{2}\big{)}T-b_{i}(T)T^{(i)}.$

  4. 4.

if $\operatorname{rw}(i,T)>\operatorname{rw}(i+1,T)$ (and
$\operatorname{cm}(i,T)<\operatorname{cm}(i+1,T)$), position
$\begin{bmatrix}\ast&i+1\\\ i&\ast\end{bmatrix}$; the formula is found in case
(3) interchanging $T$ and $T^{(i)}$, and using $b_{i}(T)=-b_{i}(T^{(i)})$.

To eliminate extra parentheses we will write $\tau(i,j)$ for $\tau((i,j))$;
where $(i,j)$ is a transposition.

There is a (unique up to constant multiple) positive Hermitian form on
$V_{\tau}$ for which $\tau$ is unitary (real orthogonal), that is
$\langle\tau(w)S_{1},S_{2}\rangle_{0}=\langle
S_{1},\tau(w)^{-1}S_{2}\rangle_{0}\allowbreak=\langle
S_{1},\tau(w)^{\ast}S_{2}\rangle_{0}$, ($S_{1},S_{2}\in V_{\tau}$,
$w\in\mathcal{S}_{N}$):

######  Definition 2.4.

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

The verification of the unitary property is based on the relation

$\displaystyle\big{\langle}T^{(i)},T^{(i)}\big{\rangle}_{0}=\big{(}1-b_{i}(T)^{2}\big{)}\langle
T,T\rangle_{0}$

when $0<b_{i}(T)\leq\frac{1}{2}$. Each $\tau(w)$ is an orthogonal matrix with
respect to the orthonormal basis $\big{\\{}\langle
T,T\rangle_{0}^{-1/2}T\colon T\in\mathcal{Y}(\tau)\big{\\}}$. The basis
vectors $T$ are simultaneous eigenvectors of the (reverse) Jucys–Murphy
elements $\omega_{i}:=\sum\limits_{j=i+1}^{N}(i,j)$ (with $\omega_{N}=0$),
which commute pairwise and $\tau(\omega_{i})T=c(i,T)T,$ for $1\leq i\leq N$
(see [11, Lemma 3.6]); as usual, $\tau$ is extended to a homomorphism of the
group algebra $\mathbb{C}\mathcal{S}_{N}$ by
$\tau\left(\sum_{w}b_{w}w\right)=\sum_{w}b_{w}\tau(w)$.

###  2.2 Vector-valued nonsymmetric Jack polynomials

The main concern of this paper is $\mathcal{P}_{\tau}=\mathcal{P}\otimes
V_{\tau}$, the space of $V_{\tau}$ valued polynomials in $x$, which is
equipped with the $\mathcal{S}_{N}$ action:

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

extended by linearity to

$\displaystyle wp(x)=\tau(w)p(xw),\qquad p\in\mathcal{P}_{\tau}.$

######  Definition 2.5.

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

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

The commutation relations analogous to the scalar case hold, that is,

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

The simultaneous eigenfunctions of $\\{\mathcal{U}_{i}\\}$ are called (vector-
valued) nonsymmetric Jack polynomials (NSJP). For generic $\kappa$ these
eigenfunctions form a basis of $\mathcal{P}_{\tau}$ (we will specify the
excluded rational values in the sequel). They have a triangularity property
with respect to the partial order $\vartriangleright$. However the structure
does not merely rely on leading terms of the type $x^{\alpha}\otimes T$. We
need the rank function:

######  Definition 2.6.

For $\alpha\in\mathbb{N}_{0}^{N}$, $1\leq i\leq N$

$\displaystyle
r_{\alpha}(i):=\\#\\{j\colon\alpha_{j}>\alpha_{i}\\}+\\#\\{j\colon 1\leq j\leq
i,\,\alpha_{j}=\alpha_{i}\\},$

then $r_{\alpha}\in\mathcal{S}_{N}.$

A consequence is that $r_{\alpha}\alpha=\alpha^{+}$, the nonincreasing
rearrangement of $\alpha$, for any $\alpha\in\mathbb{N}_{0}^{N}$ . For example
if $\alpha=(1,2,1,4)$ then $r_{\alpha}=[3,2,4,1]$ and
$r_{\alpha}\alpha=\alpha^{+}=(4,2,1,1)$ (recall
$w\alpha_{i}=\alpha_{w^{-1}(i)}$). Also $r_{\alpha}=I$ if and only if $\alpha$
is a partition ($\alpha_{1}\geq\alpha_{2}\geq\cdots\geq\alpha_{N}$).

For each $\alpha\in\mathbb{N}_{0}^{N}$ and $T\in\mathcal{Y}(\tau)$ there is a
NSJP $\zeta_{\alpha,T}$ with leading term
$x^{\alpha}\otimes\tau\big{(}r_{\alpha}^{-1}\big{)}T$, that is,

$\displaystyle\zeta_{\alpha,T}=x^{\alpha}\otimes\tau\big{(}r_{\alpha}^{-1}\big{)}T+\sum_{\alpha\vartriangleright\beta}x^{\beta}\otimes
t_{\alpha\beta}(\kappa),\qquad t_{\alpha\beta}(\kappa)\in V_{\tau},$
$\displaystyle\mathcal{U}_{i}\zeta_{\alpha,T}=\big{(}\alpha_{i}+1+\kappa
c(r_{\alpha}(i),T)\big{)}\zeta_{\alpha,T},\qquad 1\leq i\leq N.$

###  2.3 The Yang–Baxter graph

The NSJP’s can be constructed by means of a Yang–Baxter graph. The details are
in [5]; this paper has several figures illustrating some typical graphs.

A node consists of

$\displaystyle(\alpha,T,\xi_{\alpha.T},r_{\alpha},\zeta_{\alpha,T}),$

where $\alpha\in\mathbb{N}_{0}^{N}$, $\xi_{\alpha,T}$ is the spectral vector
$\xi_{\alpha,T}(i)=\alpha_{i}+1+\kappa c(r_{\alpha}(i),T)$, $1\leq i\leq N$.
The root is $\big{(}\mathbf{0},T_{0},[1+\kappa
c(i,T_{0})]_{i=1}^{N},I,1\otimes T_{0}\big{)}$ where $T_{0}$ is formed by
entering $N,N-1,\ldots,1$ column-by-column in the Ferrers diagram, for example
$\tau=(3,3,1)$

$\displaystyle T_{0}=\begin{bmatrix}7&4&2\\\ 6&3&1\\\ 5&&\end{bmatrix},\qquad
c(\cdot,T_{0})=[1,2,0,1,-2,-1,0].$

There is an adjacency relation in $\mathcal{Y}(\tau)$ based on the positions
of the pairs $\ \\{i,i+1\\}$ and an inversion counter.

######  Definition 2.7.

For $T\in\mathcal{Y}(\tau)$ set

$\displaystyle\operatorname{inv}(T):=\\#\big{\\{}(i,j)\colon
i<j,\,c(i,T)-c(j,T)\leq-2\big{\\}}.$

Recall from Section 2.1 that there are four types of positions of a given pair
$\\{i,i+1\\}$ in $T$, and in case (3) it is straightforward to check that
$\operatorname{inv}(T^{(i)})=\operatorname{inv}(T)+1$.

If $\alpha_{i}\neq\alpha_{i+1}$ then $r_{s_{i}\alpha}=r_{\alpha}s_{i}$. The
cycle $w_{0}:=(123\ldots N)$ and the affine transformation

$\displaystyle\Phi(a_{1},a_{2},\ldots,a_{N}):=(a_{2},a_{3},\ldots,a_{N},a_{1}+1)$

are fundamental parts of the construction; and
$r_{\Phi\alpha}=r_{\alpha}w_{0}$ for any $\alpha$, that is,

$\displaystyle
r_{\alpha}w_{0}(i)=r_{\alpha}(w_{0}(i))=r_{\alpha}(i+1)=r_{\Phi\alpha}(i),\qquad
1\leq i<N,$ $\displaystyle
r_{\alpha}w_{0}(N)=r_{\alpha}(w_{0}(N))=r_{\alpha}(1)=r_{\Phi\alpha}(N).$

The jumps in the graph, which raise the degree by one, are

$\displaystyle(\alpha,T,\xi_{\alpha,T},r_{\alpha},\zeta_{\alpha,T})\overset{\Phi}{\longrightarrow}\big{(}\Phi\alpha,T,\Phi\xi_{\alpha,T},r_{\alpha}w_{0},x_{N}w_{0}^{-1}\zeta_{\alpha,T}\big{)},$
(2.1) $\displaystyle\zeta_{\Phi\alpha,T}=x_{N}w_{0}^{-1}\zeta_{\alpha,T}$

the leading term is
$x^{\Phi\alpha}\otimes\tau\big{(}w_{0}^{-1}r_{\alpha}^{-1}\big{)}T$ and
$w_{0}^{-1}r_{\alpha}^{-1}=(r_{\alpha}w_{0})^{-1}$. For example:
$\alpha=(0,3,5,0)$, $r_{\alpha}=[3,2,1,4]$, $\Phi\alpha=(3,5,0,1)$,
$r_{\Phi\alpha}=[2,1,4,3]$.

There are two types of steps, labeled by $s_{i}$:

  1. 1.

If $\alpha_{i}<\alpha_{i+1}$, then

$\displaystyle(\alpha,T,\xi_{\alpha,T},r_{\alpha},\zeta_{\alpha,T})\overset{s_{i}}{\longrightarrow}(s_{i}\alpha,T,s_{i}\xi_{\alpha,T},r_{\alpha}s_{i},\zeta_{s_{i}\alpha,T}),$
$\displaystyle\zeta_{s_{i}\alpha,T}=s_{i}\zeta_{\alpha,T}-\frac{\kappa}{\xi_{\alpha,T}(i)-\xi_{\alpha,T}(i+1)}\zeta_{\alpha,T}.$

Observe that this construction is valid provided
$\xi_{\alpha,T}(i)\neq\xi_{\alpha,T}(i+1)$, that is,
$\alpha_{i+1}-\alpha_{i}\neq\kappa(c(r_{\alpha}(i),T)-c(r_{\alpha}(i+1),T))$.
The extreme values of $c(\cdot,T)$ are $\tau_{1}-1$ and $1-\ell(\tau)$, thus
$|c(r_{\alpha}(i),T)-c(r_{\alpha+1}(i),T)|\allowbreak\leq h_{\tau}-1$.
Furthermore $\alpha_{i+1}-\alpha_{i}\geq 1$ and the step is valid provided
$\kappa m\notin\\{1,2,3,\dots\\}$ for
$m=1-h_{\tau},2-h_{\tau},\ldots,h_{\tau}-1$. The bound
$-1/(h_{\tau}-1)<\kappa<1/(h_{\tau}-1)$ is sufficient.

  2. 2.

If $\alpha_{i}=\alpha_{i+1}$, and the positions of $j:=r_{\alpha}(i)$, $j+1$
in $T$ are of type (3), that is, $c(j,T)-c(j+1,T)\geq 2$ (the definition of
$r_{\alpha}$ implies $r_{\alpha}(i+1)=j+1$ and
$s_{i}r_{\alpha}^{-1}=r_{\alpha}^{-1}s_{j}$). Set

$\displaystyle
b^{\prime}=\frac{1}{c(j,T)-c(j+1,T)}=\frac{\kappa}{\xi_{\alpha,T}(i)-\xi_{\alpha,T}(i+1)};$

thus $0<b^{\prime}\leq\frac{1}{2}$, there is a step

$\displaystyle(\alpha,T,\xi_{\alpha,T},r_{\alpha},\zeta_{\alpha,T})\overset{s_{i}}{\longrightarrow}\big{(}\alpha,T^{(j)},s_{i}\xi_{\alpha,T},r_{\alpha},\zeta_{\alpha,T^{(j)}}\big{)},$
$\displaystyle\zeta_{\alpha,T^{(j)}}=s_{i}\zeta_{\alpha,T}-b^{\prime}\zeta_{\alpha,T},$

($T^{(j)}$ is the result of interchanging $j$ and $j+1$ in $T$). The leading
term is transformed
$s_{i}\big{(}x^{\alpha}\otimes\tau\big{(}r_{\alpha}^{-1}\big{)}T\big{)}=(xs_{i})^{\alpha}\otimes\tau\big{(}s_{i}r_{\alpha}^{-1}\big{)}T\allowbreak=x^{\alpha}\otimes\tau\big{(}r_{\alpha}^{-1}\big{)}\tau(s_{j})T$
and $\tau(s_{j})T=T^{(j)}+b^{\prime}T$.

There are two other possibilities corresponding to (1) and (2) for the action
of $s_{i}$ on $\zeta_{\alpha,T}$ when $\alpha_{i}=\alpha_{i+1}$ (note
$r_{\alpha}(i+1)=r_{\alpha}(i)+1$): (1)
$\operatorname{rw}(r_{\alpha}(i),T)=\operatorname{rw}(r_{\alpha}(i)+1,T)$,
then $s_{i}\zeta_{\alpha,T}=\zeta_{\alpha,T}$; (2)
$\operatorname{cm}(r_{\alpha}(i),T)=\operatorname{cm}(r_{\alpha}(i)+1,T)$,
then $s_{i}\zeta_{\alpha,T}=-\zeta_{\alpha,T}$.

The proofs that these formulae are mutually compatible for different paths in
the graph from the root $(\mathbf{0},T_{0})$ to a given node $(\alpha,T)$, use
inductive arguments based on the fact that these paths have the same length.
The number of jumps is clearly $|\alpha|$ and the number of steps is
$S(\alpha)+\operatorname{inv}(T)-\operatorname{inv}(T_{0})$, where

$\displaystyle S(\alpha):=\frac{1}{2}\sum_{1\leq i<j\leq
N}(|\alpha_{i}-\alpha_{j}|+|\alpha_{i}-\alpha_{j}+1|-1).$

##  3 Hermitian forms

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
$f,g\in\mathcal{P}_{\tau}$):

$\displaystyle\big{\langle}1\otimes T,1\otimes
T^{\prime}\big{\rangle}=\big{\langle}T,T^{\prime}\big{\rangle}_{0},\qquad
T,T^{\prime}\in\mathcal{Y}(\tau),$ (3.1) $\displaystyle\langle
wf,wg\rangle=\langle f,g\rangle,\qquad w\in\mathcal{S}_{N},$
$\displaystyle\langle x_{i}\mathcal{D}_{i}f,g\rangle=\langle
f,x_{i}\mathcal{D}_{i}g\rangle,\qquad 1\leq i\leq N.$

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 the uniqueness of the spectral vectors discussed above 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
$\\{\zeta_{\alpha,T}\\}$. We can deduce contiguity relations corresponding to
the steps described above and implied by the properties of the form. Consider
step type (1) with

$\displaystyle
s_{i}\zeta_{\alpha,T}=\zeta_{s_{i}\alpha,T}+b^{\prime}\zeta_{\alpha,T},\qquad
b^{\prime}=\frac{\kappa}{\xi_{\alpha,T}(i)-\xi_{\alpha,T}(i+1)}.$

The conditions $\langle
s_{i}\zeta_{\alpha,T},s_{i}\zeta_{\alpha,T}\rangle=\langle\zeta_{\alpha,T},\zeta_{\alpha,T}\rangle$
and $\langle\zeta_{\alpha,T},\zeta_{s_{i}\alpha,T}\rangle=0$ imply

$\displaystyle\langle\zeta_{\alpha,T},\zeta_{\alpha,T}\rangle=\big{\langle}\zeta_{s_{i}\alpha,T}+b^{\prime}\zeta_{\alpha,T},\zeta_{s_{i}\alpha,T}+b^{\prime}\zeta_{\alpha,T}\big{\rangle}=\langle\zeta_{s_{i}\alpha,T},\zeta_{s_{i}\alpha,T}\rangle+b^{\prime
2}\langle\zeta_{\alpha,T},\zeta_{\alpha,T}\rangle,$
$\displaystyle\langle\zeta_{s_{i}\alpha,T},\zeta_{s_{i}\alpha,T}\rangle=\big{(}1-b^{\prime
2}\big{)}\langle\zeta_{\alpha,T},\zeta_{\alpha,T}\rangle.$

