Populations of Solutions to Cyclotomic Bethe Equations

Populations of Solutions
to Cyclotomic Bethe Equations

Alexander VARCHENKO † and Charles A.S. YOUNG ‡

A. Varchenko and C.A.S. Young

† Department of Mathematics, University of North Carolina at Chapel Hill,
† Chapel Hill, NC 27599-3250, USA<EMAIL_ADDRESS>
‡ School of Physics, Astronomy and Mathematics, University of Hertfordshire,
‡ College Lane, Hatfield AL10 9AB, UK<EMAIL_ADDRESS>
Received June 17, 2014, in final form November 05, 2015; Published online
November 14, 2015

We study solutions of the Bethe Ansatz equations for the cyclotomic Gaudin
model of [Vicedo B., Young C.A.S., arXiv:1409.6937]. We give two
interpretations of such solutions: as critical points of a cyclotomic master
function, and as critical points with cyclotomic symmetry of a certain
“extended” master function. In finite types, this yields a correspondence
between the Bethe eigenvectors and eigenvalues of the cyclotomic Gaudin model
and those of an “extended” non-cyclotomic Gaudin model. We proceed to define
_populations_ of solutions to the cyclotomic Bethe equations, in the sense of
[Mukhin E., Varchenko A., Commun. Contemp. Math. 6 (2004), 111–163,
math.QA/0209017], for diagram automorphisms of Kac–Moody Lie algebras. In the
case of type A with the diagram automorphism, we associate to each population
a vector space of quasi-polynomials with specified ramification conditions.
This vector space is equipped with a ${\mathbb{Z}}_{2}$-gradation and a non-
degenerate bilinear form which is (skew-)symmetric on the even (resp. odd)
graded subspace. We show that the population of cyclotomic critical points is
isomorphic to the variety of isotropic full flags in this space.

Bethe equations; cyclotomic symmetry 82B23; 32S22; 17B81; 81R12

##  1 Introduction

Let ${\mathfrak{g}}$ be a complex Kac–Moody Lie algebra and
$\sigma\colon{\mathfrak{g}}\to{\mathfrak{g}}$ an automorphism of order
$M\in{\mathbb{Z}}_{\geq 1}$. Let $\omega\in{\mathbb{C}}^{\times}$ be a
primitive $M$th root of unity. We may choose a Cartan subalgebra
${\mathfrak{h}}\subset{\mathfrak{g}}$ such that
$\sigma({\mathfrak{h}})={\mathfrak{h}}$. We have the canonical pairing
$\langle\cdot,\cdot\rangle\colon{\mathfrak{h}}^{*}\otimes{\mathfrak{h}}\to{\mathbb{C}}$,
and the simple roots $\alpha_{i}\in{\mathfrak{h}}^{*}$ and coroots
$\alpha^{\vee}_{i}\in{\mathfrak{h}}$, where $i$ runs over the set $I$ of nodes
of the Dynkin diagram.

Consider the following system of equations in $m\in{\mathbb{Z}}_{\geq 0}$
variables $\bm{t}=(t_{1},\dots,t_{m})\in{\mathbb{C}}^{m}$ and labels
$\bm{\mathsf{c}}=({\mathsf{c}}(1),\dots,{\mathsf{c}}(m))\in I^{m}$:

$\displaystyle
0=\sum_{k=0}^{M-1}\sum_{i=1}^{N}\frac{\big{\langle}\sigma^{k}\Lambda_{i},\alpha^{\vee}_{{\mathsf{c}}(j)}\big{\rangle}}{t_{j}-\omega^{k}z_{i}}-\sum_{k=0}^{M-1}\sum_{\begin{subarray}{c}i=1\\\
i\neq
j\end{subarray}}^{m}\frac{\big{\langle}\sigma^{k}\alpha_{{\mathsf{c}}(i)},\alpha^{\vee}_{{\mathsf{c}}(j)}\big{\rangle}}{t_{j}-\omega^{r}t_{i}}$
$\displaystyle\hphantom{0=}{}+\frac{1}{t_{j}}\left(\big{\langle}\Lambda_{0},\alpha^{\vee}_{{\mathsf{c}}(j)}\big{\rangle}-\sum_{k=1}^{M-1}\frac{\big{\langle}\sigma^{k}\alpha_{{\mathsf{c}}(j)},\alpha_{{\mathsf{c}}(j)}^{\vee}\big{\rangle}}{1-\omega^{k}}\right),\qquad
j=1,\dots,m,$ (1.1)

where $\Lambda_{0},\Lambda_{1},\dots,\Lambda_{N}\in{\mathfrak{h}}^{*}$ are
weights (with $\sigma\Lambda_{0}=\Lambda_{0}$) and $z_{1},\dots,z_{N}$ are
non-zero points in the complex plane whose orbits, under the action of the
cyclic group $\omega^{\mathbb{Z}}$, are pairwise disjoint.

When $\sigma=\operatorname{id}$, $\omega=1$ and $\Lambda_{0}=0$, these
equations reduce to the following well-known set of equations in mathematical
physics:

$\displaystyle
0=\sum_{i=0}^{N}\frac{\big{\langle}\Lambda_{i},\alpha^{\vee}_{{\mathsf{c}}(j)}\big{\rangle}}{t_{j}-z_{i}}-\sum_{\begin{subarray}{c}i=1\\\
i\neq
j\end{subarray}}^{m}\frac{\big{\langle}\alpha_{{\mathsf{c}}(i)},\alpha^{\vee}_{{\mathsf{c}}(j)}\big{\rangle}}{t_{j}-t_{i}},\qquad
j=1,\dots,m.$ (1.2)

These are the equations for critical points of the _master functions_ [19]
which appear in the integral expressions for hypergeometric solutions to the
Knizhnik–Zamolodchikov (KZ) equations. They are also the _Bethe equations_ of
the quantum Gaudin model [1, 5, 18].

The equations (1.1) were introduced (for simple ${\mathfrak{g}}$) in the study
of cyclotomic generalizations of the Gaudin model [26, 27] – see also [3, 21,
22] – as we recall in Section 3 below. Let us call them the _cyclotomic Bethe
equations_. (Cyclotomic generalizations of the KZ equations were studied in
[2, 4], and appear in, in particular, the representation theory of cyclotomic
Hecke algebras [23].)

It is natural to ask whether the cyclotomic Bethe equations (1.1) can be
interpreted as the equations for critical points of some master function. In
the present paper we begin by giving two different such interpretations.
First, they are indeed the critical point equations for a  _cyclotomic master
function_ , which we write down in (2.2). But they are also the equations for
critical points with cyclotomic – more precisely
$S_{m}\ltimes({\mathbb{Z}}/M{\mathbb{Z}})^{m}$ – symmetry of what we call an
_extended master function_ , (2.8).

Recall that a master function is specified by a _weighted arrangement of
hyperplanes_ : that is, by a finite collection ${\mathcal{C}}$ of affine
hyperplanes in a complex affine space of finite dimension, together with an
assignment of a number $a(H)\in{\mathbb{C}}$ to each hyperplane
$H\in{\mathcal{C}}$. Indeed, for each $H\in{\mathcal{C}}$, let $\ell_{H}=0$ be
an affine equation for $H$; then the master function is
$\Phi=\sum\limits_{H\in{\mathcal{C}}}a(H)\log\ell_{H}$.

The cyclotomic master function corresponds to a hyperplane arrangement in
${\mathbb{C}}^{m}$ whose hyperplanes include $t_{i}=\omega^{k}t_{j}$, $1\leq
i<j\leq m$, for each $k\in{\mathbb{Z}}/M{\mathbb{Z}}$. By contrast, the
extended master function corresponds to a hyperplane arrangement in
${\mathbb{C}}^{mM}$, but has only those hyperplanes corresponding to the type
$A$ root system, i.e., $t_{i}=t_{j}$, $1\leq i<j\leq mM$, etc. Because the
extended master function is a master function of this standard form, its
critical point equations are the Bethe equations for a certain standard (i.e.,
non-cyclotomic) Gaudin model, which we call the _extended Gaudin model_. This
observation leads to our first result: a correspondence between the spectrum
of the cyclotomic Gaudin model and a “cyclotomic” part of the spectrum of the
extended Gaudin model. See Theorem 3.5.