A necessary condition that the form be positive-definite ($f\neq 0$ implies
$\langle f,f\rangle>0$) is that $-1<b^{\prime}<1$ in each of the possible
steps. Since (with $j=r_{\alpha}(i)$ and $\ell=r_{\alpha}(i+1)$)

$\displaystyle 1-b^{\prime
2}=\frac{[\alpha_{i+1}-\alpha_{i}+(c(\ell,T)-c(j,T)+1)\kappa][\alpha_{i+1}-\alpha_{i}+(c(\ell,T)-c(j,T)-1)\kappa]}{[\alpha_{i+1}-\alpha_{i}+(c(\ell,T)-c(j,T))\kappa]^{2}},$

the extreme values of $(c(\ell,T)-c(j,T)\pm 1)$ are $\pm h_{\tau}$, and
$\alpha_{i+1}-\alpha_{i}\geq 1$, it follows that
$-1/h_{\tau}<\kappa<1/h_{\tau}$ implies $1-b^{\prime 2}>0$. Since steps of
type (1) link any $(\alpha,T)$ to $(\alpha^{+},T)$ one can obtain (with
$\varepsilon=\pm 1$)

$\displaystyle\mathcal{E}_{\varepsilon}(\alpha,T):=\prod_{\begin{subarray}{c}1\leq
i<j\leq N\\\
\alpha_{i}<\alpha_{j}\end{subarray}}\left(1+\frac{\varepsilon\kappa}{\alpha_{j}-\alpha_{i}+\kappa(c(r_{\alpha}(j),T)-c(r_{\alpha}(i),T))}\right),$
(3.2)
$\displaystyle\langle\zeta_{\alpha,T},\zeta_{\alpha,T}\rangle=\big{(}\mathcal{E}_{1}(\alpha,T)\mathcal{E}_{-1}(\alpha,T)\big{)}^{-1}\langle\zeta_{\alpha^{+},T},\zeta_{\alpha^{+},T}\rangle.$

Similarly the steps of type (2) (with $\alpha_{i}=\alpha_{i+1}$ and
$j=r_{\alpha}(i)$, $b^{\prime}=\frac{1}{c(j,T)-c(j+1,T)}$) imply the relation

$\displaystyle\big{\langle}\zeta_{\alpha,T^{(j)}},\zeta_{\alpha,T^{(j)}}\big{\rangle}=\big{(}1-b^{\prime
2}\big{)}\langle\zeta_{\alpha,T},\zeta_{\alpha,T}\rangle.$

It was shown in [3] (this is a special case of a result of Griffeth [8,
Theorem 6.1]) that the definition for $\lambda\in\mathbb{N}_{0}^{N,+}$

$\displaystyle\langle\zeta_{\lambda,T},\zeta_{\lambda,T}\rangle=\langle
T,T\rangle_{0}\prod_{i=1}^{N}(1+\kappa c(i,T))_{\lambda_{i}}\prod_{1\leq
i<j\leq
N}\prod_{\ell=1}^{\lambda_{i}-\lambda_{j}}\left(1-\left(\frac{\kappa}{\ell+\kappa(c(i,T)-c(j,T))}\right)^{2}\right).$

together with formula (3.2) produce a Hermitian form (called the covariant
form) satisfying (3.1) and the additional property $\langle
x_{i}f,g\rangle=\langle f,\mathcal{D}_{i}g\rangle$ for all
$f,g\in\mathcal{P}_{\tau}$ and $1\leq i\leq N$ (the bound
$-1/h_{\tau}<\kappa<1/h_{\tau}$ for positivity of this form was found by
Etingof and Stoica [7]).

Here we want a Hermitian form for which multiplication by any $x_{i}$ is an
isometry, that is, $\langle x_{i}f,x_{i}g\rangle=\langle f,g\rangle$ for all
$f,g\in\mathcal{P}_{\tau}$ and $1\leq i\leq N$. Heuristically this should
involve an integral over the $N$-torus. The isometry postulate, and the
equations (3.1) determine the form uniquely, as will be shown. The postulate
$\langle x_{N}f,g\rangle=\langle f,\mathcal{D}_{N}g\rangle$ in the covariant
form is used to compute the effect of a jump
$\zeta_{\Phi\alpha,T}=x_{N}w_{0}^{-1}\zeta_{\alpha,T}$ (see (2.1)), that is,
to evaluate
$\langle\zeta_{\Phi\alpha,T},\zeta_{\Phi\alpha,T}\rangle/\langle\zeta_{\alpha,T},\zeta_{\alpha,T}\rangle$.
From the proofs in [3, Appendix, Corollary 5, Theorem 10] and [5] we see that
the factor $\prod\limits_{i=1}^{N}(1+\kappa c(i,T))_{\lambda_{i}}$ arises from
ratios of this type. This aspect (here we need the ratio to be $1$) motivates
the following:

######  Definition 3.1.

For $\lambda\in\mathbb{N}_{0}^{N,+}$, $\alpha,\beta\in\mathbb{N}_{0}^{N}$, and
$T,T^{\prime}\in\mathcal{Y}(\tau)$ the Hermitian form
$\langle\cdot,\cdot\rangle_{\mathbb{T}}$ on $\mathcal{P}_{\tau}$ is specified
by

$\displaystyle(\alpha,T)\neq(\beta,T^{\prime})\Longrightarrow\langle\zeta_{\alpha,T},\zeta_{\beta,T}\rangle_{\mathbb{T}}=0,$
$\displaystyle\langle\zeta_{\lambda,T},\zeta_{\lambda,T}\rangle_{\mathbb{T}}=\langle
T,T\rangle_{0}\prod_{1\leq i<j\leq
N}\prod_{\ell=1}^{\lambda_{i}-\lambda_{j}}\left(1-\left(\frac{\kappa}{\ell+\kappa(c(i,T)-c(j,T))}\right)^{2}\right),$
$\displaystyle\langle\zeta_{\alpha,T},\zeta_{\alpha,T}\rangle_{\mathbb{T}}=\big{(}\mathcal{E}_{1}(\alpha,T)\mathcal{E}_{-1}(\alpha,T)\big{)}^{-1}\langle\zeta_{\alpha^{+},T},\zeta_{\alpha^{+},T}\rangle_{\mathbb{T}};$

the form is extended to all of $\mathcal{P}_{\tau}$ by linearity in the second
variable and Hermitian symmetry, that is, $\langle
f,c_{1}g+c_{2}h\rangle_{\mathbb{T}}=c_{1}\langle
f,g\rangle_{\mathbb{T}}+c_{2}\langle f,h\rangle_{\mathbb{T}}$ and $\langle
f,g\rangle_{\mathbb{T}}=\overline{\langle g,f\rangle_{\mathbb{T}}}$, for
$f,g,h\in\mathcal{P}_{\tau}$ and $c_{1},c_{2}\in\mathbb{C}$.

Observe that the formula is invariant when $\lambda$ is replaced by
$\lambda+m\boldsymbol{1=}(\lambda_{1}+m,\lambda_{2}+m,\ldots$,
$\lambda_{N}+m)$ for any $m\in\mathbb{N}$. This follows easily from the
commutation (where $e_{N}:=x_{1}x_{2}\cdots x_{N}$)

$\displaystyle\mathcal{U}_{i}\big{(}e_{N}^{m}f\big{)}=me_{N}^{m}f+e_{N}^{m}\mathcal{U}_{i}f,\qquad
1\leq i\leq N,\qquad m=1,2,\ldots,$

thus $\mathcal{U}_{i}(e_{N}^{m}\zeta_{\alpha,T})=(m+\alpha_{i}+\kappa
c(r_{\alpha}(i),T))e_{N}^{m}\zeta_{\alpha,T}$, and $e_{N}^{m}\zeta_{\alpha,T}$
is a simultaneous eigenfunction of $\\{\mathcal{U}_{i}\\}$ with the same
eigenvalues and the same leading term as $\zeta_{\alpha+m\boldsymbol{1},T}$.
Hence $\zeta_{\alpha+m\boldsymbol{1},T}=e_{N}^{m}\zeta_{\alpha,T}$. We now
extend the structure of NSJP’s to $V_{\tau}$-valued Laurent polynomials,
thereby producing a basis:

######  Definition 3.2.

Suppose $\alpha\in\mathbb{Z}^{N}$ then set
$\zeta_{\alpha,T}=e_{N}^{-m}\zeta_{\alpha+m\boldsymbol{1},T}$ where
$m\in\mathbb{N}_{0}$ and satisfies $m\geq-\min_{j}\alpha_{i}$. This is valid
since $\alpha+m\boldsymbol{1}\in\mathbb{N}_{0}^{N}$ and by the relation
$\zeta_{\beta+k\boldsymbol{1},T}=e_{N}^{k}\zeta_{\beta,T}$ for
$\beta\in\mathbb{N}_{0}^{N}$ and $k\in\mathbb{N}_{0}$.

The proof that the form satisfies the properties (3.1) with respect to steps
is the same as the one in [3, Propositions 8 and 9], and it suffices to verify
the effect of a jump.

######  Theorem 3.3.

Suppose $\alpha\in\mathbb{N}_{0}^{N,+}$ then
$\langle\zeta_{\Phi\alpha,T},\zeta_{\Phi\alpha,T}\rangle_{\mathbb{T}}=\langle\zeta_{\alpha,T},\zeta_{\alpha,T}\rangle_{\mathbb{T}}$.

###### Proof.

We use (2.1) to relate
$\langle\zeta_{\Phi\alpha,T},\zeta_{\Phi\alpha,T}\rangle_{\mathbb{T}}$ to
$\langle\zeta_{\beta,T},\zeta_{\beta,T}\rangle_{\mathbb{T}}$ where
$\beta=(\Phi\alpha)^{+}=(\alpha_{1}+1,\alpha_{2}$, $\ldots,\alpha_{N})$. The
product $\mathcal{E}_{\varepsilon}(\Phi\alpha,T)$ is over the pairs
$\alpha_{i}=(\Phi\alpha)_{i-1}<(\Phi\alpha)_{N}=\alpha_{1}+1$ for $2\leq i\leq
N$, thus

$\displaystyle\mathcal{E}_{\varepsilon}(\Phi\alpha,T)=\prod\limits_{i=2}^{N}\left(1-\frac{\varepsilon\kappa}{\alpha_{1}+1-\alpha_{i}+\kappa(c(r_{\Phi\alpha}(N),T)-c(r_{\Phi\alpha}(i-1),T))}\right)$
$\displaystyle\hphantom{\mathcal{E}_{\varepsilon}(\Phi\alpha,T)}{}=\prod\limits_{i=2}^{N}\left(1-\frac{\varepsilon\kappa}{\alpha_{1}+1-\alpha_{i}+\kappa(c(1,T)-c(i,T))}\right).$

By definition

$\displaystyle\langle\zeta_{\Phi\alpha,T},\zeta_{\Phi\alpha,T}\rangle_{\mathbb{T}}=\big{(}\mathcal{E}_{1}(\Phi
a,T)\mathcal{E}_{-1}(\Phi\alpha,T)\big{)}^{-1}\langle\zeta_{\beta,T},\zeta_{\beta,T}\rangle_{\mathbb{T}}$
$\displaystyle\hphantom{\langle\zeta_{\Phi\alpha,T},\zeta_{\Phi\alpha,T}\rangle_{\mathbb{T}}}{}=\prod\limits_{i=2}^{N}\left(1-\left(\frac{\kappa}{\alpha_{1}+1-\alpha_{i}+\kappa(c(1,T)-c(i,T))}\right)^{2}\right)^{-1}$
$\displaystyle\hphantom{\langle\zeta_{\Phi\alpha,T},\zeta_{\Phi\alpha,T}\rangle_{\mathbb{T}}=}{}\times\langle
T,T\rangle_{0}\prod_{1\leq i<j\leq
N}\prod_{\ell=1}^{\beta_{i}-\beta_{j}}\left(1-\left(\frac{\kappa}{\ell+\kappa(c(i,T)-c(j,T))}\right)^{2}\right)$
$\displaystyle\hphantom{\langle\zeta_{\Phi\alpha,T},\zeta_{\Phi\alpha,T}\rangle_{\mathbb{T}}}{}=\langle
T,T\rangle_{0}\prod_{2\leq i<j\leq
N}\prod_{\ell=1}^{\alpha_{i}-\alpha_{j}}\left(1-\left(\frac{\kappa}{\ell+\kappa(c(i,T)-c(j,T))}\right)^{2}\right)$
$\displaystyle\hphantom{\langle\zeta_{\Phi\alpha,T},\zeta_{\Phi\alpha,T}\rangle_{\mathbb{T}}=}{}\times\prod\limits_{j=2}^{N}\prod_{\ell=1}^{\alpha_{1}-\alpha_{j}}\left(1-\left(\frac{\kappa}{\ell+\kappa(c(1,T)-c(j,T))}\right)^{2}\right)=\langle\zeta_{\alpha,T},\zeta_{\alpha,T}\rangle_{\mathbb{T}}.$

The terms in the product for $i=1$ and $2\leq j\leq N$,
$\ell=\beta_{1}-\beta_{j}=\alpha_{1}+1-\alpha_{j}$ are canceled out. ∎

We summarize the key results. We say $\kappa$ is generic if
$(\alpha,T)\neq(\beta,T^{\prime})$ implies the spectral vectors
$\xi_{\alpha,T}\neq\xi_{\beta,T^{\prime}}$.

######  Proposition 3.4.

For generic $\kappa$ the Hermitian form
$\langle\cdot,\cdot\rangle_{\mathbb{T}}$ satisfies

  1. $1)$

if $f$, $g$ are homogeneous and $\deg f\neq\deg g$ then $\langle
f,g\rangle_{\mathbb{T}}=0$,

  2. $2)$

$\langle wf,wg\rangle_{\mathbb{T}}=\langle f,g\rangle_{\mathbb{T}}$,
$f,g\in\mathcal{P}_{\tau}$, $w\in\mathcal{S}_{N}$,

  3. $3)$

$\langle x_{i}\mathcal{D}_{i}f,g\rangle_{\mathbb{T}}=\langle
f,x_{i}\mathcal{D}_{i}g\rangle_{\mathbb{T}}$ for $f,g\in\mathcal{P}_{\tau}$
and $1\leq i\leq N$,

  4. $4)$

$\langle x_{i}f,x_{i}g\rangle_{\mathbb{T}}=\langle f,g\rangle_{\mathbb{T}}$
for $f,g\in\mathcal{P}_{\tau}$ and $1\leq i\leq N$.