Solutions to the Bethe equations (1.2) form families called _populations_.
Populations were first introduced in [15, 20], where a  _generation procedure_
was given which produces families of new solutions to the Bethe equations
starting from a given solution. A population is then defined to be the Zariski
closure of the set of all solutions to the Bethe equations obtained by
repeated application of this generation procedure, starting from a given
solution. It is known that if ${\mathfrak{g}}$ is simple then every population
is isomorphic to the flag variety of the Langlands dual Lie algebra
${}^{L}{\mathfrak{g}}$. This was shown in [15] for types $A$, $B$, $C$ and in
all finite types in [6, 16]. (A population can also be understood as the
variety of Miura opers with a given underlying oper; see [6, 16].)

In the present work our main goal is to initiate the study of _cyclotomic
populations_ : populations of solutions to the equations (1.1).

We formulate in Section 4 a definition of cyclotomic populations for
${\mathfrak{g}}$ a general Kac–Moody Lie algebra and $\sigma$ any diagram
automorphism of ${\mathfrak{g}}$ satisfying the _linking condition_. (We also
place certain restrictions on the weight $\Lambda_{0}$; see Section 4.1.) The
linking condition [7] states that, for every node $i\in I$, the restriction of
the Dynkin diagram to the orbit $\sigma^{\mathbb{Z}}(i)$ consists either of
disconnected nodes (in which case $i$ has _linking number $L_{i}=1$_), or of a
number of disconnected copies of the ${\mathrm{A}}_{2}$ Dynkin diagram (in
which case $i$ has _linking number $L_{i}=2$_). What the linking condition
ensures is that it is possible to “fold” the Dynkin diagram by the
automorphism $\sigma$. See Section 2.3 and [7].

In Section 4 we define the cyclotomic population to be the Zariski closure of
the set of all cyclotomic critical points obtained by repeated application of
a certain “cyclotomic generation procedure”, starting from a given cyclotomic
critical point. So the key ingredient is this generation procedure. Let us
describe it, in outline. There is an “elementary cyclotomic generation” step
associated to each orbit $\sigma^{\mathbb{Z}}(i)$. There are two cases:
$L_{i}=1$ and $L_{i}=2$.

First, suppose $i\in I$ is a node with linking number $L_{i}=1$. A critical
point $(\bm{t},\bm{\mathsf{c}})$ is represented by a tuple of polynomials,
$\bm{y}=(y_{i}(x))_{i\in I}$, where the roots of the polynomial $y_{i}(x)$,
$i\in I$, are the Bethe variables $t_{s}$ of “colour” $i$, i.e., those such
that ${\mathsf{c}}(s)=i$. Following [15], one defines a function of $x$,

$\displaystyle
y_{i}^{(i)}(x;c):=y_{i}(x)\int^{x}\xi^{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}T_{i}(\xi)\prod_{j\in
I}y_{j}(\xi)^{-\langle\alpha_{j},\alpha^{\vee}_{i}\rangle}d\xi+cy_{i}(x),$
(1.3)

depending on a parameter $c\in{\mathbb{C}}$ . Here $T_{i}(x)$, $i\in I$, are
certain functions encoding the “frame” data, i.e., the points
$z_{1},\dots,z_{N}$ and the weights $\Lambda_{1},\dots,\Lambda_{N}$; see
(4.5). The Bethe equations ensure that $y_{i}^{(i)}(x;c)$ is in fact a
polynomial, and moreover that if we consider the new tuple $\bm{y}^{(i)}(c)$
in which $y_{i}(x)$ is replaced by $y_{i}^{(i)}(x;c)$, then for almost all
values of $c$ this new tuple again represents a solution to the Bethe
equations. Call the replacement $\bm{y}\mapsto\bm{y}^{(i)}(c)$ _elementary
generation in direction $i$_. Now suppose the initial tuple $\bm{y}$
represents a cyclotomic point. That means

$\displaystyle y_{\sigma j}(\omega x)\simeq y_{j}(x),\qquad j\in I;$

see Lemma 4.5. Since the orbit $\sigma^{\mathbb{Z}}(i)$ consists of
disconnected nodes of the Dynkin diagram, the operations of elementary
generation in the directions $\sigma^{\mathbb{Z}}(i)$ commute. By performing
each of them once, in any order, we can arrange to arrive at a new cyclotomic
point. See Theorem 4.6.

Next, suppose $i\in I$ is a node with linking number $L_{i}=2$. Then for every
copy of the ${\mathrm{A}}_{2}$ diagram, with nodes say $j$ and
${\bar{\jmath}}$, one must perform the sequence of generation steps $j$,
${\bar{\jmath}}$, $j$. Doing this for each copy of ${\mathrm{A}}_{2}$ in turn,
in any order, we can arrange to arrive at a new cyclotomic point. See Theorem
4.20.

When $L_{i}=2$ there is a subtlety coming from our assumptions about the
weight at the origin, $\Lambda_{0}$. Throughout Section 4, motivated by [26],
we assume that $\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle$ is non-integral
when $L_{i}=2$. That means that the expression (1.3) develops a branch point
at the origin. The upshot is that at certain intermediate steps, the weight at
the origin is shifted to ${\mathsf{s}}_{i}\cdot\Lambda_{0}$, before eventually
being shifted back to $\Lambda_{0}$. See Proposition 4.10 and compare [17].

In either case, $L_{i}=1$ or $L_{i}=2$, we write $\bm{y}^{(i,\sigma)}(c)$ for
the tuple of polynomials representing the new cyclotomic critical point. It
depends on a single parameter $c$. The replacement
$\bm{y}\mapsto\bm{y}^{(i,\sigma)}(c)$ is the elementary cyclotomic generation,
in the direction of the orbit $\sigma^{\mathbb{Z}}(i)$.

To a critical point $(\bm{t},\bm{\mathsf{c}})$ represented by a tuple of
polynomials $\bm{y}$ one can associate a weight $\Lambda_{\infty}$. See (2.4)
and (4.10). For fixed $\Lambda_{0},\Lambda_{1},\dots,\Lambda_{N}$, we may
regard $\Lambda_{\infty}$ as encoding the number of roots $t_{s}$ of each
“colour” $i\in I$, i.e., the degrees of the polynomials $y_{i}(x)$. It is
known that $\Lambda_{\infty}(\bm{y}^{(i)}(c))$ is equal either to
$\Lambda_{\infty}(\bm{y})$ or to
${\mathsf{s}}_{i}\cdot\Lambda_{\infty}(\bm{y})$, where
${\mathsf{s}}_{i}\cdot{}$ denotes the shifted action of the Weyl reflection in
root $\alpha_{i}$. See [15]. We have an analogous statement in the cyclotomic
case. Namely, there is a “folded” Weyl group $W^{\sigma}$ with generators
${\mathsf{s}}_{i}^{\sigma}$. See Section 2.3. And we show that
$\Lambda_{\infty}(\bm{y}^{(i,\sigma)}(c))$ is equal either to
$\Lambda_{\infty}(\bm{y})$ or to
${\mathsf{s}}_{i}^{\sigma}\cdot\Lambda_{\infty}(\bm{y})$. For the precise
statement see Theorems 4.6 and 4.20.

We proceed in Section 5 to treat in detail the case of type A with the diagram
automorphism.

Recall first from [15] the structure of populations in type $A_{R}$,
$R\in{\mathbb{Z}}_{\geq 1}$, for the master functions associated to marked
points $z_{1},\dots,z_{N}$ and integral dominant weights
$\Lambda_{1},\dots,\Lambda_{N}$. In that setting, every population of critical
points is isomorphic to a variety of full flags in a certain
$(R+1)$-dimensional vector space ${\mathcal{K}}$ of polynomials. The
ramification points of ${\mathcal{K}}$ are $z_{1},\dots,z_{N}$ and $\infty$,
and the ramification data at these points are specified by the weights
$\Lambda_{1},\dots,\Lambda_{N}$ and an integral dominant weight
$\tilde{\Lambda}_{\infty}$. Given a full flag ${\mathcal{F}}=\\{0=F_{0}\subset
F_{1}\subset F_{2}\subset\dots\subset F_{R+1}={\mathcal{K}}\\}$ in
${\mathcal{K}}$, pick any basis $(u_{i}(x))_{i=1}^{R+1}$ of polynomials
adjusted to this flag, i.e., such that
$F_{k}=\operatorname{span}_{\mathbb{C}}(u_{1}(x),\dots,u_{k}(x))$. Then define
a tuple of functions $\bm{y}^{\mathcal{F}}=(y^{\mathcal{F}}_{k}(x))_{k=1}^{R}$
by

$\displaystyle
y^{\mathcal{F}}_{k}(x)=\operatorname{Wr}(u_{1}(x),\dots,u_{k}(x))/\big{(}T_{1}^{k-1}(x)T_{2}^{k-2}(x)\dots
T_{k-1}(x)\big{)},$

where – as in (1.3) above – the $(T_{i}(x))_{i=1}^{R}$ are functions encoding
the “frame” data $z_{1},\dots,z_{N}$ and $\Lambda_{1},\dots,\Lambda_{N}$, and
where $\operatorname{Wr}(u_{1}(x),\dots,u_{k}(x))$ denotes the Wronskian
determinant. The ramification properties of ${\mathcal{K}}$ ensure that the
$y^{\mathcal{F}}_{k}(x)$ are in fact polynomials. Moreover the map
${\mathcal{F}}\mapsto\bm{y}^{\mathcal{F}}$ is an isomorphism of varieties from
the variety of full flags in ${\mathcal{K}}$ to the population associated with
${\mathcal{K}}$. The space ${\mathcal{K}}$ is the kernel of a certain linear
differential operator ${\mathcal{D}}$ of order $R+1$ (essentially a type $A$
oper). This operator ${\mathcal{D}}$ can be defined in terms of the
$(T_{i}(x))_{i=1}^{R}$ together with the polynomials $(y_{i}(x))_{i=1}^{R}$ of
(any) point in the population. (See Section 5.4.)

Now let us discuss how the picture changes in our present setting. For us, the
weight at the origin $\Lambda_{0}$ need not be integer dominant. We assume it
satisfies weaker assumptions given in (5.1). These assumptions mean that we
are led to consider vector spaces ${\mathcal{K}}$ of _quasi-_ polynomials:
that is, polynomials in $x^{\frac{1}{2}}$. The local behaviour of these quasi-
polynomials near the origin is encoded in $\Lambda_{0}$. The remaining
ramification points are $z_{1},\dots,z_{N}$, $-z_{1},\dots,-z_{N}$, and
$\infty$. See Definition 5.2.

The space of quasi-polynomials ${\mathcal{K}}$ admits a natural
${\mathbb{Z}}_{2}$ gradation
${\mathcal{K}}={\mathcal{K}}_{\textup{O}}\oplus{\mathcal{K}}_{\textup{Sp}}$.
We call flags which respect this gradation _decomposable_. Decomposable full
flags are classified by their _type_ ; see Section 5.3. In particular the
flags ${\mathcal{F}}\in F\\!L_{S}({\mathcal{K}})$ of a certain preferred type
$S$, (5.9), are sent to polynomials under the map
${\mathcal{F}}\mapsto\bm{y}^{\mathcal{F}}$. This map of varieties
$F\\!L_{S}({\mathcal{K}})\to{\mathbb{P}}({\mathbb{C}}[x])^{R}$ is an
isomorphism onto its image. The _cyclotomic population_ is then the set of
cyclotomic tuples in this image, i.e., the set of tuples
$\bm{y}^{\mathcal{F}}$, ${\mathcal{F}}\in F\\!L_{S}({\mathcal{K}})$, such that
$y_{i}(x)\simeq y_{R+1-i}(-x)$, $i=1,\dots,R$. The question is: which flags in
$F\\!L_{S}({\mathcal{K}})$ map to cyclotomic tuples?

To answer this question we introduce the notion of a _cyclotomically self-dual
space of quasi-polynomials_. The space ${\mathcal{K}}$ has a natural dual
space ${\mathcal{K}}^{\dagger}$ of quasi-polynomials – see Section 5.5 – and
we say ${\mathcal{K}}$ is cyclotomically self-dual if for all
$v(x)\in{\mathcal{K}}$, $v(-x)\in{\mathcal{K}}^{\dagger}$. (Compare the very
similar notion of a self-dual space of polynomials in [15].) We show that a
sufficient condition for ${\mathcal{K}}$ to be cyclotomically self-dual is
that there exists at least one full flag ${\mathcal{F}}$ in ${\mathcal{K}}$
such that $\bm{y}^{\mathcal{F}}$ is cyclotomic (Theorem 5.14). If
${\mathcal{K}}$ is cyclotomically self-dual then it admits a canonical non-
degenerate bilinear form $B$. We show that, for all full flags ${\mathcal{F}}$
in ${\mathcal{K}}$, the tuple $\bm{y}^{\mathcal{F}}$ is cyclotomic if and only
if ${\mathcal{F}}$ is isotropic with respect to $B$ (Theorem 5.17).

Therefore the cyclotomic population is isomorphic to the variety
$F\\!L_{S}^{\perp}({\mathcal{K}})$ of isotropic flags of type $S$ in
${\mathcal{K}}$. The bilinear form $B$ is symmetric on
${\mathcal{K}}_{\textup{O}}$ and skew-symmetric on
${\mathcal{K}}_{\textup{Sp}}$, and these subspaces are mutually orthogonal
with respect to $B$ (Theorem 5.23). Hence this variety
$F\\!L_{S}^{\perp}({\mathcal{K}})$ is isomorphic to the direct product of
spaces of isotropic flags $F\\!L^{\perp}({\mathcal{K}}_{\textup{Sp}})\times
F\\!L^{\perp}({\mathcal{K}}_{\textup{O}})$.

##  2 Master functions and cyclotomic symmetry

###  2.1 Kac–Moody algebras

Let $I$ be a finite set of indices and $A=(a_{i,j})_{i,j\in I}$ a generalized
Cartan matrix, i.e., $a_{i,i}=2$ and $a_{i,j}\in{\mathbb{Z}}_{\leq 0}$
whenever $i\neq j$, with $a_{i,j}=0$ if and only if $a_{j,i}=0$. Let
${\mathfrak{g}}:={\mathfrak{g}}(A)$ be the corresponding complex Kac–Moody Lie
algebra [11, Section 1], ${\mathfrak{h}}\subset{\mathfrak{g}}$ a Cartan
subalgebra, and

$\displaystyle{\mathfrak{g}}={\mathfrak{n}}_{-}\oplus{\mathfrak{h}}\oplus{\mathfrak{n}}_{+}$

a triangular decomposition. Let $\alpha_{i}\in{\mathfrak{h}}^{*}$,
$\alpha^{\vee}_{i}\in{\mathfrak{h}}$, $i\in I$ be collections of simple roots
and coroots respectively. We have $\dim{\mathfrak{h}}=|I|+\dim\ker
A=2|I|-\operatorname{rank}A$. By definition,

$\displaystyle\langle\alpha_{i},\alpha^{\vee}_{j}\rangle=a_{j,i},$

where
$\langle\cdot,\cdot\rangle\colon{\mathfrak{h}}^{*}\otimes{\mathfrak{h}}\to{\mathbb{C}}$
is the canonical pairing.