###### Proof.

For generic $\kappa$ the NSJP’s $\zeta_{\alpha,T}$ with $|\alpha|=n$ form a
basis for $\mathcal{P}_{\tau,n}$; this immediately implies (1). For (2) the
fact that $\langle
s_{i}\zeta_{\alpha,T},s_{i}\zeta_{\beta,T^{\prime}}\rangle_{\mathbb{T}}=\langle\zeta_{\alpha,T},\zeta_{\beta,T^{\prime}}\rangle_{\mathbb{T}}$
for $1\leq i<N$ follows from the corresponding results in [3, Propositions 8
and 9, Corollary 3] when $|\alpha|=|\beta|$, otherwise from Definition 3.1.
This suffices for (2) since $\\{s_{i}\\}$ generates $\mathcal{S}_{N}$. The
definition of NSJP’s implies trivially that
$\langle\mathcal{U}_{i}\zeta_{\alpha,T},\zeta_{\beta,T^{\prime}}\rangle_{\mathbb{T}}=\langle\zeta_{\alpha,T},\mathcal{U}_{i}\zeta_{\beta,T^{\prime}}\rangle_{\mathbb{T}}$
for all $i$ and $(\alpha,T)$, $(\beta,T^{\prime})$ because both sides vanish
if $(\alpha,T)\neq(\beta,T^{\prime})$, otherwise equal
$\xi_{\alpha,T}(i)\langle\zeta_{\alpha,T},\zeta_{\alpha,T}\rangle_{\mathbb{T}}$.
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 $\langle(i,j)f,g\rangle_{\mathbb{T}}=\langle
f,(i,j)g\rangle_{\mathbb{T}}$ from (2) show that $\langle
x_{i}\mathcal{D}_{i}f,g\rangle_{\mathbb{T}}=\langle
f,x_{i}\mathcal{D}_{i}g\rangle_{\mathbb{T}}$ for $1\leq i\leq N$. For part (4)
(recall $w_{0}=(123\ldots N)$)
$\zeta_{\Phi\alpha,T}=x_{N}w_{0}^{-1}\zeta_{\alpha,T}$ and
$\zeta_{\Phi\beta,T^{\prime}}=x_{N}w_{0}^{-1}\zeta_{\beta,T^{\prime}}$. By
Theorem 3.3
$\langle\zeta_{\Phi\alpha,T},\zeta_{\Phi\beta,T^{\prime}}\rangle_{\mathbb{T}}=\langle\zeta_{\alpha,T},\zeta_{\beta,T^{\prime}}\rangle_{\mathbb{T}}$
(if $(\alpha,T)\neq(\beta,T^{\prime})$ then
$(\Phi\alpha,T)\neq(\Phi\beta,T^{\prime})$). Thus for each $(\alpha,T)$,
$(\beta,T^{\prime})$

$\displaystyle\big{\langle}x_{N}\big{(}w_{0}^{-1}\zeta_{\alpha,T}\big{)},x_{N}\big{(}w_{0}^{-1}\zeta_{\beta,T^{\prime}}\big{)}\big{\rangle}_{\mathbb{T}}=\langle\zeta_{\alpha,T},\zeta_{\beta,T^{\prime}}\rangle_{\mathbb{T}}=\big{\langle}w_{0}^{-1}\zeta_{\alpha,T},w_{0}^{-1}\zeta_{\beta,T^{\prime}}\big{\rangle}_{\mathbb{T}}.$

The set $\big{\\{}w_{0}^{-1}\zeta_{\alpha,T}:(\alpha,T)\big{\\}}$ is a basis
for $\mathcal{P}_{\tau}$ thus $\langle
x_{N}f,x_{N}g\rangle_{\mathbb{T}}=\langle f,g\rangle_{\mathbb{T}}$ for all
$f,g\in\mathcal{P}_{\tau}$. For any $i$

$\displaystyle\langle x_{i}f,x_{i}g\rangle_{\mathbb{T}}$
$\displaystyle=\langle(i,N)x_{i}f,(i,N)x_{i}g\rangle_{\mathbb{T}}=\langle
x_{N}(i,N)f,x_{N}(i,N)g\rangle_{\mathbb{T}}$
$\displaystyle=\langle(i,N)f,(i,N)g\rangle_{\mathbb{T}}=\langle
f,g\rangle_{\mathbb{T}};$

and this completes the proof. ∎

This lays the abstract foundation for the next developments.

##  4 Fourier–Stieltjes coefficients on the torus

The torus $\mathbb{T}^{N}:=\big{\\{}x\in\mathbb{C}^{N}\colon|x_{i}|=1,\,1\leq
i\leq N\big{\\}}$ is a multiplicative compact abelian group with dual group
$\mathbb{Z}^{N}$. We will use this property to find the measure of
orthogonality for the NSJP’s on the torus. First we produce the
Fourier–Stieltjes coefficients of the hypothetical measure and then use a
matrix version of a theorem of Bochner to deduce the existence of the measure.

When $\kappa$ is generic the NSJP’s form a basis for $\mathcal{P}_{\tau}$ and
it is possible to make the definition

$\displaystyle\widetilde{A}\big{(}\alpha,\beta,T,T^{\prime}\big{)}:=\big{(}\langle
T,T\rangle_{0}\big{\langle}T^{\prime},T^{\prime}\big{\rangle}_{0}\big{)}^{-1/2}\big{\langle}x^{\alpha}\otimes
T,x^{\beta}\otimes T^{\prime}\big{\rangle}_{\mathbb{T}}$

for $\alpha,\beta\in\mathbb{N}_{0}^{N}$ and
$T,T^{\prime}\in\mathcal{Y}(\tau)$. In effect this uses the orthonormal basis
of $V_{\tau}$. By the symmetry of the form
$\widetilde{A}(\alpha,\beta,T,T^{\prime})=\widetilde{A}(\beta,\alpha,T^{\prime},T)$.
By Proposition 3.4 $|\alpha|\neq|\beta|$ implies
$\widetilde{A}(\alpha,\beta,T,T^{\prime})=0$. Another consequence is
$\widetilde{A}(\mathbf{0},\mathbf{0},T,T^{\prime})=\delta_{T,T^{\prime}}$.

######  Definition 4.1.

For each $\gamma\in\mathbb{Z}^{N}$ with $\sum_{i=1}^{N}\gamma_{i}=0$ let
$\gamma_{i}^{\pi}=\max(\gamma_{i},0)$ and
$\gamma_{i}^{\nu}=-\min(\gamma_{i},0)$ for $1\leq i\leq N$; then
$\gamma=\gamma^{\pi}-\gamma^{\nu}$ and
$\gamma^{\pi},\gamma^{\nu}\in\mathbb{N}_{0}^{N}$. Furthermore
$|\gamma^{\pi}|=|\gamma^{\nu}|$ and
$\sum_{i}|\gamma_{i}|=|\gamma^{\pi}|+|\gamma^{\nu}|$ is even.

Introduce the index set $\boldsymbol{Z}_{N}$ and its graded components by

$\displaystyle\boldsymbol{Z}_{N}:=\left\\{\alpha\in\mathbb{Z}^{N}\colon\sum_{i=1}^{N}\alpha_{i}=0\right\\},$
$\displaystyle\boldsymbol{Z}_{N,n}:=\left\\{\alpha\in\boldsymbol{Z}_{N}\colon\sum_{i=1}^{N}|\alpha_{i}|=2n\right\\},\qquad
n=0,1,2,\ldots.$

Formula (A.1) for $\\#\boldsymbol{Z}_{N,n}$ is in Appendix A.

######  Definition 4.2.

For $\gamma\in\mathbb{Z}^{N}$ the matrix $A_{\gamma}$ (of size
$\\#\mathcal{Y}(\tau)\times\\#\mathcal{Y}(\tau)$) is given by

$\displaystyle(A_{\gamma})_{T,T^{\prime}}=\widetilde{A}\big{(}\gamma^{\pi},\gamma^{\nu},T,T^{\prime}\big{)},\qquad
T,T^{\prime}\in\mathcal{Y}(\tau),\qquad\gamma\in\boldsymbol{Z}_{N},$
$\displaystyle A_{\gamma}=0,\qquad\gamma\notin\boldsymbol{Z}_{N}.$

######  Proposition 4.3.

Suppose $\alpha,\beta\in\mathbb{N}_{0}^{N}$ and
$T,T^{\prime}\in\mathcal{Y}(\tau)$ then
$\widetilde{A}(\alpha,\beta,T,T^{\prime})=(A_{\alpha-\beta})_{T,T^{\prime}}$.

###### Proof.

If $|\alpha|\neq|\beta|$ then $\sum\limits_{i=1}^{N}(\alpha_{i}-\beta_{i})\neq
0$, $\widetilde{A}(\alpha,\beta,T,T^{\prime})=0$ and $A_{\alpha-\beta}=0$ by
definition. If $|\alpha|=|\beta|$ let $\zeta_{i}=\min(\alpha_{i},\beta_{i})$
for $1\leq i\leq N$ then $x^{\zeta}$ is a factor of both $x^{\alpha}$ and
$x^{\beta}$; by Proposition 3.4 $\langle x^{\alpha}\otimes T,x^{\beta}\otimes
T^{\prime}\rangle_{\mathbb{T}}=\langle x^{\alpha-\zeta}\otimes
T,x^{\beta-\zeta}\otimes T^{\prime}\rangle_{\mathbb{T}}$ . By construction
$(\alpha-\beta)^{\pi}=\alpha-\zeta,(\alpha-\beta)^{\nu}=\beta-\zeta$. It
follows that

$\displaystyle\big{\langle}x^{\alpha}\otimes T,x^{\beta}\otimes
T^{\prime}\big{\rangle}=T^{\ast}A_{\alpha-\beta}T^{\prime}=\big{(}\langle
T,T\rangle_{0}\big{\langle}T^{\prime},T^{\prime}\big{\rangle}_{0}\big{)}^{1/2}(A_{\alpha-\beta})_{T,T^{\prime}}.$
∎

For a formal Laurent series
$h(x)=\sum\limits_{\alpha\in\mathbb{Z}^{N}}c_{\alpha}x^{\alpha}$ let
$\operatorname{CT}(h(x))=c_{\mathbf{0}}$, the constant term. Then

$\displaystyle\big{\langle}x^{\alpha}\otimes T,x^{\beta}\otimes
T^{\prime}\big{\rangle}=\big{(}\langle
T,T\rangle_{0}\big{\langle}T^{\prime},T^{\prime}\big{\rangle}_{0}\big{)}^{1/2}\operatorname{CT}\bigg{(}x^{-\alpha}\sum_{\gamma\in\mathbb{Z}^{N}}(A_{\gamma})_{T,T^{\prime}}x^{\gamma}x^{\beta}\bigg{)}.$

In the next section we investigate analytical properties of the formal series,
but first we consider algebraic properties, that is, those not needing any
convergence results.

######  Theorem 4.4.

Suppose $\gamma\in\boldsymbol{Z}_{N}$ and $w\in\mathcal{S}_{N}$ then
$A_{-\gamma}=A_{\gamma}^{\ast}$ and
$A_{w\gamma}=\tau(w)A_{\gamma}\tau\big{(}w^{-1}\big{)}$.

###### Proof.

The relation
$\widetilde{A}(\alpha,\beta,T,T^{\prime})=\widetilde{A}(\beta,\alpha,T^{\prime},T)$
shows $(A_{\alpha-\beta})_{T,T^{\prime}}=(A_{\beta-\alpha})_{T^{\prime},T}$.
By definition

$\displaystyle\big{\langle}w\big{(}x^{\alpha}\otimes
T\big{)},w\big{(}x^{\beta}\otimes
T^{\prime}\big{)}\big{\rangle}_{\mathbb{T}}=\big{\langle}x^{w\alpha}\otimes\tau(w)T,x^{w\beta}\otimes\tau(w)T^{\prime}\big{\rangle}_{\mathbb{T}}$
$\displaystyle\hphantom{\big{\langle}w\big{(}x^{\alpha}\otimes
T\big{)},w\big{(}x^{\beta}\otimes
T^{\prime}\big{)}\big{\rangle}_{\mathbb{T}}}{}=\big{(}\langle
T,T\rangle_{0}\big{\langle}T^{\prime},T^{\prime}\big{\rangle}_{0}\big{)}^{1/2}T^{\ast}\tau(w)^{\ast}A_{w\alpha-w\beta}\tau(w)T^{\prime}$
$\displaystyle\hphantom{\big{\langle}w\big{(}x^{\alpha}\otimes
T\big{)},w\big{(}x^{\beta}\otimes
T^{\prime}\big{)}\big{\rangle}_{\mathbb{T}}}{}=\big{\langle}x^{\alpha}\otimes
T,x^{\beta}\otimes T^{\prime}\big{\rangle}_{\mathbb{T}}=\big{(}\langle
T,T\rangle_{0}\big{\langle}T^{\prime},T^{\prime}\big{\rangle}_{0}\big{)}^{1/2}T^{\ast}A_{\alpha-\beta}T^{\prime}$

and thus $A_{\gamma}=\tau(w)^{-1}A_{w\gamma}\tau(w)$ (recall $\tau$ is real-
orthogonal so $\tau(w)^{\ast}=\tau\big{(}w^{-1}\big{)}$). ∎

Summing over the graded components $\boldsymbol{Z}_{N,n}$ produces Laurent
polynomials with good properties, such as analyticity in
$(\mathbb{C}\backslash\\{0\\})^{N}$. The maps $a\mapsto w\alpha$
($w\in\mathcal{S}_{N}$) and $\alpha\mapsto-\alpha$ act as permutations on each
$\boldsymbol{Z}_{N,n}$.

######  Definition 4.5.

For $n=0,1,2,\ldots$ let

$\displaystyle
H_{n}(x):=\sum_{\alpha\in\boldsymbol{Z}_{N,n}}A_{\alpha}x^{\alpha},$

a Laurent polynomial with matrix coefficients.

For complex Laurent polynomials
$f(x)=\sum\limits_{\alpha\in\mathbb{Z}^{N}}c_{\alpha}x^{\alpha}$ (finite sum)
define
$f(x)^{\ast}=\sum\limits_{\alpha\in\mathbb{Z}^{N}}\overline{c_{\alpha}}x^{-\alpha}$;
if the coefficients $\\{c_{\alpha}\\}$ are matrices then
$f(x)^{\ast}=\sum\limits_{\alpha\in\mathbb{Z}^{N}}c_{\alpha}^{\ast}x^{-\alpha}$.
There is a slight abuse of notation here: if $x\in\mathbb{T}^{N}$ then
$\overline{(x^{\alpha})}=x^{-\alpha}$ and $f(x)^{\ast}$ agrees with the
adjoint of the matrix $f(x)$.

######  Proposition 4.6.

Suppose $n=0,1,2,\ldots$ and $w\in\mathcal{S}_{N}$ then
$H_{n}(xw)=\tau(w)^{-1}H_{n}(x)\tau(w)$ and $H_{n}(x)^{\ast}=H_{n}(x)$.

###### Proof.

Compute

$\displaystyle H_{n}(xw)$
$\displaystyle=\sum_{\alpha\in\boldsymbol{Z}_{N,n}}A_{\alpha}(wx)^{\alpha}=\sum_{\alpha\in\boldsymbol{Z}_{N,n}}A_{\alpha}x^{w\alpha}=\sum_{\beta\in\boldsymbol{Z}_{N,n}}A_{w^{-1}\beta}x^{\beta}$
$\displaystyle=\tau\big{(}w^{-1}\big{)}\sum_{\beta\in\boldsymbol{Z}_{N,n}}A_{\beta}x^{\beta}\tau(w)=\tau(w)^{-1}H_{n}(x)\tau(w).$

Also
$H_{n}(x)^{\ast}=\sum\limits_{\alpha\in\boldsymbol{Z}_{N,n}}A_{\alpha}^{\ast}x^{-\alpha}=\sum\limits_{\alpha\in\boldsymbol{Z}_{N,n}}A_{-\alpha}x^{-\alpha}=H_{n}(x)$.
∎

As a consequence we find an important commutation satisfied by a particular
point value of $H_{n}(x)$ (recall the $N$-cycle $w_{0}=(1,2,\ldots,N)$).

######  Corollary 4.7.

Suppose $n=1,2,3,\ldots$ then
$\tau(w_{0})^{-1}H_{n}(x_{0})\tau(w_{0})=H_{n}(x_{0})$, where
$x_{0}=\big{(}1,\omega,\ldots,\omega^{N-1}\big{)}$,
$\omega=\exp\frac{2\pi\mathrm{i}}{N}$.