We assume that $A$ is symmetrizable, i.e., there exists a diagonal matrix
$D=\operatorname{diag}(d_{i})_{i\in I}$, whose entries are coprime positive
integers, such that the matrix $B=DA$ is symmetric. Let $(\cdot,\cdot)$ be the
associated symmetric bilinear form on ${\mathfrak{h}}^{*}$. We have
$(\alpha_{i},\alpha_{j})=d_{i}a_{i,j}$ and

$\displaystyle\langle\lambda,\alpha^{\vee}_{i}\rangle=2(\lambda,\alpha_{i})/(\alpha_{i},\alpha_{i})\qquad\text{for
all $\lambda\in{\mathfrak{h}}^{*}$}.$

The form $(\cdot,\cdot)$ is non-degenerate. Therefore it gives an
identification ${\mathfrak{h}}\cong_{\mathbb{C}}{\mathfrak{h}}^{*}$ and hence
a non-degenerate symmetric bilinear form on ${\mathfrak{h}}$ which we also
write as $(\cdot,\cdot)$.

Let
${\mathcal{P}}:=\\{\lambda\in{\mathfrak{h}}^{*}\colon\langle\lambda,\alpha^{\vee}_{i}\rangle\in{\mathbb{Z}}\\}$
be the integral weight lattice and
${\mathcal{P}}_{+}:=\\{\lambda\in{\mathfrak{h}}^{*}\colon\langle\lambda,\alpha^{\vee}_{i}\rangle\in{\mathbb{Z}}_{\geq
0}\\}$ the set of dominant integral weights.

Let $W\subset\operatorname{End}({\mathfrak{h}}^{*})$ be the Weyl group. It is
generated by the reflections ${\mathsf{s}}_{i}$, $i\in I$, given by

$\displaystyle{\mathsf{s}}_{i}(\lambda):=\lambda-\langle\lambda,\alpha^{\vee}_{i}\rangle\alpha_{i},\qquad\lambda\in{\mathfrak{h}}^{*}.$

Let $\rho\in{\mathfrak{h}}^{*}$ be a vector such that
$\langle\rho,\alpha^{\vee}_{i}\rangle=1$ for $i\in I$. We use $\cdot$ to
denote the _shifted action_ of the Weyl group, i.e.,

$\displaystyle{\mathsf{s}}\cdot\lambda:=w(\lambda+\rho)-\rho,\qquad{\mathsf{s}}\in
W,\quad\lambda\in{\mathfrak{h}}^{*}.$

###  2.2 Diagram automorphism

Suppose $\sigma$ is an automorphism of the Dynkin diagram [11, Section 4.7] of
$A$. That is, $\sigma$ is a permutation of the index set $I$ such that

$\displaystyle a_{\sigma i,\sigma j}=a_{i,j}.$

Let $M$ be the order of $\sigma$ and let $\omega\in{\mathbb{C}}^{\times}$ be a
primitive $M$th root of unity.

To such a permutation is associated a _diagram automorphism_
${\mathfrak{g}}\to{\mathfrak{g}}$ of the Kac–Moody Lie algebra [7], which we
shall also write as $\sigma$. We have

$\displaystyle\sigma E_{i}=E_{\sigma i},\qquad\sigma F_{i}=F_{\sigma
i},\qquad\sigma\alpha^{\vee}_{i}=\alpha^{\vee}_{\sigma i},\qquad i\in I,$

where $E_{i}\in{\mathfrak{n}}$, $F_{i}\in{\mathfrak{n}}^{-}$, $i\in I$, are a
set of Chevalley generators of $[{\mathfrak{g}},{\mathfrak{g}}]$. This defines
$\sigma$ on the derived subalgebra $[{\mathfrak{g}},{\mathfrak{g}}]$ of
${\mathfrak{g}}$. For the action of $\sigma$ on the derivations, i.e., on a
complement of $[{\mathfrak{g}},{\mathfrak{g}}]$ in ${\mathfrak{g}}$, see [7,
Section 3.2]. This action may be chosen to ensure that
$\sigma\colon{\mathfrak{g}}\to{\mathfrak{g}}$ has order $M$ and respects the
bilinear form $(\cdot,\cdot)$ on ${\mathfrak{h}}$:

$\displaystyle(\sigma X,\sigma Y)=(X,Y)\qquad\text{for all}\quad
X,Y\in{\mathfrak{h}}.$

The action of $\sigma$ on ${\mathfrak{h}}^{*}$ is defined by
$\sigma\lambda:=\lambda\circ\sigma^{-1}$ so that $\langle\sigma\lambda,\sigma
X\rangle=\langle\lambda,X\rangle$ for all $\lambda\in{\mathfrak{h}}^{*}$,
$X\in{\mathfrak{h}}$. Note that then $\sigma\alpha_{i}=\alpha_{\sigma i}$ for
all $i\in I$.

Let ${\mathfrak{g}}^{\sigma}\subset{\mathfrak{g}}$ be the Lie subalgebra of
elements invariant under $\sigma$. We have

$\displaystyle{\mathfrak{g}}^{\sigma}={\mathfrak{n}}_{-}^{\sigma}\oplus{\mathfrak{h}}^{\sigma}\oplus{\mathfrak{n}}_{+}^{\sigma}$

with
${\mathfrak{n}}_{\pm}^{\sigma}={\mathfrak{g}}^{\sigma}\cap{\mathfrak{n}}_{\pm}$
and ${\mathfrak{h}}^{\sigma}={\mathfrak{g}}^{\sigma}\cap{\mathfrak{h}}$.

###  2.3 The linking condition and the folded diagram

For any $i\in I$ let

$\displaystyle M_{i}:=\big{|}\big{\\{}i,\sigma
i,\sigma^{2}i,\dots,\sigma^{M-1}i\big{\\}}\big{|}$

be the length of the orbit of the node $i$ under the automorphism $\sigma$ of
the Dynkin diagram $A$. Define

$\displaystyle L_{i}:=1-\sum_{k=1}^{M_{i}-1}a_{\sigma^{k}i,i}.$

Note that $L_{i}\geq 1$. Following [7], we say that $\sigma$ obeys the
_linking condition_ if and only if

$\displaystyle L_{i}\leq 2\qquad\text{for all}\quad i\in I.$ (2.1)

To understand the meaning of this condition, consider the restriction of the
Dynkin diagram to the orbit of the node $i$. If $L_{i}=1$ then this induced
subgraph has no edges at all. If $L_{i}=2$ then it consists of $M_{i}/2$
disconnected copies of the type ${\mathrm{A}}_{2}$ Dynkin diagram.

######  Remark 2.1.

If $A$ is of finite type, then all diagram automorphisms obey the linking
condition. Moreover, in all finite types except ${\mathrm{A}}_{2n}$,
$n\in{\mathbb{Z}}_{\geq 1}$, we in fact have $L_{i}=1$ for every node $i$:
that is, no two distinct nodes in the same $\sigma$-orbit are ever linked by
an edge of the Dynkin diagram. In type ${\mathrm{A}}_{2n}$ the non-trivial
diagram automorphism gives $L_{i}=2$ for $i\in\\{n,n+1\\}$ and $L_{i}=1$
otherwise:

$n-1$$\phantom{1}n\phantom{1}$$n+1$$n+2$$2n$$1$

######  Remark 2.2.

If $A$ is of affine type then all diagram automorphisms obey the linking
condition with the following exception. In type $\mathrm{A}_{n}^{(1)}$,
$n\in{\mathbb{Z}}_{\geq 2}$, let $R$ be a generator of the cyclic subgroup
$C_{n+1}$ of the full automorphism group of the Dynkin diagram (which is the
dihedral group $D_{n+1}$). Then $R$ does not obey the linking condition.
Indeed, the $R$-orbit of any node $i$ is the whole diagram, and $L_{i}=1+n$.

Given any diagram automorphism satisfying the linking condition it is possible
to define a  _folded_ Dynkin diagram. Let us make a choice of subset

$\displaystyle I_{\sigma}\subseteq I$

consisting of exactly one representative of each $\sigma$-orbit. Then the
Cartan matrix $A^{\sigma}=(a^{\sigma}_{i,j})_{i,j\in I_{\sigma}}$ of the
folded diagram is given by

$\displaystyle a^{\sigma}_{i,j}=L_{i}\sum_{k=0}^{M_{i}-1}a_{\sigma^{k}i,j}.$

######  Remark 2.3.

Compare Section 3.3 of [7], noting that our convention
$a_{j,i}=\langle\alpha_{i},\alpha^{\vee}_{j}\rangle$ differs from that of [7].

######  Lemma 2.4 ([7]).

If $\sigma$ obeys the linking condition then $A^{\sigma}$ $($and its
transpose$)$ is a symmetrizable Cartan matrix whose type $($finite, affine, or
indefinite$)$ is the same as that of $A$.

For each $i\in I_{\sigma}$ let us define also

$\displaystyle\alpha^{\vee,\sigma}_{i}:=L_{i}\sum_{k=0}^{M_{i}-1}\alpha^{\vee}_{\sigma^{k}i}\qquad\text{and}\qquad
E^{\sigma}_{i}:=\sum_{k=0}^{M_{i}-1}E_{\sigma i},\qquad
F^{\sigma}_{i}:=L_{i}\sum_{k=0}^{M_{i}-1}F_{\sigma i}.$

Then we have

$\displaystyle\left[E^{\sigma}_{i},F^{\sigma}_{j}\right]=\delta_{i,j}\alpha^{\vee,\sigma}_{i},\qquad\left[\alpha^{\vee,\sigma}_{i},E^{\sigma}_{j}\right]=E^{\sigma}_{j}a^{\sigma}_{j,i},\qquad\left[\alpha^{\vee,\sigma}_{i},F^{\sigma}_{j}\right]=-F^{\sigma}_{j}a^{\sigma}_{j,i}\qquad
i,j\in I_{\sigma}.$

Thus $\alpha^{\vee,\sigma}_{i}$, $E^{\sigma}_{i}$, $F^{\sigma}_{i}$, $i\in
I_{\sigma}$ generate a copy of (the derived subalgebra of) the Kac–Moody Lie
algebra ${\mathfrak{g}}({A^{\sigma}})$ inside
${\mathfrak{g}}^{\sigma}:=\\{X\in{\mathfrak{g}}\colon\sigma X=X\\}$. Next, for
all $i\in I_{\sigma}$, if we let

$\displaystyle\alpha^{\sigma}_{i}:=\frac{L_{i}}{M_{i}}\sum_{k=0}^{M_{i}-1}\alpha_{\sigma^{k}i}\in{\mathfrak{h}}^{*}$

then
$\langle\alpha^{\sigma}_{i},\alpha^{\vee,\sigma}_{j}\rangle=a^{\sigma}_{j,i}$.
Define $W^{\sigma}$ to be the group generated by the elements
${\mathsf{s}}^{\sigma}_{i}\in\operatorname{End}({\mathfrak{h}}^{*})$ given by

$\displaystyle{\mathsf{s}}^{\sigma}_{i}(\lambda):=\lambda-\langle\lambda,\alpha^{\vee,\sigma}_{i}\rangle\alpha^{\sigma}_{i},\qquad
i\in I_{\sigma}.$

######  Lemma 2.5.

$W^{\sigma}$ is a subgroup of $W$. Indeed, we have

$\displaystyle{\mathsf{s}}^{\sigma}_{i}=\begin{cases}\displaystyle\prod_{k=0}^{M_{i}-1}{\mathsf{s}}_{\sigma^{k}i},&L_{i}=1,\vspace{1mm}\\\
\displaystyle\left(\prod_{k=0}^{M_{i}/2-1}{\mathsf{s}}_{\sigma^{k}i}\right)\left(\prod_{k=0}^{M_{i}/2-1}{\mathsf{s}}_{\sigma^{k+M_{i}/2}i}\right)\left(\prod_{k=0}^{M_{i}/2-1}{\mathsf{s}}_{\sigma^{k}i}\right),&L_{i}=2.\end{cases}$

###  2.4 The cyclotomic master function

Let $\bm{\Lambda}=(\Lambda_{i})_{i=1}^{N}$ be a collection of
$N\in{\mathbb{Z}}_{\geq 0}$ integral dominant weights
$\Lambda_{i}\in{\mathcal{P}}_{+}$. Let $\bm{z}=(z_{i})_{i=1}^{N}$ be a
collection of nonzero points $z_{i}\in{\mathbb{C}}^{\times}$ such that
$\omega^{\mathbb{Z}}z_{i}\cap\omega^{\mathbb{Z}}z_{j}=\varnothing$ whenever
$i\neq j$. We shall call $\Lambda_{i}$ _the weight at $z_{i}$_.

In addition, we pick a weight $\Lambda_{0}\in{\mathfrak{h}}^{\sigma,*}$. We
call $\Lambda_{0}$ _the weight at the origin_.

Let $\bm{\mathsf{c}}=({\mathsf{c}}(j))_{j=1}^{m}$ be an $m$-tuple of elements
of $I$, and introduce variables $\bm{t}=(t_{j})_{j=1}^{m}$. We shall say that
$t_{j}$ is a _variable of colour_ ${\mathsf{c}}(j)$.

We define the _cyclotomic master function_
$\Phi=\Phi_{{\mathfrak{g}},\sigma}(\bm{t};\bm{\mathsf{c}};\bm{z};\bm{\Lambda},\Lambda_{0})$
associated to these data to be

$\displaystyle\Phi:=\sum_{i=1}^{N}\left(\frac{1}{2}\sum_{k=1}^{M-1}\big{(}\Lambda_{i},\sigma^{k}\Lambda_{i}\big{)}+(\Lambda_{i},\Lambda_{0})\right)\log
z_{i}+\sum_{k=0}^{M-1}\sum_{1\leq i<j\leq
n}\big{(}\Lambda_{i},\sigma^{k}\Lambda_{j}\big{)}\log\big{(}z_{i}-\omega^{k}z_{j}\big{)}$
$\displaystyle\hphantom{\Phi:=}{}-\sum_{k=0}^{M-1}\sum_{i=1}^{N}\sum_{j=1}^{m}\big{(}\alpha_{{\mathsf{c}}(j)},\sigma^{k}\Lambda_{i}\big{)}\log\big{(}t_{j}-\omega^{k}z_{i}\big{)}$
$\displaystyle\hphantom{\Phi:=}{}+\sum_{k=0}^{M-1}\sum_{1\leq i<j\leq
m}\big{(}\alpha_{{\mathsf{c}}(i)},\sigma^{k}\alpha_{\mathsf{c}}(j)\big{)}\log\big{(}t_{i}-\omega^{k}t_{j}\big{)}$
$\displaystyle\hphantom{\Phi:=}{}+\sum_{i=1}^{m}\left(\frac{1}{2}\sum_{k=1}^{M-1}\big{(}\alpha_{{\mathsf{c}}(i)},\sigma^{k}\alpha_{{\mathsf{c}}(i)}\big{)}-(\alpha_{{\mathsf{c}}(i)},\Lambda_{0})\right)\log
t_{i}.$ (2.2)

A point $\bm{t}$ with complex coordinates is called a _critical point_ of the
cyclotomic master function if

$\displaystyle\frac{\partial\Phi}{\partial t_{i}}=0,\qquad i=1,\dots,m,$

or equivalently (in view of Lemma 2.6 below) if the following equations are
satisfied:

$\displaystyle
0=\sum_{k=0}^{M-1}\sum_{i=1}^{N}\frac{\big{(}\alpha_{{\mathsf{c}}(j)},\sigma^{k}\Lambda_{i}\big{)}}{t_{j}-\omega^{k}z_{i}}-\sum_{k=0}^{M-1}\sum_{\begin{subarray}{c}i=1\\\
i\neq
j\end{subarray}}^{m}\frac{\big{(}\alpha_{{\mathsf{c}}(j)},\sigma^{k}\alpha_{{\mathsf{c}}(i)}\big{)}}{t_{j}-\omega^{r}t_{i}}$
$\displaystyle\hphantom{0=}{}+\frac{1}{t_{j}}\left(-\sum_{k=1}^{M-1}\frac{\big{(}\alpha_{{\mathsf{c}}(j)},\sigma^{k}\alpha_{{\mathsf{c}}(j)}\big{)}}{1-\omega^{k}}+(\alpha_{{\mathsf{c}}(j)},\Lambda_{0})\right)$
(2.3)

for $j=1,\dots,m$. Call this system of equations the _cyclotomic Bethe
equations_.