###### Proof.

By definition $x_{0}w_{0}=\big{(}\omega,\ldots,\omega^{N-1},1\big{)}=\omega
x_{0}$. Each monomial $x^{\alpha}$ for $\alpha\in\boldsymbol{Z}_{N}$ is
homogeneous of degree zero (suppose $c\in\mathbb{C}\backslash\\{0\\}$ then
$(cx)^{\alpha}=c^{\alpha_{1}+\cdots+\alpha_{N}}x^{\alpha}=x^{\alpha}$) thus
$H_{n}(x_{0}w_{0})=H_{n}(\omega x_{0})=H_{n}(x_{0})$ and
$H_{n}(x_{0}w_{0})=\tau(w_{0})^{-1}H_{n}(x_{0})\tau(w_{0})$. ∎

We turn to the harmonic analysis significance of the matrices
$\\{A_{\alpha}\\}$. For an integrable function $f$ on $\mathbb{T}^{N}$ the
Fourier transform (coefficient) is

$\displaystyle\widehat{f}(\alpha)=\int_{\mathbb{T}^{N}}f(x)x^{-\alpha}\mathrm{d}m(x),\qquad\alpha\in\mathbb{Z}^{N},$

where $x:=(\exp(\mathrm{i}\theta_{1}),\ldots,\exp(\mathrm{i}\theta_{N}))$ and
$\mathrm{d}m(x)=(2\pi)^{-N}\mathrm{d}\theta_{1}\cdots\mathrm{d}\theta_{N}$;
and for a Baire measure $\mu$ on $\mathbb{T}^{N}$ the Fourier–Stieltjes
transform is

$\displaystyle\widehat{\mu}(\alpha)=\int_{\mathbb{T}^{N}}x^{-\alpha}\mathrm{d}\mu(x),\qquad\alpha\in\mathbb{Z}^{N}.$

We will show that there is a matrix-valued measure $\mu$, positive in a
certain sense, such that $\widehat{\mu}(\alpha)=A_{\alpha}$ for all
$\alpha\in\mathbb{Z}^{N}$ provided that $-1/h_{\tau}<\kappa<1/h_{\tau}$. There
is a version of a theorem of Bochner about positive-definite functions on a
locally compact abelian group which proves this claim. The details of the
proof and some consequences are in Appendix A.

Let $n=\dim V_{\tau}$ and identify $V_{\tau}$ with $\mathbb{C}^{n}$ whose
elements are considered as column vectors (in effect we use indices $1\leq
i\leq n$ instead of $\\{T\in\mathcal{Y}(\tau)\\}$). The inner product on
$\mathbb{C}^{n}$ is $\langle
u,v\rangle:=\sum\limits_{i=1}^{n}\overline{u_{i}}v_{i}$, and the norm is
$|v|=\sqrt{\langle v,v\rangle}$. Note that $\langle u,Av\rangle=u^{\ast}Av$. A
positive-definite matrix $P$ satisfies $\langle u,Pu\rangle\geq 0$ for all
$u\in\mathbb{C}^{n}$ (this implies $P^{\ast}=P$).

######  Definition 4.8.

A function $F\colon\mathbb{Z}^{N}\rightarrow M_{n}(\mathbb{C})$ is positive-
definite if

$\displaystyle\sum\limits_{\alpha,\beta\in\mathbb{Z}^{N}}f(\alpha)^{\ast}F(\alpha-\beta)f(\beta)\geq
0$

for any finitely supported $\mathbb{C}^{n}$-valued function $f$ on
$\mathbb{Z}^{N}$.

######  Theorem 4.9.

Suppose $F$ is positive-definite then there exist Baire measures
$\\{\mu_{jk}\colon 1\leq j,k\leq n\\}$ on $\mathbb{T}^{N}$ such that

$\displaystyle\int_{\mathbb{T}^{N}}x^{-\alpha}\mathrm{d}\mu_{jk}(x)=F(\alpha)_{jk},\qquad\alpha\in\mathbb{Z}^{N},\qquad
1\leq j,k\leq n.$

Furthermore each $\mu_{jj}$ is positive and

$\displaystyle\left\langle
f,g\right\rangle_{F}:=\sum_{i,j=1}^{n}\int_{\mathbb{T}^{N}}\overline{f(x)_{i}}g(x)_{j}\mathrm{d}\mu_{ij}(x)$

defines a positive-semidefinite Hermitian form on
$C\big{(}\mathbb{T}^{N};\mathbb{C}^{n}\big{)}$ $($continuous
$\mathbb{C}^{n}$-valued functions on $\mathbb{T}^{N})$ satisfying $|\langle
f,g\rangle_{F}|\leq B\|f\|_{\infty}\|g\|_{\infty}$ for $f,g\in
C\big{(}\mathbb{T}^{N};\mathbb{C}^{n}\big{)}$ with $B<\infty$.

The proof is in Appendix A.1 and Theorem A.3. (In general the measures
$\mu_{jk}$ are not real-valued for $j\neq k$.) For notational simplicity we
introduce

$\displaystyle\int_{\mathbb{T}^{N}}f(x)^{\ast}\mathrm{d}\mu(x)g(x):=\sum_{i,j=1}^{n}\int_{\mathbb{T}^{N}}\overline{f(x)_{i}}g(x)_{j}\mathrm{d}\mu_{ij}(x).$
(4.1)

To show that $\alpha\mapsto A_{\alpha}$ is positive-definite let $f$ be a
finitely supported $\mathbb{C}^{n}$-valued function $f$ on $\mathbb{Z}^{N}$
and let $p(x)=\sum\limits_{\alpha,T}\langle
T,T\rangle_{0}^{-1/2}f_{T}(\alpha)x^{\alpha}\otimes T$ be the associated
Laurent polynomial (now we use the $T$ indices on $\mathbb{C}^{n}$). Because
this is a finite sum there is a nonnegative integer $m$ such that
$e_{N}^{m}p(x)$ is polynomial (no negative powers). Then for
$-1/h_{\tau}<\kappa<1/h_{\tau}$

$\displaystyle
0\leq\big{\langle}e_{N}^{m}p,e_{N}^{m}p\big{\rangle}_{\mathbb{T}}=\\!\sum_{\alpha,\beta\in\mathbb{Z}^{N}}\sum_{T,T^{\prime}}\big{(}\langle
T,T\rangle_{0}\big{\langle}T^{\prime},T^{\prime}\big{\rangle}_{0}\big{)}^{-1/2}\overline{f_{T}(\alpha)}f_{T^{\prime}}(\beta)\big{\langle}x^{\alpha+m\boldsymbol{1}}\otimes
T,x^{\beta+m\boldsymbol{1}}\otimes T^{\prime}\big{\rangle}$
$\displaystyle\hphantom{0\leq\big{\langle}e_{N}^{m}p,e_{N}^{m}p\big{\rangle}_{\mathbb{T}}}=\sum_{\alpha,\beta\in\mathbb{Z}^{N}}\sum_{T,T^{\prime}}\overline{f_{T}(\alpha)}f_{T^{\prime}}(\beta)(A_{\alpha-\beta})_{T,T^{\prime}}.$

Let $\mu=[\mu_{T,T^{\prime}}]$ be the matrix of measures produced by the
theorem, that is,

$\displaystyle(A_{\alpha})_{T,T^{\prime}}=\int_{\mathbb{T}^{N}}x^{-\alpha}\mathrm{d}\mu_{T,T^{\prime}}(x),\qquad\alpha\in\mathbb{Z}^{N},\qquad
T,T^{\prime}\in\mathcal{Y}(\tau).$

######  Theorem 4.10.

For $-1/h_{\tau}<\kappa<1/h_{\tau}$ there exists a matrix of Baire measures
$\mu=[\mu_{T,T^{\prime}}]$ on $\mathbb{T}^{N}$ such that

$\displaystyle\langle
f,g\rangle_{\mathbb{T}}=\int_{\mathbb{T}^{N}}f(x)^{\ast}\mathrm{d}\mu(x)g(x)$

for all Laurent polynomials $f$, $g$ with coefficients in $V_{\tau}$, in
particular for all NSJP’s $f$, $g$.

Of course we want more detailed information about these measures. The first
step is to apply an approximate identity, a tool from the convolution
structure for measures and functions on the torus. We consider Cesàro
summation of the series $\sum_{\alpha}A_{\alpha}x^{\alpha}$ based on summing
first over each $\boldsymbol{Z}_{N,n}$. Set

$\displaystyle S_{n}(x):=\sum_{\alpha\in\boldsymbol{Z}_{N,n}}x^{\alpha},$

a Laurent polynomial, and the corresponding $(C,\delta)$-kernel (for
$\delta>0$) is defined to be (the Pochhammer symbol is
$(t)_{m}=\prod\limits_{i=1}^{m}(t+i-1)$)

$\displaystyle\sigma_{n}^{\delta}(x):=\sum_{k=0}^{n}\frac{(-n)_{k}}{(-n-\delta)_{k}}S_{k}(x).$

The point is that
$\lim\limits_{n\rightarrow\infty}\frac{(-n)_{k}}{(-n-\delta)_{k}}=1$ for fixed
$k$. In terms of convolution

$\displaystyle\sigma_{n}^{\delta}\ast\mu(x)=\int_{\mathbb{T}^{N}}\sigma_{n}^{\delta}\big{(}xy^{-1}\big{)}\mathrm{d}\mu(y),$
$\displaystyle\widehat{(\sigma_{n}^{\delta}\ast\mu)}(\alpha)=\int_{\mathbb{T}^{N}}\int_{\mathbb{T}^{N}}x^{-\alpha}\sigma_{n}^{\delta}\big{(}xy^{-1}\big{)}\mathrm{d}\mu(y)\mathrm{d}m(x)$
$\displaystyle\hphantom{\widehat{(\sigma_{n}^{\delta}\ast\mu)}(\alpha)}{}=\int_{\mathbb{T}^{N}}\int_{\mathbb{T}^{N}}(xy)^{-\alpha}\sigma_{n}^{\delta}(x)\mathrm{d}\mu(y)\mathrm{d}m(x)=A_{\alpha}\widehat{\sigma_{n}^{\delta}}(\alpha),$

and $\widehat{\sigma_{n}^{\delta}}(\alpha)=\frac{(-n)_{k}}{(-n-\delta)_{k}}$
for $\alpha\in\boldsymbol{Z}_{N,k}$ for $0\leq k\leq n$ and $=0$ for
$|\alpha|>2n$ (or $\alpha\notin\boldsymbol{Z}_{N}$). Thus
$\sigma_{n}^{\delta}\ast\mu(x)=\sum\limits_{k=0}^{n}\frac{(-n)_{k}}{(-n-\delta)_{k}}H_{k}(x)$.
In fact $\sigma_{n}^{N-1}(x)\geq 0$ for all $x\in\mathbb{T}^{N}$ (Corollary
4.12 below) which implies $\sigma_{n}^{N-1}\ast\mu$ converges to $\mu$ in a
useful sense (weak-$\ast$, see Theorem 4.17(4)) and
$\sigma_{n}^{N-1}\ast\mu(x)$ is a Laurent polynomial all of whose point values
are positive-semidefinite matrices. Also
$\big{\|}\sigma_{n}^{N-1}\big{\|}_{1}:=\int_{\mathbb{T}^{N}}\big{|}\sigma_{n}^{N-1}\big{|}\mathrm{d}m=1$.

The complete symmetric polynomial in $N$ variables and degree $n$ is given by

$\displaystyle
h_{n}(x):=\sum\left\\{x^{\alpha}\colon\alpha\in\mathbb{N}_{0}^{N}\colon\sum_{i=1}^{N}\alpha_{i}=n\right\\}.$

Recall

$\displaystyle\\#\left\\{\alpha\in\mathbb{N}_{0}^{N}\colon\sum_{i=1}^{N}\alpha_{i}=m\right\\}=\frac{(N)_{m}}{m!}\qquad\text{for}\quad
m=0,1,2,3,\ldots.$

######  Theorem 4.11.

For $n\geq 0$

$\displaystyle
h_{n}\left(\frac{1}{x_{1}},\frac{1}{x_{2}},\ldots,\frac{1}{x_{N}}\right)h_{n}(x_{1},\ldots,x_{N})=\frac{(N)_{n}}{n!}\sigma_{n}^{N-1}(x).$
(4.2)

###### Proof.

The product is a sum of terms $x^{\alpha-\beta}$ with
$\alpha,\beta\in\mathbb{N}_{0}^{N}$ and $|\alpha|=n=|\beta|$. For example the
term $x^{0}=1$ appears exactly $\frac{(N)_{n}}{n!}$ times, because the number
of terms in $h_{n}$ is $\frac{(N)_{n}}{n!}$. Consider a fixed
$\gamma\in\boldsymbol{Z}_{N,m}$ for some $m$ with $0\leq m\leq n$. The term
$x^{\gamma}$ appears in the product for each pair $(\alpha,\beta)$ with

$\displaystyle\alpha=\gamma^{\pi}+\alpha^{\prime},\qquad\beta=\gamma^{\nu}+\alpha^{\prime},\qquad\gamma=\alpha-\beta,$

where $\alpha^{\prime}\in\mathbb{N}_{0}^{N}$ and
$\sum\limits_{i=1}^{N}\alpha_{i}^{{}^{\prime}}=n-m$. Recall
$\gamma_{i}^{\pi}=\max(\gamma_{i},0)$ and
$\gamma_{i}^{\nu}=-\min(0,\gamma_{i})=\max(0,-\gamma_{i})$; thus
$\sum\limits_{i=1}^{N}\gamma_{i}^{\pi}=m$ and
$\sum\limits_{i=1}^{N}\alpha_{i}=n$. Therefore the coefficient of $x^{\gamma}$
is

$\displaystyle\\#\left\\{\alpha^{\prime}\in\mathbb{N}_{0}^{N}\colon\sum\limits_{i=1}^{N}\alpha_{i}^{{}^{\prime}}=n-m\right\\}=\frac{(N)_{n-m}}{(n-m)!}.$

Hence

$\displaystyle
h_{n}\left(\frac{1}{x_{1}},\frac{1}{x_{2}},\ldots,\frac{1}{x_{N}}\right)h_{n}(x_{1},\ldots,x_{N})=\sum_{m=0}^{N}\frac{(N)_{n-m}}{(n-m)!}S_{m}(x).$

To finish the proof multiply this relation by $\frac{n!}{(N)_{n}}$ and compute

$\displaystyle\frac{n!}{(N)_{n}}\frac{(N)_{n-m}}{(n-m)!}=(-1)^{m}(-n)_{m}\frac{(N)_{n-m}}{(N)_{n-m}(N+n-m)_{m}}$
$\displaystyle\hphantom{\frac{n!}{(N)_{n}}\frac{(N)_{n-m}}{(n-m)!}}{}=(-1)^{m}\frac{(-n)_{m}}{(N+n-m)_{m}}=\frac{(-n)_{m}}{(1-N-n)_{m}}.$
∎

######  Corollary 4.12.

$\sigma_{n}^{N-1}(x)\geq 0$ for all $x\in\mathbb{T}^{N}$.

###### Proof.

$\overline{h_{n}(x_{1},\ldots,x_{N})}=h_{n}\left(\frac{1}{x_{1}},\frac{1}{x_{2}},\ldots,\frac{1}{x_{N}}\right)$
for $x\in\mathbb{T}^{N}$. ∎

Observe that this kernel applies to the quotient space
$\mathbb{T}^{N}/\mathbb{D}$ where

$\displaystyle\mathbb{D}:=\\{(u,u,\ldots,u)\colon u\in\mathbb{C},\,|u|=1\\},$

is the diagonal subgroup. That is, each $S_{n}(x)$ is homogeneous of degree
zero, constant on sets $\\{(ux_{1},ux_{2},\ldots,ux_{N})\colon|u|=1\\}$ for
fixed $x\in\mathbb{T}^{N}$.

Here are approximate identity properties of $\sigma_{n}^{N-1}$; we use
$\mathbb{T}^{N}/\mathbb{D}$ to refer to functions homogeneous of degree zero.
There is a standard formula:

######  Lemma 4.13.

Suppose $g,h\in C\big{(}\mathbb{T}^{N}\big{)}$ and $\nu$ is a Baire measure on
$\mathbb{T}^{N}$ then for $h^{{\dagger}}(x):=h\big{(}x^{-1}\big{)}$

$\displaystyle\int_{\mathbb{T}^{N}}g(x)(h\ast\nu)(x)\mathrm{d}m(x)=\int_{\mathbb{T}^{N}}\big{(}g\ast
h^{{\dagger}}\big{)}(y)\mathrm{d}\nu(y).$

###### Proof.

The left side equals

$\displaystyle\int_{\mathbb{T}^{N}}\int_{\mathbb{T}^{N}}g(x)h\big{(}xy^{-1}\big{)}\mathrm{d}\nu(y)\mathrm{d}m(x)=\int_{\mathbb{T}^{N}}\int_{\mathbb{T}^{N}}g(x)h^{{\dagger}}\big{(}yx^{-1}\big{)}\mathrm{d}m(x)\mathrm{d}\nu(y)$

(by Fubini’s theorem) which equals the right side. ∎

The following is a standard result on approximate identities.

######  Proposition 4.14.

Suppose $f\in C\big{(}\mathbb{T}^{N}/\mathbb{D}\big{)}$ then
$\big{\|}f-f\ast\sigma_{n}^{N-1}\big{\|}_{\infty}\rightarrow 0$ as
$n\rightarrow\infty$.

###### Proof.

For $\varepsilon>0$ there exists a Laurent polynomial $p$ on
$\mathbb{T}^{N}/\mathbb{D}$ such that $\|f-p\|_{\infty}<\varepsilon$. Then

$\displaystyle f-f\ast\sigma_{n}^{N-1}=(f-p)+\big{(}p-\sigma_{n}^{N-1}\ast
p\big{)}+(p-f)\ast\sigma_{n}^{N-1},$

and
$\big{\|}(p-f)\ast\sigma_{n}^{N-1}\big{\|}_{\infty}\leq\|p-f\|_{\infty}\big{\|}\sigma_{n}^{N-1}\big{\|}_{1}<\varepsilon$.
Let
$p(x)=\sum\limits_{m=0}^{M}\sum\limits_{\alpha\in\boldsymbol{Z}_{N,m}}c_{\alpha}x^{\alpha}$
for some coefficients $c_{\alpha}$ (and finite $M$); thus

$\displaystyle\big{(}p-\sigma_{n}^{N-1}\ast
p\big{)}(x)=\sum\limits_{m=0}^{M}\sum\limits_{\alpha\in\boldsymbol{Z}_{N,m}}\left(1-\frac{(-n)_{m}}{(1-N-n)_{m}}\right)c_{\alpha}x^{\alpha},$

which tends to zero in norm as $n\rightarrow\infty$. ∎

######  Corollary 4.15.

Suppose $\nu$ is a Baire measure on $\mathbb{T}^{N}$ and $f\in
C\big{(}\mathbb{T}^{N}/\mathbb{D}\big{)}$ then

$\displaystyle\lim_{n\rightarrow\infty}\int_{\mathbb{T}^{N}}f(x)\big{(}\sigma_{n}^{N-1}\ast\nu\big{)}(x)\mathrm{d}m(x)=\int_{\mathbb{T}^{N}}f(x)\mathrm{d}\nu(x).$

###### Proof.

By Lemma 4.13

$\displaystyle\int_{\mathbb{T}^{N}}f(x)\big{(}\sigma_{n}^{N-1}\ast\nu\big{)}(x)\mathrm{d}m(x)=\int_{\mathbb{T}^{N}}\big{(}f\ast\sigma_{n}^{N-1}\big{)}(x)\mathrm{d}\nu(x),$

since $\big{(}\sigma_{n}^{N-1}\big{)}^{{\dagger}}=\sigma_{n}^{N-1}$, and
$f\ast\sigma_{n}^{N-1}$ converges uniformly to $f$ as $n\rightarrow\infty$. ∎

######  Definition 4.16.

Define the $\mu$-approximating Laurent polynomials

$\displaystyle
K_{n}(x):=\sigma_{n}^{N-1}\ast\mu(x)=\sum_{m=0}^{n}\frac{(-n)_{m}}{(1-N-n)_{m}}\sum_{\alpha\in\boldsymbol{Z}_{N,n}}A_{\alpha}x^{\alpha}.$

Note $\frac{(-n)_{m}}{(1-N-n)_{m}}=\frac{(n-m+1)_{N-1}}{(n+1)_{N-1}}$ for
$0\leq m\leq n$; for example with $N=3$,
$\frac{(-n)_{m}}{(-2-n)_{m}}=\big{(}1-\frac{m}{n+1}\big{)}\big{(}1-\frac{m}{n+2}\big{)}$,
and $=0$ for $m>n$.

######  Theorem 4.17.

For $-1/h_{\tau}<\kappa<1/h_{\tau}$ and $n=1,2,3,\ldots$ the following hold:

  1. $(1)$

$K_{n}(x)$ is positive semi-definite for each $x\in\mathbb{T}^{N}$,

  2. $(2)$

$K_{n}(xw)=\tau(w)^{-1}K_{n}(x)\tau(w)$ for each
$x\in\mathbb{T}^{N},w\in\mathcal{S}_{N}$,

  3. $(3)$

$K_{n}(x_{0})\tau(w_{0})=\tau(w_{0})K_{n}(x_{0})$,

  4. $(4)$

$\lim\limits_{n\rightarrow\infty}\int_{\mathbb{T}^{N}}f(x)^{\ast}K_{n}(x)g(x)\mathrm{d}m(x)=\langle
f,g\rangle_{\mathbb{T}}$ for all $f,g\in\mathcal{P}_{\tau}$; the limit exists
for any $f,g\in C\big{(}\mathbb{T}^{N};V_{\tau}\big{)}$ and defines a
$\|\cdot\|_{\infty}$-bounded positive Hermitian form.

###### Proof.

Part (1) is a consequence of Theorem A.4. Parts (2) and (3) follow from the
properties of $H_{m}$ in Proposition 4.6. For part (4) there is an
intermediate step of averaging over the diagonal group $\mathbb{D}$. Define
the operator $\rho\colon C\big{(}\mathbb{T}^{N}\big{)}\rightarrow
C\big{(}\mathbb{T}^{N}/\mathbb{D}\big{)}$ by

$\displaystyle\rho(p)(x):=\frac{1}{2\pi}\int_{-\pi}^{\pi}p\big{(}e^{\mathrm{i}\theta}x\big{)}\mathrm{d}\theta,\qquad
p\in C\big{(}\mathbb{T}^{N}\big{)}.$

Clearly $\|\rho(p)\|_{\infty}\leq\|p\|_{\infty}$; in effect $\rho$ is the
projection onto Fourier series supported by $\boldsymbol{Z}_{N}$. Then

$\displaystyle\begin{split}&\int_{\mathbb{T}^{N}}p(x)\mathrm{d}\mu_{T,T^{\prime}}(x)=\int_{\mathbb{T}^{N}}\rho(p)(x)\mathrm{d}\mu_{T,T^{\prime}}(x),\\\
&\int_{\mathbb{T}^{N}}p(x)(K_{n}(x))_{T,T^{\prime}}\mathrm{d}m(x)=\int_{\mathbb{T}^{N}}\rho(p)(x)(K_{n}(x))_{T,T^{\prime}}\mathrm{d}m(x),\qquad
T,T^{\prime}\in\mathcal{Y}(\tau).\end{split}$

To extend this to the form $\langle\cdot,\cdot\rangle_{\mathbb{T}}$ express
the typical sum
$\sum\limits_{i,j=1}^{n}\overline{f_{i}}B_{ij}g_{j}=\operatorname{tr}\big{(}(f\otimes
g^{\ast})^{\ast}B\big{)}$ where $\operatorname{tr}$ denotes the trace and
$(f\otimes g^{\ast})_{ij}=f_{i}\overline{g_{j}}$ ($1\leq i,j\leq n$). Then
($\rho$ is applied to matrices entry-wise) for $f,g\in
C\big{(}\mathbb{T}^{N};V_{\tau}\big{)}$

$\displaystyle\int_{\mathbb{T}^{N}}f(x)^{\ast}\mathrm{d}\mu(x)g(x)=\int_{\mathbb{T}^{N}}\operatorname{tr}\big{[}(f(x)\otimes
g(x)^{\ast})^{\ast}\mathrm{d}\mu(x)\big{]}=\int_{\mathbb{T}^{N}}\operatorname{tr}\big{[}\\{\rho(f(x)\otimes
g(x)^{\ast})\\}^{\ast}\mathrm{d}\mu(x)\big{]},$
$\displaystyle\int_{\mathbb{T}^{N}}f(x)^{\ast}K_{n}(x)g(x)\mathrm{d}m(x)=\int_{\mathbb{T}^{N}}\operatorname{tr}\big{[}(f(x)\otimes
g(x)^{\ast})^{\ast}K_{n}(x)\big{]}\mathrm{d}m(x)$
$\displaystyle\hphantom{\int_{\mathbb{T}^{N}}f(x)^{\ast}K_{n}(x)g(x)\mathrm{d}m(x)}{}=\int_{\mathbb{T}^{N}}\operatorname{tr}\big{[}\\{\rho(f(x)\otimes
g(x)^{\ast})\\}^{\ast}K_{n}(x)\big{]}\mathrm{d}m(x).$

The convergence properties of Proposition 4.14 imply part (4). ∎

##  5 Recurrence relations

As a simple illustration consider $\boldsymbol{Z}_{N,1}$ where it suffices to
find $A_{1,-1,0\ldots,0}$. Introduce the unit basis vectors $\varepsilon_{i}$
for $\mathbb{Z}^{N}$ (with $(\varepsilon_{i})_{j}=\delta_{ij}$), so that
$(1,-1,0,\ldots)=\varepsilon_{1}-\varepsilon_{2}$. The relation
$(\varepsilon_{2}-\varepsilon_{1})=(1,2)(\varepsilon_{1}-\varepsilon_{2})$
implies
$A_{\varepsilon_{2}-\varepsilon_{1}}=\tau(1,2)A_{\varepsilon_{1}-\varepsilon_{2}}\tau(1,2)=A_{\varepsilon_{1}-\varepsilon_{2}}^{\ast}$.
From $\langle x_{1}\mathcal{D}_{1}f,g\rangle_{\mathbb{T}}=\langle
f,x_{1}\mathcal{D}_{1}g\rangle_{\mathbb{T}}$ (Proposition 3.4(3)) we find

$\displaystyle x_{1}\mathcal{D}_{1}(x_{1}\otimes T)=x_{1}\otimes T+\kappa
x_{1}\sum_{j=2}^{N}\frac{x_{1}-(x(1,j))_{1}}{x_{1}-x_{j}}\tau(1,j)T$
$\displaystyle\hphantom{x_{1}\mathcal{D}_{1}(x_{1}\otimes T)}{}=x_{1}\otimes
T+\kappa
x_{1}\otimes\tau(\omega_{1})T=x_{1}\otimes(I+\kappa\tau(\omega_{1}))T,$
$\displaystyle x_{1}\mathcal{D}_{1}(x_{2}\otimes T^{\prime})=\kappa
x_{1}\sum_{j=2}^{N}\frac{x_{2}-(x(2,j))_{1}}{x_{1}-x_{j}}\tau(1,j)T^{\prime}=-\kappa
x_{1}\tau(1,2)T^{\prime};$

recall the Jucys–Murphy elements $\omega_{i}:=\sum\limits_{j=i+1}^{N}(i,j)$
and the action $\tau(\omega_{i})T=c(i,T)T$ for $T\in\mathcal{Y}(\tau)$. Next
the equation $\langle x_{1}\mathcal{D}_{1}(x_{1}\otimes T),x_{2}\otimes
T^{\prime}\rangle_{\mathbb{T}}=\langle x_{1}\otimes
T,x_{1}\mathcal{D}_{1}(x_{2}\otimes T^{\prime})\rangle_{\mathbb{T}}$ yields

$\displaystyle
T^{\ast}(I+\kappa\tau(\omega_{1}))^{\ast}A_{\varepsilon_{1}-\varepsilon_{2}}T^{\prime}=-\kappa
T^{\ast}A_{\mathbf{0}}\tau(1,2)T^{\prime},$

and $A_{\mathbf{0}}=I$. This holds for arbitrary $T$, $T^{\prime}$, and
$\tau(\omega_{1})$ is diagonal with the entry at $(T,T)$ being $c(1,T)$ thus

$\displaystyle(I+\kappa\tau(\omega_{1}))A_{\varepsilon_{1}-\varepsilon_{2}}=-\kappa\tau(1,2),\qquad
A_{\varepsilon_{1}-\varepsilon_{2}}=-\kappa(I+\kappa\tau(\omega_{1}))^{-1}\tau(1,2),$

provided $\kappa c(1,T)\neq-1$ for all $T\in\mathcal{Y}(\tau)$.

######  Lemma 5.1.

For $\alpha\in\mathbb{N}_{0}^{N}$, $T\in\mathcal{Y}(\tau)$ and $1\leq i\leq N$

$\displaystyle\begin{split}&x_{i}\mathcal{D}_{i}\big{(}x^{\alpha}\otimes
T\big{)}=\alpha_{i}x^{\alpha}\otimes
T-\kappa\sum_{\alpha_{j}>\alpha_{i}}\sum_{\ell=1}^{\alpha_{j}-\alpha_{i}}x^{\alpha+\ell(\varepsilon_{i}-\varepsilon_{j})}\otimes\tau(i,j)T\\\
&\hphantom{x_{i}\mathcal{D}_{i}\big{(}x^{\alpha}\otimes
T\big{)}=}{}+\kappa\sum_{\alpha_{i}>\alpha_{j}}\sum_{\ell=0}^{\alpha_{i}-\alpha_{j}-1}x^{\alpha+\ell(\varepsilon_{j}-\varepsilon_{i})}\otimes\tau(i,j)T.\end{split}$

###### Proof.

This follows from performing the division in
$\frac{x^{\alpha}-x^{(i,j)\alpha}}{x_{i}-x_{j}}$. ∎

######  Proposition 5.2.

For $\alpha,\beta\in\mathbb{N}_{0}^{N}$ such that $|\alpha|=|\beta|$ and
$1\leq i\leq N$

$\displaystyle(\alpha_{i}-\beta_{i})A_{\alpha-\beta}=\kappa\sum_{\alpha_{j}>\alpha_{i}}\sum_{\ell=1}^{\alpha_{j}-\alpha_{i}}\tau(i,j)A_{\alpha+\ell(\varepsilon_{i}-\varepsilon_{j})-\beta}-\kappa\sum_{\alpha_{i}>\alpha_{j}}\sum_{\ell=0}^{\alpha_{i}-\alpha_{j}-1}\tau(i,j)A_{\alpha+\ell(\varepsilon_{j}-\varepsilon_{i})-\beta}$
$\displaystyle\hphantom{(\alpha_{i}-\beta_{i})A_{\alpha-\beta}=}{}-\kappa\sum_{\beta_{j}>\beta_{i}}\sum_{\ell=1}^{\beta_{j}-\beta_{i}}A_{\alpha-\ell(\varepsilon_{i}-\varepsilon_{j})-\beta}\tau(i,j)$
$\displaystyle\hphantom{(\alpha_{i}-\beta_{i})A_{\alpha-\beta}=}{}+\kappa\sum_{\beta_{i}>\beta_{j}}\sum_{\ell=0}^{\beta_{i}-\beta_{j}-1}A_{\alpha-\ell(\varepsilon_{j}-\varepsilon_{i})-\beta}\tau(i,j).$
(5.1)

###### Proof.

The statement follows from the equation

$\displaystyle\big{\langle}x_{i}\mathcal{D}_{i}(x^{\alpha}\otimes
T),x^{\beta}\otimes T^{\prime}\big{\rangle}=\big{\langle}x^{\alpha}\otimes
T,x_{i}\mathcal{D}_{i}\big{(}x^{\beta}\otimes T^{\prime}\big{)}\big{\rangle}$

and the lemma. ∎

The following is one of the main results of this section. Note that it is
important to involve the multiplicity of the first part of $\gamma$.