######  Lemma 2.6.

For any $\lambda\in{\mathfrak{h}}^{*}$,

$\displaystyle\sum_{k=1}^{M-1}\frac{(\lambda,\sigma^{k}\lambda)}{1-\omega^{k}}=\frac{1}{2}\sum_{k=1}^{M-1}\left(\frac{(\lambda,\sigma^{k}\lambda)}{1-\omega^{k}}+\frac{(\lambda,\sigma^{k}\lambda)}{1-\omega^{-k}}\right)=\frac{1}{2}\sum_{k=1}^{M-1}(\lambda,\sigma^{k}\lambda).$

Define $\Lambda_{\infty}$, _the weight at infinity_ , to be

$\displaystyle\Lambda_{\infty}:=\Lambda_{0}+\sum_{k=0}^{M-1}\sum_{i=1}^{N}\Lambda_{\sigma^{k}(i)}-\sum_{k=0}^{M-1}\sum_{i=1}^{m}\alpha_{\sigma^{k}{\mathsf{c}}(i)}.$
(2.4)

The group $S_{m}$ acts on pairs of $m$-tuples $(\bm{t},\bm{\mathsf{c}})$ by
permuting indices:

$\displaystyle\rho.(\bm{t},\bm{\mathsf{c}})=\big{(}\big{(}t_{\rho^{-1}(1)},\dots,t_{\rho^{-1}(m)}\big{)},\big{(}{\mathsf{c}}\big{(}\rho^{-1}(1)\big{)},\dots,{\mathsf{c}}\big{(}\rho^{-1}(m)\big{)}\big{)}\big{)}.$

The group ${\mathbb{Z}}/M{\mathbb{Z}}$ acts on pairs
$(t,{\mathsf{c}})\in{\mathbb{C}}\times I$ by
$k.(t,{\mathsf{c}})=(\omega^{k}t_{i},\sigma^{k}{\mathsf{c}})$. This gives rise
to an action of the wreath product
$S_{m}\wr({\mathbb{Z}}/M{\mathbb{Z}}):=S_{m}\ltimes({\mathbb{Z}}/M{\mathbb{Z}})^{m}$
on pairs of tuples $(\bm{t},\bm{\mathsf{c}})\in{\mathbb{C}}^{m}\times I^{m}$.

######  Lemma 2.7.

Up to an additive constant, the cyclotomic master function $\Phi$ is invariant
under the pull-back of this action of $S_{m}\wr({\mathbb{Z}}/M{\mathbb{Z}})$.
In particular, if $\bm{t}$ is a critical point of
$\Phi(\bm{t};\bm{\mathsf{c}})$ then $X.\bm{t}$ is a critical point of
$\Phi(X.\bm{t};X.\bm{\mathsf{c}})$, for all $X\in
S_{m}\wr({\mathbb{Z}}/M{\mathbb{Z}})$.

###  2.5 The extended master function

The equations (2.3) admit another, closely related, interpretation. Recall the
definition of the (usual) master function [19]. Namely, let
$\tilde{\bm{\Lambda}}=(\tilde{\Lambda}_{i})_{i=0}^{\tilde{N}}$ be a collection
of $\tilde{N}+1\in{\mathbb{Z}}_{\geq 0}$ weights
$\tilde{\Lambda}_{i}\in{\mathfrak{h}}^{*}$, and let
$\tilde{\bm{z}}=(\tilde{z}_{i})_{i=0}^{\tilde{N}}$ be a collection of nonzero
points $\tilde{z}_{i}\in{\mathbb{C}}^{\times}$. Pick
$\tilde{m}\in{\mathbb{Z}}_{\geq 0}$, let
$\bm{\mathsf{c}}=({\mathsf{c}}(j))_{j=1}^{\tilde{m}}$ be an $\tilde{m}$-tuple
of elements of $I$ and introduce variables $\bm{t}=(t_{j})_{j=1}^{\tilde{m}}$.
The _master function_ associated to these data is

$\displaystyle{\widetilde{\Phi}}:=\sum_{0\leq
i<j\leq\tilde{N}}\big{(}\tilde{\Lambda}_{i},\tilde{\Lambda}_{j}\big{)}\log(\tilde{z}_{i}-\tilde{z}_{j})-\sum_{i=0}^{\tilde{N}}\sum_{j=1}^{\tilde{m}}\big{(}\alpha_{{\mathsf{c}}(j)},\tilde{\Lambda}_{i}\big{)}\log(t_{j}-\tilde{z}_{i})$
$\displaystyle\hphantom{{\widetilde{\Phi}}:=}{}+\sum_{1\leq
i<j\leq\tilde{m}}\big{(}\alpha_{{\mathsf{c}}(i)},\alpha_{{\mathsf{c}}(j)}\big{)}\log(t_{i}-t_{j}).$
(2.5)

It is a function of the variables $\bm{t}$, depending on the parameters
$\bm{\mathsf{c}}$, $\tilde{\bm{z}}$ and $\tilde{\bm{\Lambda}}$. The critical
points of the master function are those points $\bm{t}$ with complex
coordinates such that $\partial{\widetilde{\Phi}}/\partial t_{j}=0$ for
$j=1,\dots,\tilde{m}$, i.e., those points such that the following equations
are satisfied:

$\displaystyle
0=\sum_{i=0}^{\tilde{N}}\frac{(\alpha_{{\mathsf{c}}(j)},\tilde{\Lambda}_{i})}{t_{j}-\tilde{z}_{i}}-\sum_{\begin{subarray}{c}i=1\\\
i\neq
j\end{subarray}}^{\tilde{m}}\frac{(\alpha_{{\mathsf{c}}(j)},\alpha_{{\mathsf{c}}(i)})}{t_{j}-t_{i}},\qquad
j=1,\dots,\tilde{m}.$ (2.6)

In this paper we are concerned with the following special case. Let
$\tilde{N}=NM$, choose $(\tilde{z}_{i})_{i=0}^{NM}$ to be

$\displaystyle\tilde{z}_{0}=0,\qquad\tilde{z}_{k+Mi}=\omega^{k}z_{i},\qquad
k=0,1,\dots,M-1,\quad i=1,\dots,N,$ (2.7a) and choose the weights at these
points to be
$\displaystyle\tilde{\Lambda}_{0}=\Lambda_{0},\qquad\tilde{\Lambda}_{k+Mi}=\sigma^{k}\Lambda_{i},$
(2.7b)

where $z_{i}$, $\Lambda_{i}$, $i=1,\dots,N$, and $\Lambda_{0}$ are as in
Section 2.4. We call the master function in this case the _extended master
function_ ,
$\widehat{\Phi}=\widehat{\Phi}_{{\mathfrak{g}},\sigma}(\bm{t};\bm{\mathsf{c}};\bm{z};\bm{\Lambda};\Lambda_{0})$.
It is given by