######  Theorem 5.3.

Suppose $\gamma\in\boldsymbol{Z}_{N,n}$ such that $\gamma^{\pi}$ is a
partition and $\gamma_{1}^{\pi}=\gamma_{m}^{\pi}>\gamma_{m+1}$ then

$\displaystyle\left(\gamma_{1}I+\kappa\sum_{\ell=m+1}^{N}\tau(1,\ell)\right)A_{\gamma}=-\kappa\sum_{j=m+1,\,\gamma_{j}\geq
0}^{N}\sum_{\ell=1}^{\gamma_{1}-\gamma_{j}-1}\tau(1,j)A_{\gamma+\ell(\varepsilon_{j}-\varepsilon_{1})}$
$\displaystyle\qquad{}-\kappa\sum_{j=m+1,\,\gamma_{j}<0}^{N}\left\\{\tau(1,j)\sum_{\ell=1}^{\gamma_{1}-1}A_{\gamma+\ell(\varepsilon_{j}-\varepsilon_{1})}+\sum_{\ell=1}^{-\gamma_{j}}A_{\gamma-\ell(\varepsilon_{1}-\varepsilon_{j})}\tau(1,j)\right\\}.$
(5.2)

Each of the coefficients $A_{\delta}$ appearing on the second line of the
equation satisfies $\delta\in\boldsymbol{Z}_{N,s}$ for some $s<n$ and for
$A_{\delta}$ on the right-hand side of the first line
$\delta=\delta^{\pi}-\gamma^{\nu}$ where
$\delta^{\pi}\vartriangleleft\gamma^{\pi}$.

###### Proof.

The formula follows from equation (5.1) by setting $i=1$, $\beta_{1}=0$ and
omitting the case $\alpha_{j}>\alpha_{i}$. Suppose $\gamma_{j}^{\nu}=0$ for
$j\leq k$, and $\gamma_{j}^{\pi}=0$ for $j>k$. The typical multi-index in the
second line is

$\displaystyle\big{(}\gamma_{1}-\ell,\gamma_{2}^{\pi},\ldots,\gamma_{k}^{\pi},-\gamma_{k+1}^{\nu},\ldots,-\gamma_{j}^{\nu}+\ell,\ldots\big{)}$

with $1\leq\ell\leq\gamma_{1}-1$ or $1\leq\ell\leq\gamma_{j}^{\nu}$. If
$\gamma_{1}\geq\ell$ then the sum of the nonnegative components is $n-\ell<n$;
and if $\gamma_{1}<\ell$ (possible if $\gamma_{j}^{\nu}>\gamma_{1}$) then the
sum of the nonnegative components is $n-\gamma_{1}$. In both cases the multi-
index is in $\bigcup\limits_{s<n}\boldsymbol{Z}_{N,s}$. From [6, Lemma 10.1.3]
$\gamma^{\pi}\succ(\gamma^{\pi}+\ell(\varepsilon_{j}-\varepsilon_{1}))^{+}$
for $1\leq\ell\leq\gamma_{1}-\gamma_{j}^{\pi}-1$, thus the multi-indices on
the right-hand side of the first line satisfy
$\delta=\delta^{\pi}-\gamma^{\nu}$ with $(\delta^{\pi})^{+}\prec\gamma^{\pi}$,
that is, $\delta^{\pi}\vartriangleleft\gamma^{\pi}$. ∎

If $\gamma$ is $\vartriangleright$-minimal with fixed $\gamma^{\nu}$ then
there are no terms on the right side of the first line (that is,
$\gamma_{j}\geq 0$ implies $\gamma_{1}\geq\gamma_{j}\geq\gamma_{1}-1$).

######  Proposition 5.4.

Among $\gamma\in\boldsymbol{Z}_{N,n}$ such that $\gamma^{\nu}=\beta$ for some
fixed $\beta$ with $|\beta|=n$ and such that $\beta_{j}>0$ exactly when $j>k$
the minimal multi-index for the order
$\gamma^{(1)\pi}\vartriangleright\gamma^{(2)\pi}$ is
$\gamma^{(0)}=\big{(}p+1,\ldots,\overset{(m)}{p+1},p,\ldots,\overset{(k)}{p},-\beta_{k+1},\ldots,-\beta_{N}\big{)}$
where $p=\big{\lfloor}\frac{n}{k}\big{\rfloor}$ and $m=n-kp$ $($so $0\leq
m<k)$. For this multi-index the right-hand side of (5.2) contains only
$A_{\delta}$ with $\delta\in\bigcup\limits_{s=0}^{n-1}\boldsymbol{Z}_{N,s}$.

The proof is technical and is presented as Proposition A.5.

######  Theorem 5.5.

The coefficients $A_{\alpha}$ are rational functions of $\kappa$ and are
finite provided

$\displaystyle\kappa\notin\left\\{-\frac{m}{c}\colon m,c\in\mathbb{N},\,1\leq
c\leq\tau_{1}-1\right\\}\cup\left\\{\frac{m}{c}\colon m,c\in\mathbb{N},\,1\leq
c\leq\ell(\tau)-1\right\\}.$

Also $A_{-\alpha}=A_{\alpha}^{\ast}$ and
$\tau(w)^{\ast}A_{w\alpha}\tau(w)=A_{\alpha}$ for all
$\alpha\in\boldsymbol{Z}_{N}$, $w\in\mathcal{S}_{N}$ and permitted values of
$\kappa$.

###### Proof.

The NSJP $\zeta_{\alpha,T}$ is a rational function of $\kappa$ with no poles
in $-1/h_{\tau}<\kappa<1/h_{\tau}$. The coefficients $A_{\alpha}$ are defined
in terms of all the NSJP’s and are also rational in $\kappa$. In equation
(5.2) the operator on the left of $A_{\gamma}$ is

$\displaystyle\left(\gamma_{1}I+\kappa\sum_{\ell=m+1}^{N}\tau(1,\ell)\right)=\tau(1,m)(\gamma_{1}I+\kappa\tau(\omega_{m}))\tau(1,m),$

where $\omega_{m}$ is the Jucys–Murphy element
$\sum\limits_{\ell=m+1}^{N}(m,\ell)$; the action $\tau(\omega_{m})T=c(m,T)T$
for all $T\in\mathcal{Y}(\tau)$ shows that the eigenvalues of the operator are
$\\{\gamma_{1}+\kappa c(m,T)\colon T\in\mathcal{Y}(\tau)\\}$ and the operator
is invertible provided $\kappa c(m,T)\notin\\{-1,-2,-3,\ldots\\}$ for $1\leq
m\leq N$. The set of values of $c(m,T)$ is $\\{j\in\mathbb{Z}\colon
1-\ell(\tau)\leq j\leq\tau_{1}-1\\}$. Thus an inductive argument based on $n$
in $\boldsymbol{Z}_{N,n}$, the order in Proposition 5.4, and formula (5.2)
shows there are unique solutions for $\\{A_{\alpha}\\}$ provided that the
possible poles at $n+\kappa c(i,T)=0$ are excluded. The relations
$A_{-\alpha}=A_{\alpha}^{\ast}$ and
$\tau(w)^{\ast}A_{w\alpha}\tau(w)=A_{\alpha}$ hold at least in an interval
hence for all $\kappa$, excluding the poles. ∎

The largest interval around $0$ without poles is
$-\frac{1}{\tau_{1}-1}<\kappa<\frac{1}{\ell(\tau)-1}$. As illustration we
describe $A_{\gamma}$ for $\gamma\in\boldsymbol{Z}_{N,2}$. Above we showed

$\displaystyle A_{\varepsilon_{1}-\varepsilon_{j}}=-\kappa\
(I+\kappa\tau(\omega_{1}))^{-1}\tau(1,j),\qquad 2\leq j\leq N.$

Next for $\alpha=\varepsilon_{1}+\varepsilon_{2}$ and $\beta=2\varepsilon_{j}$
for $3\leq j\leq N$ we find

$\displaystyle\left(I+\kappa\sum_{i=3}^{N}\tau(1,i)\right)A_{\varepsilon_{1}+\varepsilon_{2}-2\varepsilon_{j}}=-\kappa(A_{\varepsilon_{2}-\varepsilon_{1}}+A_{\varepsilon_{2}-\varepsilon_{j}})\tau(1,j).$

For $\alpha=\varepsilon_{1}+\varepsilon_{2}$ and
$\beta=\varepsilon_{j}+\varepsilon_{j+1}$ with $3\leq j\leq N-1$

$\displaystyle\left(I+\kappa\sum_{i=3}^{N}\tau(1,i)\right)A_{\varepsilon_{1}+\varepsilon_{2}-\varepsilon_{j}-\varepsilon_{j+1}}=-\kappa\big{(}A_{\varepsilon_{2}-\varepsilon_{j}}\tau(1,j+1)+A_{\varepsilon_{2}-\varepsilon_{j+1}}\tau(1,j)\big{)}.$

For $\alpha=2\varepsilon_{1}$ and $\beta=2\varepsilon_{N}$ we obtain

$\displaystyle(2I+\kappa\tau(\omega_{1}))A_{2\varepsilon_{1}-2\varepsilon_{N}}$
$\displaystyle\qquad{}=-\kappa\left\\{\sum_{\ell=2}^{N-1}\tau(1,\ell)A_{\varepsilon_{1}+\varepsilon_{\ell}-2\varepsilon_{N}}+\tau(1,N)A_{\varepsilon_{1}-\varepsilon_{N}}+(A_{\varepsilon_{1}-\varepsilon_{N}}+I)\tau(1,N)\right\\}.$

The other coefficients for $n=2$ are obtained using the relations

$\displaystyle
A_{-\alpha}=A_{\alpha}^{\ast}\qquad\text{and}\qquad\tau(w)^{\ast}A_{w\alpha}\tau(w)=A_{\alpha}.$

##  6 The differential equation

We will show that $\mu$ satisfies a differential system in a distributional
sense. Let $\mathbb{T}_{\rm
reg}^{N}:=\mathbb{T}^{N}\backslash\bigcup\limits_{1\leq i<j\leq N}\\{x\colon
x_{i}=x_{j}\\}$ (this avoids the singularities of the system) and
$\partial_{j}:=\frac{\partial}{\partial x_{j}}$ for $1\leq j\leq N$. The
system is

$\displaystyle x_{i}\partial_{i}K(x)=\kappa\sum_{j\neq
i}\frac{x_{j}}{x_{i}-x_{j}}\tau(i,j)K(x)+\kappa K(x)\sum_{j\neq
i}\tau(i,j)\frac{x_{i}}{x_{i}-x_{j}},\qquad 1\leq i\leq N.$ (6.1)

Any solution of this system satisfies
$\sum\limits_{i=1}^{N}x_{i}\partial_{i}K\left(x\right)=0$ and thus is
homogeneous of degree zero.

The relation $\langle x_{i}\mathcal{D}_{i}f,g\rangle_{\mathbb{T}}=\langle
f,x_{i}\mathcal{D}_{i}g\rangle_{\mathbb{T}}$ extends to
$C^{(1)}\big{(}\mathbb{T}^{N};V_{\tau}\big{)}$, the continuously
differentiable $V_{\tau}$-valued functions, because Laurent polynomials are
dense in this space. We have shown

$\displaystyle\lim_{n\rightarrow\infty}\int_{\mathbb{T}^{N}}\big{\\{}(x_{i}\mathcal{D}_{i}f(x))^{\ast}K_{n}(x)g(x)-f(x)^{\ast}K_{n}(x)x_{i}\mathcal{D}_{i}g(x)\big{\\}}\mathrm{d}m(x)=0.$

Suppose $p,q\in C^{(1)}\big{(}\mathbb{T}^{N}\big{)}$ (scalar
$\mathbb{C}$-valued) then by periodicity
$\int_{\mathbb{T}^{N}}\frac{\partial}{\partial\theta_{j}}(pq)\mathrm{d}m=0$
thus

$\displaystyle\int_{\mathbb{T}^{N}}\left(\frac{\partial}{\partial\theta_{j}}p\right)q\mathrm{d}m=-\int_{\mathbb{T}^{N}}p\left(\frac{\partial}{\partial\theta_{j}}q\right)\mathrm{d}m.$

Also from
$\frac{\partial}{\partial\theta_{j}}f(x)=\mathrm{i}e^{\mathrm{i}\theta_{j}}\partial_{j}f(x)=\mathrm{i}x_{j}\partial_{j}f(x)$
and $\partial_{j}f^{\ast}(x)=-x_{j}^{-2}(\partial_{j}f)^{\ast}(x)$ we obtain

$\displaystyle\int_{\mathbb{T}^{N}}(x_{j}\partial_{j}f(x))^{\ast}g(x)\mathrm{d}m=-\int_{\mathbb{T}^{N}}x_{j}^{-1}\big{(}{-}x_{j}^{2}\big{)}\partial_{j}f^{\ast}(x)g(x)\mathrm{d}m=\int_{\mathbb{T}^{N}}f^{\ast}(x)x_{j}\partial_{j}g(x)\mathrm{d}m.$

That is, $x_{j}\partial_{j}$ is self-adjoint with respect to $\langle
f,g\rangle:=\int_{\mathbb{T}^{N}}f^{\ast}(x)g(x)\mathrm{d}m$. The result
extends to $C^{(1)}\big{(}\mathbb{T}^{N};V_{\tau}\big{)}$.

Specialize to a closed $\mathcal{S}_{N}$-invariant subset
$E\subset\mathbb{T}_{\rm reg}^{N}$ which is the closure of its interior (for
example
$E_{\varepsilon}=\big{\\{}x\in\mathbb{T}^{N}\colon\min_{i<j}|x_{i}-x_{j}|\geq\varepsilon\big{\\}}$
for $\varepsilon>0$), and let $f,g\in
C^{(1)}\big{(}\mathbb{T}^{N};V_{\tau}\big{)}$ have supports contained in $E$,
that is, $f(x)=0=g(x)$ for $x\notin E$. For fixed $f$, $g$, $i$ let

$\displaystyle
I_{n}=\int_{\mathbb{T}^{N}}\big{\\{}(x_{i}\mathcal{D}_{i}f(x))^{\ast}K_{n}(x)g(x)-f(x)^{\ast}K_{n}(x)x_{i}\mathcal{D}_{i}g(x)\big{\\}}\mathrm{d}m(x)$
$\displaystyle\hphantom{I_{n}}{}=\int_{\mathbb{T}^{N}}\big{\\{}(x_{i}\partial_{i}f(x))^{\ast}K_{n}(x)g(x)-f(x)^{\ast}K_{n}(x)x_{i}\mathcal{\partial}_{i}g(x)\big{\\}}$
$\displaystyle\hphantom{I_{n}=}{}+\kappa\int_{\mathbb{T}^{N}}\sum_{j\neq
i}\left(\tau(i,j)x_{i}\frac{f(x)-f(x(i,j))}{x_{i}-x_{j}}\right)^{\ast}K_{n}(x)g(x)\mathrm{d}m(x)$
$\displaystyle\hphantom{I_{n}=}{}-\kappa\int_{\mathbb{T}^{N}}f(x)^{\ast}K_{n}(x)\sum_{j\neq
i}\left(\tau(i,j)x_{i}\frac{g(x)-g(x(i,j))}{x_{i}-x_{j}}\right)\mathrm{d}m(x).$

By using
$\big{(}\frac{x_{i}}{x_{i}-x_{j}})^{\ast}=-\frac{x_{j}}{x_{i}-x_{j}}$,
$\tau(i,j)^{\ast}=\tau(i,j)$, and rearranging the sums we obtain

$\displaystyle
I_{n}=\int_{\mathbb{T}^{N}}\big{\\{}(x_{i}\partial_{i}f(x))^{\ast}K_{n}(x)g(x)-f(x)^{\ast}K_{n}(x)x_{i}\mathcal{\partial}_{i}g(x)\big{\\}}\mathrm{d}m(x)$
$\displaystyle\hphantom{I_{n}=}{}-\kappa\int_{\mathbb{T}^{N}}f(x)^{\ast}\sum_{j\neq
i}\left\\{\frac{x_{j}}{x_{i}-x_{j}}\tau(i,j)K_{n}(x)+K_{n}(x)\tau(i,j)\frac{x_{i}}{x_{i}-x_{j}}\right\\}g(x)\mathrm{d}m(x)$
$\displaystyle\hphantom{I_{n}=}{}+\kappa\sum_{j\neq
i}\int_{\mathbb{T}^{N}}\big{\\{}x_{j}f(x(i,j))^{\ast}\tau(i,j)K_{n}(x)g(x)+x_{i}f(x)^{\ast}K_{n}(x)\tau(i,j)g(x(i,j))\big{\\}}\frac{\mathrm{d}m(x)}{x_{i}-x_{j}}.$

Each integral in the third line is finite because $f$ and $g$ vanish on a
neighborhood of $\bigcup\limits_{1\leq i<j\leq N}\\{x\colon$ $x_{i}=x_{j}\\}$.
The terms inside $\\{\cdot\\}$ are invariant under the change of variable
$x\mapsto x(i,j)$, because $\tau(i,j)K_{n}(x(i,j))=K_{n}(x)\tau(i,j)$, but the
denominator $(x_{i}-x_{j})$ changes sign; thus the integrand is odd under
$(i,j)$ and the integral vanishes.

Because terms like $\tau(i,j)\frac{x_{i}}{x_{i}-x_{j}}g(x)$ are in
$C^{(1)}\big{(}\mathbb{T}^{N};V_{\tau}\big{)}$ (assumption on the support of
$g$) we can take the limit as $n\rightarrow\infty$ in the second line. By the
adjoint property of $x_{i}\partial_{i}$ we find

$\displaystyle\int_{\mathbb{T}^{N}}(x_{i}\partial_{i}f(x))^{\ast}K_{n}(x)g(x)\mathrm{d}m(x)=\int_{\mathbb{T}^{N}}f(x)^{\ast}x_{i}\mathcal{\partial}_{i}\\{K_{n}(x)g(x)\\}\mathrm{d}m(x)$
$\displaystyle\qquad{}=\int_{\mathbb{T}^{N}}f(x)^{\ast}\big{\\{}K_{n}(x)x_{i}\mathcal{\partial}_{i}g(x)+(x_{i}\partial_{i}K_{n}(x))g(x)\big{\\}}\mathrm{d}m(x).$

Thus the fact that $\lim\limits_{n\rightarrow\infty}I_{n}=0$ implies (recall
the matrix-valued integral notation (4.1))

$\displaystyle\lim_{n\rightarrow\infty}\int_{\mathbb{T}^{N}}f(x)^{\ast}(x_{i}\partial_{i}K_{n}(x))g(x)\mathrm{d}m(x)$
$\displaystyle\qquad{}=\kappa\int_{\mathbb{T}^{N}}f(x)^{\ast}\sum_{j\neq
i}\frac{1}{x_{i}-x_{j}}\big{\\{}x_{j}\tau(i,j)\mathrm{d}\mu(x)+\mathrm{d}\mu(x)\tau(i,j)x_{i}\big{\\}}g(x).$
(6.2)

This statement is valid for all $f,g\in
C^{(1)}\big{(}\mathbb{T}^{N};V_{\tau}\big{)}$ that vanish on a neighborhood of
$\bigcup\limits_{1\leq i<j\leq N}\\{x\colon$ $x_{i}=x_{j}\\}$. The first line
of the equation can be written as

$\displaystyle\lim_{n\rightarrow\infty}\int_{\mathbb{T}^{N}}\big{\\{}(x_{i}\partial_{i}f(x))^{\ast}K_{n}(x)g(x)-f(x)^{\ast}K_{n}(x)x_{i}\mathcal{\partial}_{i}g(x)\big{\\}}\mathrm{d}m(x)$
$\displaystyle\qquad{}=\int_{\mathbb{T}^{N}}\big{\\{}(x_{i}\partial_{i}f(x))^{\ast}\mathrm{d}\mu(x)g(x)-f(x)^{\ast}\mathrm{d}\mu(x)x_{i}\mathcal{\partial}_{i}g(x)\big{\\}},$

which expresses the distributional derivative of $\mathrm{d}\mu$. Thus the
distribution-sense differential system is satisfied by $\mathrm{d}\mu$ on
closed subsets of $\mathbb{T}_{\rm reg}^{N}$.

In the scalar case ($\tau=(N)$) the orthogonality weight is known to be (due
to [1] for the symmetric Jack polynomials)

$\displaystyle K(x)=\prod\limits_{1\leq i<j\leq
N}\big{\\{}(x_{i}-x_{j})\big{(}x_{i}^{-1}-x_{j}^{-1}\big{)}\big{\\}}^{\kappa}$

and the differential equation system reduces to (note $\tau(w)=1$)

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

The differential system (6.1) could be the subject for an article all by
itself, but we can sketch a result about the absolutely continuous part of
$\mu$, namely that $\mathrm{d}\mu(x)=L(x)^{\ast}BL(x)\mathrm{d}m(x)$,
$x\in\mathbb{T}_{\rm reg}^{N}$ where $B$ is a locally constant positive matrix
and $L(x)$ is a fundamental solution of

$\displaystyle\partial_{i}L(x)=\kappa L(x)\left\\{\sum_{j\neq
i}\frac{1}{x_{i}-x_{j}}\tau(i,j)-\frac{\gamma}{x_{i}}I\right\\},\qquad 1\leq
i\leq N,$ (6.3)
$\displaystyle\gamma:=\frac{1}{2N}\sum_{i=1}^{\ell(\tau)}\tau_{i}(\tau_{i}-2i+1)=\frac{1}{N}\sum_{j=1}^{N}c(j,T_{0}).$

The effect of the term $\frac{\gamma}{x_{i}}I$ is to make $L(x)$ homogeneous
of degree zero, that is, $\sum\limits_{i=1}^{N}x_{i}\partial_{i}L(x)=0$,
because

$\displaystyle\sum\limits_{1\leq i<j\leq
N}\tau(i,j)=\left\\{\sum\limits_{j=1}^{N}c(j,T_{0})\right\\}I$

(the sum of the contents in the diagram of $\tau$).

The differential system is Frobenius integrable; this means that in the system
$\partial_{i}L(x)=\kappa L(x)M_{i}(x)$, $1\leq i\leq N$, the two formal
differentiations

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

are equal to each other for all $i$, $j$ (see [2]). The system is analytic and
thus any local solution can be continued analytically to any point in
$\mathbb{C}_{\rm
reg}^{N}:=(\mathbb{C}\backslash\\{0\\})^{N}\backslash\bigcup\limits_{1\leq
i<j\leq N}\\{x\colon x_{i}=x_{j}\\}$. When restricted to $\mathbb{T}_{\rm
reg}^{N}$ there are solutions defined on each connected component. This is
possible because $L(x)$ is constant on $\\{ux\colon|u|=1\\}$ for fixed
$x\in\mathbb{T}_{\rm reg}^{N}$ and each component is homotopic to a circle.
Denote the component containing all the points
$\big{\\{}\big{(}e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N}}\big{)}\colon-\pi<\theta_{1}<\theta_{2}<\cdots<\theta_{N}<\pi\big{\\}}$
by $\mathcal{C}_{0}$. Since $x$ and $ux$ are in the same component for $|u|=1$
we see that $x_{0}\in\mathcal{C}_{0}$ (recall
$x_{0}=\big{(}1,\omega,\ldots,\omega^{N-1}\big{)}$,
$\omega=e^{2\pi\mathrm{i}/N})$. The components are $\\{\mathcal{C}_{0}w\colon
w\in\mathcal{S}_{N},\,w(1)=1\\}$ (corresponding to the $(N-1)!$ circular
permutations of $(1,2,\ldots,N)$). Now fix the unique solution $L(x)$ such
that $L(x_{0}w)=I$ for each $w$ with $w(1)=1$. By differentiating
$L(x)^{-1}L(x)=I$ obtain the system satisfied by $L^{-1}$:

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

The goal here is to replace $f$, $g$ in formula (6.2) by $L^{-1}f$, $L^{-1}g$
and deduce the desired result. By use of
$x_{i}\partial_{i}(L^{-1\ast})=-\big{(}x_{i}\partial_{i}L^{-1}\big{)}^{\ast}$
we obtain

$\displaystyle\big{(}x_{i}\partial_{i}\big{(}L(x)^{-1\ast}\big{)}\big{)}\mathrm{d}\mu(x)L(x)^{-1}+L(x)^{-1\ast}\mathrm{d}\mu(x)\big{(}x_{i}\partial_{i}L(x)^{-1}\big{)}$
$\displaystyle\qquad{}=\kappa L^{-1\ast}\bigg{\\{}\sum_{j\neq
i}\frac{-x_{j}}{x_{i}-x_{j}}\tau(i,j)-\gamma I\bigg{\\}}\mathrm{d}\mu
L^{-1}-\kappa L^{-1\ast}\mathrm{d}\mu\bigg{\\{}\sum_{j\neq
i}\frac{x_{i}}{x_{i}-x_{j}}\tau(i,j)-\gamma I\bigg{\\}}L^{-1}$
$\displaystyle\qquad{}=-\kappa L^{-1\ast}\sum_{j\neq
i}\left\\{\frac{x_{j}}{x_{i}-x_{j}}\tau(i,j)\mathrm{d}\mu+\mathrm{d}\mu\tau(i,j)\frac{x_{i}}{x_{i}-x_{j}}\right\\}L^{-1}.$

Substitute this relation in formula (6.2)

$\displaystyle\lim_{n\rightarrow\infty}\int_{\mathbb{T}^{N}}f(x)^{\ast}L(x)^{-1\ast}(x_{i}\partial_{i}K_{n}(x))L(x)^{-1}g(x)\mathrm{d}m(x)$
$\displaystyle\qquad{}+\int_{\mathbb{T}^{N}}f(x)^{\ast}(x_{i}\partial_{i})\big{(}L(x)^{-1\ast}\big{)}\mathrm{d}\mu(x)L(x)^{-1}g(x)$
$\displaystyle\qquad{}+\int_{\mathbb{T}^{N}}f(x)^{\ast}L(x)^{-1\ast}\mathrm{d}\mu(x)x_{i}\partial_{i}L(x)^{-1}g(x)=0.$

The formula is valid because $L^{-1}f,L^{-1}g\in C^{(1)}\big{(}\mathbb{T}_{\rm
reg}^{N};V_{\tau}\big{)}$ and vanish for $x\notin E$. The first line of the
equation is the distributional derivative of $\mu$ so the equation is
equivalent to

$\displaystyle\int_{\mathbb{T}^{N}}f(x)^{\ast}\big{[}x_{i}\partial_{i}\big{(}L(x)^{-1\ast}\mathrm{d}\mu(x)L(x)^{-1}\big{)}\big{]}g(x)=0,\qquad
1\leq i\leq N.$

Thus all the partial derivatives of the distribution $L^{-1\ast}\mathrm{d}\mu
L^{-1}$ vanish and $L^{-1\ast}\mathrm{d}\mu L^{-1}=B\mathrm{d}m$, where $B$ is
constant on each component of $\mathbb{T}_{\rm reg}^{N}$. We conclude
$\mathrm{d}\mu(x)=L(x)^{\ast}BL(x)\mathrm{d}m(x)$ for $x\in\mathbb{T}_{\rm
reg}^{N}$. Part (1) of Theorem 4.17 implies $B$ is a positive matrix.

We have to point out that this result provides no information about the
behavior of $\mu$ on the singular set $\bigcup\limits_{i<j}\\{x\colon
x_{i}=x_{j}\\}$. We conjecture that $\mu$ does not have a singular part. This
question seems a worthy topic for further investigations. Some of the
difficulties in this problem come from the behavior of the solutions of system
(6.3) in neighborhoods of the sets $\\{x\colon x_{i}=x_{j}\\}$; there are
singularities of order $|x_{i}-x_{j}|^{\pm\kappa}$. Both signs appear because
the eigenvalues of $\tau(i,j)$ are $1$ and $-1$; a consequence of the
assumption that $\tau$ is not one-dimensional.

##  Appendix A Appendix

###  A.1 The matrix Bochner theorem

For a matrix $A\in M_{n}(\mathbb{C})$ the operator norm is

$\displaystyle\left\|A\right\|:=\sup\\{|Av|\colon|v|=1\\}=\sup\\{|\langle
u,Av\rangle|\colon|u|=1=|v|\\}.$

######  Proposition A.1.

Suppose $F$ is a positive-definite $M_{n}(\mathbb{C})$-valued function on
$\mathbb{Z}^{N}$ then

  1. $(1)$

$F(\mathbf{0})$ is positive-definite,

  2. $(2)$

$F(-\alpha)=F(\alpha)^{\ast}$ and $\|F(\alpha)\|\leq\|F(\mathbf{0})\|$ for all
$\alpha\in\mathbb{Z}^{N}$.

###### Proof.

Part (1) follows immediately from taking $f(\mathbf{0})=u$, $f(\alpha)=0$ for
$\alpha\neq 0$. For part (2) fix $\alpha\neq 0$ and let $f(\mathbf{0})=u$,
$f(\alpha)=v$ and $f(\beta)=0$ otherwise. By definition

$\displaystyle
u^{\ast}F(\mathbf{0})u+v^{\ast}F(\mathbf{0})v+v^{\ast}F(\alpha)u+u^{\ast}F(-\alpha)v\geq
0.$

Thus $\operatorname{Im}(v^{\ast}F(\alpha)u+u^{\ast}F(-\alpha)v)=0$ for all
$u$, $v$. For $1\leq j,k\leq n$ let $u=\varepsilon_{j}$, $v=c\varepsilon_{k}$,
$c\in\mathbb{C}$, then

$\displaystyle
0=\operatorname{Im}\big{(}\overline{c}F(\alpha)_{kj}+cF(-\alpha)_{jk}\big{)}.$

Set $c=1$ and $c=\mathrm{i}$ to show
$F(-\alpha)_{jk}=\overline{F(\alpha)_{kj}}$, that is,
$F(-\alpha)=F(\alpha)^{\ast}$. Thus
$\overline{u^{\ast}F(-\alpha)v}=v^{\ast}F(\alpha)u$ and

$\displaystyle-2\operatorname{Re}(v^{\ast}F(\alpha)u)\leq
u^{\ast}F(\mathbf{0})u+v^{\ast}F(\mathbf{0})v.$

Specialize to vectors $u$, $v$ such that $|u|=1=|v|$ and
$v^{\ast}F(\alpha)u=\langle v,F(\alpha)u\rangle=-\|F(\alpha)\|$. Since
$F(\mathbf{0})$ is positive-definite it follows that
$u^{\ast}F(\mathbf{0})u+v^{\ast}F(\mathbf{0})v\leq 2\|F(\mathbf{0})\|$, and so
$\|F(\alpha)\|\leq\|F(\mathbf{0})\|$. ∎

######  Proposition A.2.

Suppose $\alpha,\beta\in\mathbb{Z}^{N}$ and $\alpha,\beta\neq\mathbf{0}$ then

$\displaystyle\|F(\alpha)-F(\beta)\|^{2}\leq
2\|F(\mathbf{0})\|\\{v^{\ast}F(\mathbf{0})v-\operatorname{Re}(v^{\ast}F(\alpha-\beta)v)\\},$

where $|v|=1$ and $|(F(\alpha)-F(\beta))^{\ast}v|=\|F(\alpha)-F(\beta)\|$.

###### Proof.

Assume $\alpha\neq\beta$ and let $f(\mathbf{0})=u$, $f(\alpha)=w$,
$f(\beta)=-w$, and $f(\gamma)=0$ otherwise. By definition

$\displaystyle
u^{\ast}F(\mathbf{0})u+2w^{\ast}F(\mathbf{0})w+w^{\ast}(F(\alpha)-F(\beta))u+u^{\ast}(F(-\alpha)-F(-\beta))w$
$\displaystyle\qquad{}-w^{\ast}F(\alpha-\beta)w-w^{\ast}F(\beta-\alpha)w\geq
0.$

Let $u,v\in\mathbb{C}^{n}$ satisfy $|u|=1=|v|$ and
$|(F(\alpha)-F(\beta))^{\ast}v|=v^{\ast}(F(\alpha)-F(\beta))u=\|F(\alpha)-F(\beta)\|$.
Set $w=tv$ with $t\in\mathbb{R}$. The inequality becomes

$\displaystyle
u^{\ast}F(\mathbf{0})u+2t^{2}v^{\ast}F(\mathbf{0}t)v+2t\|F(\alpha)-F(\beta)\|-2t^{2}\operatorname{Re}\big{(}v^{\ast}F(\alpha-\beta)v\big{)}\geq
0.$

The discriminant of the quadratic polynomial in $t$ must be nonpositive and
this implies the stated inequality, since
$u^{\ast}F(\mathbf{0})u\leq\|F(\mathbf{0})\|$. ∎

This is a sort of uniform continuity.