$\displaystyle\widehat{\Phi}:=\sum_{k=0}^{M-1}\sum_{i=1}^{N}\big{(}\Lambda_{0},\sigma^{k}\Lambda_{i}\big{)}\log\big{(}{-}\omega^{k}z_{i}\big{)}+\sum_{k,\,l=0}^{M-1}\sum_{1\leq
i<j\leq
n}\big{(}\sigma^{k}\Lambda_{i},\sigma^{l}\Lambda_{j}\big{)}\log\big{(}\omega^{k}z_{i}-\omega^{l}z_{j}\big{)}$
$\displaystyle\hphantom{\widehat{\Phi}:=}{}+\sum_{0\leq k<l\leq
T-1}\sum_{i=1}^{N}\big{(}\sigma^{k}\Lambda_{i},\sigma^{l}\Lambda_{i}\big{)}\log\big{(}\omega^{k}-\omega^{l}\big{)}z_{i}-\sum_{j=1}^{\tilde{m}}(\alpha_{{\mathsf{c}}(j)},\Lambda_{0})\log(t_{j})$
$\displaystyle\hphantom{\widehat{\Phi}:=}{}-\sum_{k=0}^{M-1}\sum_{i=1}^{N}\sum_{j=1}^{\tilde{m}}\big{(}\alpha_{{\mathsf{c}}(j)},\omega^{k}\Lambda_{i}\big{)}\log\big{(}t_{j}-\omega^{k}z_{i}\big{)}+\sum_{1\leq
i<j\leq\tilde{m}}(\alpha_{{\mathsf{c}}(i)},\alpha_{{\mathsf{c}}(j)})\log(t_{i}-t_{j}),$
(2.8)

and the critical point equations (2.6) take the form

$\displaystyle
0=\sum_{k=0}^{M-1}\sum_{i=1}^{N}\frac{\big{(}\alpha_{{\mathsf{c}}(j)},\sigma^{k}\Lambda_{i}\big{)}}{t_{j}-\omega^{k}z_{i}}+\frac{(\alpha_{{\mathsf{c}}(j)},\Lambda_{0})}{t_{j}}-\sum_{\begin{subarray}{c}i=1\\\
i\neq
j\end{subarray}}^{\tilde{m}}\frac{(\alpha_{{\mathsf{c}}(j)},\alpha_{{\mathsf{c}}(i)})}{t_{j}-t_{i}},\qquad
j=1,\dots,\tilde{m}.$ (2.9)

The group $S_{\tilde{m}}$ acts on pairs of $\tilde{m}$-tuples
$(\bm{t},\bm{\mathsf{c}})$ by permuting indices:

$\displaystyle\rho.(\bm{t},\bm{\mathsf{c}})=\big{(}\big{(}t_{\rho^{-1}(1)},\dots,t_{\rho^{-1}(\tilde{m})}\big{)},\big{(}{\mathsf{c}}\big{(}\rho^{-1}(1)\big{)},\dots,{\mathsf{c}}\big{(}\rho^{-1}(\tilde{m})\big{)}\big{)}\big{)}.$
(2.10)

######  Lemma 2.8.

Any master function of the form (2.5) is invariant under the pull-back of this
action of $S_{\tilde{m}}$. In particular the extended master function (2.8) is
invariant.

Let us call a point $(\bm{t},\bm{\mathsf{c}})\in{\mathbb{C}}^{\tilde{m}}\times
I^{\tilde{m}}$ a _cyclotomic point_ if we have $\tilde{m}=Mm$ for some
$m\in{\mathbb{Z}}_{\geq 0}$ and, by acting with some permutation in
$S_{\tilde{m}}$, we can arrange that

$\displaystyle
t_{i+mk}=\omega^{k}t_{i}\qquad{\mathsf{c}}(i+mk)=\sigma^{k}{\mathsf{c}}(i),\qquad
i=1,\dots,m,\quad k=0,\dots,M-1.$ (2.11)

######  Lemma 2.9.

This point $(t_{i})_{i=1}^{\tilde{m}}$ is a critical point of the extended
master function if and only if $(t_{i})_{i=1}^{m}$ is a critical point of the
cyclotomic master function, i.e., $(t_{i})_{i=1}^{m}$ obeys (2.3).

###### Proof.

Given (2.11), the equation (2.9) for $t_{j}$ is nothing but the corresponding
equation in (2.3) and the equation for $t_{j+km}$, $k=1,\dots,M-1$, is
actually the same equation up to an overall factor of $\omega^{-k}$. (To see
this one must use the compatibility of $\sigma$ with the inner product:
$(\sigma x,y)=(x,\sigma^{-1}y)$.) ∎

Thus, the cyclotomic Bethe equations (2.3) are also the equations for
cyclotomic critical points of the extended master function.

##  3 Gaudin models and the Bethe ansatz equations

Our first result, Theorem 3.5, concerns the relationship between critical
points and the eigenvalues of Gaudin Hamiltonians. Suppose, for this section
only, that the Cartan matrix is of finite type, i.e., that ${\mathfrak{g}}$ is
semisimple, and that $\sigma$ is an automorphism of ${\mathfrak{g}}$ of order
$M>1$. Recall [8, 9] that the _quadratic Gaudin Hamiltonians_ are the
following $\tilde{N}+1$ elements of
$U({\mathfrak{g}})^{\otimes(\tilde{N}+1)}$:

$\displaystyle\tilde{\mathcal{H}}^{(i)}:=\sum_{\begin{subarray}{c}j=0\\\ j\neq
i\end{subarray}}^{\tilde{N}}\sum_{a=1}^{\dim{\mathfrak{g}}}\frac{I^{a(i)}I_{a}^{(j)}}{\tilde{z}_{i}-\tilde{z}_{j}},\qquad
i=0,1,\dots,\tilde{N},$

where $I_{a}$, $a=1,\dots,\dim{\mathfrak{g}}$, is a basis of ${\mathfrak{g}}$,
$I^{a}$ is the dual basis with respect to the non-degenerate invariant
bilinear form
$(\cdot,\cdot)\colon{\mathfrak{g}}\times{\mathfrak{g}}\to{\mathbb{C}}$, and we
write ${X}^{(i)}$ for $X$ acting in the $i$th tensor factor. (For convenience
we number these factors starting from $0$.)

For $\Lambda\in{\mathfrak{h}}^{*}$, let $M_{\Lambda}$ denote the Verma module
over ${\mathfrak{g}}$ with highest weight $\Lambda$,
$M_{\Lambda}:=\operatorname{Ind}_{{\mathfrak{h}}\oplus{\mathfrak{n}}_{+}}^{\mathfrak{g}}{\mathbb{C}}\mathsf{v}_{\Lambda}$.
Let us represent the $\tilde{\mathcal{H}}^{(i)}$ as linear maps in
$\operatorname{End}\big{(}\bigotimes_{i=0}^{\tilde{N}}M_{\tilde{\Lambda}_{i}}\big{)}$.
Then the following can be shown using the techniques of the Bethe Ansatz.