For any fixed $u\in\mathbb{C}^{n}$ the scalar function
$g_{u}(\alpha)=u^{\ast}F(\alpha)u$ is positive-definite in the classical sense
and thus by Bochner’s theorem (see [13, pp. 17–21]) there exists a unique
positive Baire measure $\mu_{u}$ on $\mathbb{T}^{N}$ such that

$\displaystyle\int_{\mathbb{T}^{N}}x^{-\alpha}\mathrm{d}\mu_{u}(x)=u^{\ast}F(\alpha)u,\qquad\alpha\in\mathbb{Z}^{N}.$

In particular, for $1\leq j\leq n$ let $u=\varepsilon_{j}$ then
$u^{\ast}F(\alpha)u=F(\alpha)_{jj}$ and set $\mu_{jj}=\mu_{\varepsilon_{j}}$.
For $1\leq j,k\leq n$ (with $j\neq k$) let
$u=\varepsilon_{j}+\varepsilon_{k}$,
$v=\varepsilon_{j}+\mathrm{i}\varepsilon_{k}$

$\displaystyle\int_{\mathbb{T}^{N}}x^{-\alpha}\mathrm{d}\mu_{u}(x)=F(\alpha)_{jj}+F(\alpha)_{jk}+F\left(\alpha\right)_{kj}+F(\alpha)_{kk},$
$\displaystyle\int_{\mathbb{T}^{N}}x^{-\alpha}\mathrm{d}\mu_{v}(x)=F(\alpha)_{jj}+\mathrm{i}F(\alpha)_{jk}-\mathrm{i}F(\alpha)_{kj}+F(\alpha)_{kk}.$

Thus

$\displaystyle\int_{\mathbb{T}^{N}}x^{-\alpha}\mathrm{d}(\mu_{u}-\mathrm{i}\mu_{v})=2F(\alpha)_{jk}+(1-\mathrm{i})\int_{\mathbb{T}^{N}}x^{-\alpha}\mathrm{d}(\mu_{jj}+\mu_{kk}),$

and define

$\displaystyle\mu_{jk}=\frac{1}{2}(\mu_{u}-\mathrm{i}\mu_{v})-\frac{1-\mathrm{i}}{2}(\mu_{jj}+\mu_{kk}),$

with the result

$\displaystyle\int_{\mathbb{T}^{N}}x^{-\alpha}\mathrm{d}\mu_{jk}(x)=F(\alpha)_{jk},\alpha\in\mathbb{Z}^{N}.$

From the above equations we also obtain

$\displaystyle\int_{\mathbb{T}^{N}}x^{-\alpha}\mathrm{d}(\mu_{u}+\mathrm{i}\mu_{v})=2F(\alpha)_{kj}+(1+\mathrm{i})\int_{\mathbb{T}^{N}}x^{-\alpha}\mathrm{d}(\mu_{jj}+\mu_{kk}).$

The formula

$\displaystyle\mu_{kj}=\frac{1}{2}(\mu_{u}+\mathrm{i}\mu_{v})-\frac{1+\mathrm{i}}{2}(\mu_{jj}+\mu_{kk}),$

is consistent with the previous one and it can be directly verified that
$\widehat{\mu_{kj}}(-\alpha)=\overline{\widehat{\mu_{jk}}(\alpha)}$ from the
general equation
$\widehat{\mu_{u}}(-\alpha)=\overline{\widehat{\mu_{u}}(\alpha)}$. This is a
restatement of $F(-\alpha)=F(\alpha)^{\ast}$.

From $\|\mu_{u}\|=u^{\ast}F(\mathbf{0})u$ (measure/total variation norm) we
obtain $\|\mu_{jj}\|=F(\mathbf{0})_{jj}$ and

$\displaystyle\|\mu_{jk}\|\leq\frac{1}{\sqrt{2}}(F(\mathbf{0})_{jj}+F(\mathbf{0})_{kk})+F(\mathbf{0})_{jj}+F(\mathbf{0})_{kk}+|\operatorname{Re}F(\mathbf{0})_{jk}|+|\operatorname{Im}F(\mathbf{0})_{jk}|.$

In particular, if $F(\mathbf{0})=I$ then $\|\mu_{jk}\|\leq 2+\sqrt{2}$. For
the space $C\big{(}\mathbb{T}^{N};\mathbb{C}^{n}\big{)}$ (continuous functions
on $\mathbb{T}^{N}$, values in $\mathbb{C}^{n}$) define an inner product

$\displaystyle\langle
f,g\rangle_{F}:=\sum_{i,j=1}^{n}\int_{\mathbb{T}^{N}}\overline{f(x)_{i}}g(x)_{j}\mathrm{d}\mu_{ij}(x),$

then for $\alpha,\beta\in\mathbb{Z}^{N}$, and $1\leq i,j\leq n$ (with the
standard unit basis vectors $\varepsilon_{i}$ of $\mathbb{C}^{n}$)

$\displaystyle\big{\langle}x^{\alpha}\varepsilon_{i},x^{\beta}\varepsilon_{j}\big{\rangle}_{F}=\int_{\mathbb{T}^{N}}x^{-\alpha}x^{\beta}\mathrm{d}\mu_{ij}(x)=F(\alpha-\beta)_{ij}.$

The following summarizes the above results.

######  Theorem A.3.

The Hermitian form $\langle\cdot,\cdot\rangle_{F}$ is bounded on
$C\big{(}\mathbb{T}^{N};\mathbb{C}^{n}\big{)}$, that is $|\langle
f,g\rangle_{F}|\leq B\|f\|_{\infty}\|g\|_{\infty}$ for some $B<\infty$, is
positive-semidefinite, and if $f$, $g$ are finitely supported functions on
$\mathbb{Z}^{N}$ with values in $\mathbb{C}^{n}$ then for
$\widehat{f}(x):=\sum_{\alpha}f(\alpha)x^{\alpha}$ and
$\widehat{g}(x):=\sum_{\alpha}g(\alpha)x^{\alpha}$

$\displaystyle\big{\langle}\widehat{f},\widehat{g}\big{\rangle}_{F}=\sum_{\alpha,\beta}f(\alpha)^{\ast}F(\alpha-\beta)g(\beta).$

###### Proof.

The bound follows from the uniform bound on $\|\mu_{jk}\|$ for all $j$, $k$,
depending only on $F(\mathbf{0})$. Suppose $\widehat{f}$, $\widehat{g}$ are
trigonometric (Laurent) polynomials (equivalent to finite support on
$\mathbb{Z}^{N}$), then

$\displaystyle\big{\langle}\widehat{f},\widehat{g}\big{\rangle}_{F}=\sum_{\alpha,\beta}\sum_{i,j=1}^{n}\int_{\mathbb{T}^{N}}\overline{f(\alpha)_{i}}x^{-\alpha}g(\beta)_{j}x^{\beta}\mathrm{d}\mu_{ij}(x)$
$\displaystyle\hphantom{\big{\langle}\widehat{f},\widehat{g}\big{\rangle}_{F}}{}=\sum_{\alpha,\beta}\sum_{i,j=1}^{n}\overline{f(\alpha)_{i}}F(\alpha-\beta)_{ij}g(\beta)_{j}=\sum_{\alpha,\beta}f(\alpha)^{\ast}F(\alpha-\beta)g(\beta).$

By definition $\big{\langle}\widehat{f},\widehat{f}\big{\rangle}_{F}\geq 0$
and by the density of the trigonometric polynomials in
$C\big{(}\mathbb{T}^{N};\mathbb{C}^{n}\big{)}$ it follows that
$\langle\cdot,\cdot\rangle_{F}$ is a positive semidefinite (it is possible
that $\big{\langle}\widehat{f},\widehat{f}\big{\rangle}_{F}=0$ for some $f\neq
0$) Hermitian form. ∎

The next result is used for the approximate identity arguments.

######  Theorem A.4.

Suppose $\sigma\in C\big{(}\mathbb{T}^{N}\big{)}$ and $\sigma(x)\geq 0$ for
all $x\in\mathbb{T}^{N}$ then each $\sigma\ast\mu_{ij}\in
C\big{(}\mathbb{T}^{N}\big{)}$ and $[\sigma\ast\mu_{ij}(x)]_{i,j=1}^{n}\in
M_{n}(\mathbb{C})$ is positive semidefinite for each $x\in\mathbb{T}^{N}$.

###### Proof.

Suppose $(f_{i})_{i=1}^{n}\in C\big{(}\mathbb{T}^{N};\mathbb{C}^{n}\big{)}$. A
convolution formula (similar to that in Lemma 4.13) shows that

$\displaystyle\sum_{i,j=1}^{n}\int_{\mathbb{T}^{N}}\overline{f(x)_{i}}\sigma\ast\mu_{ij}(x)f(x)_{j}\mathrm{d}m(x)=\sum_{i,j=1}^{n}\int_{\mathbb{T}^{N}}\int_{\mathbb{T}^{N}}\sigma\big{(}xy^{-1}\big{)}\overline{f(x)_{i}}f(x)_{j}\mathrm{d}\mu_{ij}(y)\mathrm{d}m(x)$
$\displaystyle\qquad{}=\int_{\mathbb{T}^{N}}\sigma(u)\int_{\mathbb{T}^{N}}\sum_{i,j=1}^{n}\overline{f(yu)_{i}}f(yu)_{j}\mathrm{d}\mu_{ij}(y)\mathrm{d}m(u)\geq
0,$

(change-of-variable $x=yu$) because $y\mapsto f(yu)\in
C\big{(}\mathbb{T}^{N};\mathbb{C}^{n}\big{)}$ and the double sum is continuous
in $u$ and nonnegative by the above theorem. Now let $f(x)=g(x)v$ where
$v\in\mathbb{C}^{n}$ and $g\in C\big{(}\mathbb{T}^{N}\big{)}$, $g\geq 0$,
$\int_{\mathbb{T}^{N}}g^{2}\mathrm{d}m=1$ and $g=0$ off an
$\varepsilon$-neighborhood of a fixed $z\in\mathbb{T}^{N}$ (i.e.,
$|x-z|\geq\varepsilon$ implies $g(x)=0$) then

$\displaystyle\sum_{i,j=1}^{n}\int_{\mathbb{T}^{N}}\overline{f(x)_{i}}\sigma\ast\mu_{ij}(x)f(x)_{j}\mathrm{d}m(x)=\int_{\mathbb{T}^{N}}g(x)^{2}\sum_{i,j=1}^{n}\overline{v_{i}}(\sigma\ast\mu_{ij}(x))v_{j}\mathrm{d}m(x)\geq
0.$

Let $\varepsilon\rightarrow 0$ then the integral tends to
$\sum\limits_{i,j=1}^{n}\overline{v_{i}}(\sigma\ast\mu_{ij}(z))v_{j}$, and
this completes the proof. ∎

###  A.2 Some results for the index set

For $n=1,2,3,\ldots$

$\displaystyle\\#\boldsymbol{Z}_{N,n}=\sum_{j=1}^{N-1}\binom{N}{j}\binom{n-1}{j-1}\binom{N-j+n-1}{n};$
(A.1)

in each $j$-subset (there are $\binom{N}{j}$) of $[1,2,\ldots,N]$ take
$j$-tuples $\alpha_{i_{1}},\ldots,\alpha_{i_{j}}$ with
$\sum\limits_{\ell=1}^{j}\alpha_{i_{\ell}}=n$ and each $\alpha_{i_{\ell}}\geq
1$ ($\binom{n-1}{j-1}$ possibilities) and in the complement take
$(N-j)$-tuples with $\sum\limits_{\ell=1}^{N-j}\alpha_{i_{\ell}}=-n$ and each
$\alpha_{i_{\ell}}\leq 0$ ($\binom{N-j+n-1}{n}$ possibilities). For example
when $n\geq 1$

$\displaystyle\\#\boldsymbol{Z}_{2,n}=2,\quad\\#\boldsymbol{Z}_{3,n}=6n,\quad\\#\boldsymbol{Z}_{4,n}=10n^{2}+2,\quad\\#\boldsymbol{Z}_{5,n}=\frac{5n}{3}\big{(}7n^{2}+5\big{)},\quad\ldots.$

######  Proposition A.5.

Among $\gamma\in\boldsymbol{Z}_{N,n}$ such that $\gamma^{\nu}=\beta$ for some
fixed $\beta$ with $|\beta|=n$ and such that $\beta_{j}>0$ exactly when $j>k$
the minimal multi-index for the order
$\gamma^{(1)\pi}\vartriangleright\gamma^{(2)\pi}$ is
$\gamma^{(0)}=\big{(}p+1,\ldots,\overset{(m)}{p+1},p,\ldots,\overset{(k)}{p},-\beta_{k+1},\ldots,-\beta_{N}\big{)}$
where $p=\big{\lfloor}\frac{n}{k}\big{\rfloor}$ and $m=n-kp$ $($so $0\leq
m<k)$.

###### Proof.

The claim is that $\big{(}\gamma_{i}^{(0)}\big{)}_{i=1}^{k}$ is
$\prec$-minimal among partitions $\alpha$ of length $\leq k$ and $|\alpha|=n$.
Argue by induction on the length. The statement is obviously true when $k=1$.
Suppose it is true for $k$ and let $\alpha$ be a partition of length $\leq
k+1$. Let

$\displaystyle n_{1}=\sum_{i=1}^{k}\alpha_{i},\qquad
n_{2}=n_{1}+\alpha_{k+1},\qquad
p_{1}=\left\lfloor\frac{n_{1}}{k}\right\rfloor,\qquad
p_{2}=\left\lfloor\frac{n_{2}}{k+1}\right\rfloor,$ $\displaystyle
m_{1}=n_{1}-kp_{1},\qquad m_{2}=n_{2}-(k+1)p_{2}.$

Define $\gamma^{(1)}$ and $\gamma^{(2)}$ analogously to the above
($\gamma_{i}^{(s)}=p_{s}+1$ for $1\leq i\leq m_{s}$ and
$\gamma_{i}^{(1)}=p_{1}$ for $m_{1}<i\leq k$, $\gamma^{(2)}=p_{2}$ for
$m_{2}<i\leq k+1$). By the inductive hypothesis
$\sum\limits_{j=1}^{i}\alpha_{j}\geq\sum\limits_{j=1}^{i}\gamma_{j}^{(1)}$.
This implies $\alpha_{k+1}\leq\alpha_{k}\leq\gamma_{k}^{(1)}$. Thus
$(k+1)p_{2}\leq n_{1}+\alpha_{k+1}\leq m_{1}+kp_{1}+\alpha_{k+1}\leq
m_{1}+(k+1)p_{1}$ and $p_{2}\leq\frac{m_{1}}{k+1}+p_{1}$. Since $p_{1}$,
$p_{2}$ are integers this implies $p_{2}\leq p_{1}$. If $p_{1}=p_{2}$ then
$m_{2}=m_{1}-(p_{1}-\alpha_{k+1})$, and clearly
$\sum\limits_{j=1}^{i}\gamma_{j}^{(1)}\geq\sum\limits_{j=1}^{i}\gamma_{j}^{(2)}$
for $1\leq i\leq k$. If $p_{2}<p_{1}$ then $\gamma_{j}^{(2)}\leq p_{2}+1\leq
p_{1}\leq\gamma_{j}^{(1)}$ for $1\leq j\leq k$. Thus
$\alpha\succeq\gamma^{(2)}$. ∎

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