######  Theorem 3.1 ([1, 18]).

To any critical point $\bm{t}$ of the master function ${\widetilde{\Phi}}$,
i.e., to any solution to the equations (2.6), there corresponds a simultaneous
eigenvector $\tilde{\psi}_{\bm{t}}$ of the linear operators
$\tilde{\mathcal{H}}^{(i)}\in\operatorname{End}\big{(}\bigotimes_{i=0}^{\tilde{N}}M_{\tilde{\Lambda}_{i}}\big{)}$.
For each $i=0,\dots,\tilde{N}$ the eigenvalue of $\tilde{\mathcal{H}}^{(i)}$
on $\tilde{\psi}_{\bm{t}}$ is

$\displaystyle\tilde{E}^{(i)}:=\frac{\partial{\widetilde{\Phi}}}{\partial\tilde{z}_{i}}=\sum_{\begin{subarray}{c}j=0\\\
j\neq
i\end{subarray}}^{\tilde{N}}\frac{(\tilde{\Lambda}_{i},\tilde{\Lambda}_{j})}{\tilde{z}_{i}-\tilde{z}_{j}}-\sum_{j=1}^{\tilde{m}}\frac{(\Lambda_{i},\alpha_{{\mathsf{c}}(j)})}{\tilde{z}_{i}-t_{j}}.$
(3.1)

The eigenvector $\tilde{\psi}_{\bm{t}}$ is given explicitly by

$\displaystyle\tilde{\psi}_{\bm{t}}=\sum_{\bm{n}\in
P_{\tilde{m},\tilde{N}+1}}\bigotimes_{i=0}^{\tilde{N}}\frac{F_{c(n^{i}_{1})}F_{c(n^{i}_{2})}\cdots
F_{c(n^{i}_{p_{i}-1})}F_{c(n^{i}_{p_{i}})}\mathsf{v}_{\tilde{\Lambda}_{i}}}{\big{(}w_{n^{i}_{1}}-w_{n^{i}_{2}}\big{)}\cdots\big{(}w_{n^{i}_{p_{i}-1}}-w_{n^{i}_{p_{i}}}\big{)}\big{(}w_{n^{i}_{p_{i}}}-z_{i}\big{)}},$
(3.2)

where the sum $\bm{n}\in P_{\tilde{m},\tilde{N}+1}$ is over ordered partitions
of the labels $\\{1,\dots,\tilde{m}\\}$ into $\tilde{N}+1$ parts.

(The fact that this simultaneous eigenvector is nonzero is proved for
${\mathfrak{g}}={\mathfrak{sl}}_{n}$ nondegenerate critical points in [14],
for ${\mathfrak{g}}={\mathfrak{sl}}_{n}$ isolated critical points in [13], and
for semisimple ${\mathfrak{g}}$ and isolated critical points in [25]. See also
[24].)

In [26]111In [26] $\sigma\colon{\mathfrak{g}}\to{\mathfrak{g}}$ is allowed to
be any automorphism commuting with the Cartan involution, not necessarily a
diagram involution. A posteriori the Bethe equations and energy eigenvalues
depend on the inner part of $\sigma$ only through the definition of
$\Lambda_{0}$, (3.5)., B. Vicedo and one of the present authors defined
_cyclotomic Gaudin Hamiltonians_. The quadratic cyclotomic Gaudin Hamiltonians
are the elements of $U({\mathfrak{g}})^{\otimes N}$ given by

$\displaystyle\mathcal{H}_{i}:=\sum_{p=0}^{M-1}\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{N}\sum_{a=1}^{\dim{\mathfrak{g}}}\frac{I^{a(i)}\sigma^{p}I_{a}^{(j)}}{z_{i}-\omega^{-p}z_{j}}+\frac{1}{z_{i}}\sum_{p=1}^{M-1}\sum_{a=1}^{\dim{\mathfrak{g}}}\frac{I^{a(i)}\sigma^{p}I_{a}^{(i)}}{(1-\omega^{p})},\qquad
i=1,\ldots,n.$ (3.3)

######  Remark 3.2.

These Hamiltonians can be understood in a number of ways. Physically, one
thinks of them as describing the dynamics of a “long-range spin chain” in
which the “spin” at $z_{i}$ interacts not only directly with the other spins
at the points $z_{j}$, $j\neq i$, but also with their images under rotations
of the spectral plane [3]. At the level of the Lax matrix, this corresponds to
replacing the usual rational skew-symmetric solution to the classical
Yang–Baxter equation, $r(u,v)=I^{a}\otimes I_{a}/(u-v)$, by a certain _non_
-skew-symmetric solution – see [21, 22] and discussion in [26]. The motivation
for such models comes in part from physics, where in certain important cases
the Lax matrix has cyclotomic symmetry in the spectral variable [12, 28].

Let us assign to the point $z_{i}$ the Verma module $M_{\Lambda_{i}}$,
$\Lambda_{i}\in{\mathfrak{h}}^{*}$. In other words, let us represent the
Hamiltonians (3.3) as linear maps

$\displaystyle\mathcal{H}^{(i)}\in\operatorname{End}\left(\bigotimes_{i=1}^{N}M_{\Lambda_{i}}\right),\qquad
i=1,\dots,N.$ (3.4)

Let (in this section, Section 3) $\Lambda_{0}\in{\mathfrak{h}}^{\sigma,*}$ be
the weight given by

$\displaystyle\Lambda_{0}(h):=\sum_{r=1}^{M-1}\frac{\operatorname{tr}_{\mathfrak{n}}(\sigma^{-r}\operatorname{ad}_{h})}{1-\omega^{r}}.$
(3.5)

######  Theorem 3.3 ([26]).

To any critical point of the cyclotomic master function, i.e., to any solution
$\bm{t}$ to the equations (2.3), there corresponds a simultaneous eigenvector
$\psi_{\bm{t}}$ of the linear operators $\mathcal{H}^{(i)}$, $i=1,\dots,N$.
The eigenvalue of $\mathcal{H}^{(i)}$ on $\psi_{\bm{t}}$ is

$\displaystyle E^{(i)}:=\frac{\partial\Phi}{\partial
z_{i}}=\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{N}\sum_{s=0}^{M-1}\frac{(\Lambda_{i},\sigma^{s}\Lambda_{j})}{z_{i}-\omega^{s}z_{j}}-\sum_{j=1}^{m}\sum_{s=0}^{M-1}\frac{(\Lambda_{i},\sigma^{s}\alpha_{{\mathsf{c}}(j)})}{z_{i}-\omega^{s}t_{j}}$
$\displaystyle\hphantom{E^{(i)}:=}{}+\frac{1}{z_{i}}\left((\Lambda_{i},\Lambda_{0})+\sum_{s=1}^{M-1}\frac{(\Lambda_{i},\sigma^{s}\Lambda_{i})}{1-\omega^{s}}\right).$
(3.6)

The explicit form of the eigenvector $\psi_{\bm{t}}$ is

$\displaystyle\psi_{\bm{t}}=$ (3.7)
$\displaystyle=\sum_{\begin{subarray}{c}\bm{n}\in P_{m,N}\\\
(k_{1},\dots,k_{m})\in{\mathbb{Z}}_{M}^{m}\end{subarray}}\bigotimes_{i=1}^{N}\frac{\check{\sigma}^{k_{n^{i}_{1}}}\big{(}F_{c(n^{i}_{1})}\big{)}\check{\sigma}^{k_{n^{i}_{2}}}\big{(}F_{c(n^{i}_{2})}\big{)}\cdots\check{\sigma}^{k_{n^{i}_{p_{i}-1}}}\big{(}F_{c(n^{i}_{p_{i}-1})}\big{)}\check{\sigma}^{k_{n^{i}_{p_{i}}}}\big{(}F_{c(n^{i}_{p_{i}})}\big{)}\mathsf{v}_{\Lambda_{i}}}{\big{(}\omega^{k_{n^{i}_{1}}}w_{n^{i}_{1}}-\omega^{k_{n^{i}_{2}}}w_{n^{i}_{2}}\big{)}\cdots\big{(}\omega^{k_{n^{i}_{p_{i}-1}}}w_{n^{i}_{p_{i}-1}}-\omega^{k_{n^{i}_{p_{i}}}}w_{n^{i}_{p_{i}}}\big{)}\big{(}\omega^{k_{n^{i}_{p_{i}}}}w_{n^{i}_{p_{i}}}-z_{i}\big{)}},$

where $\check{\sigma}(X):=\omega\sigma(X)$.

(It is an interesting open problem to determine under what circumstances the
vector $\psi_{\bm{t}}$ is non-zero.)

On the other hand, consider the (usual) quadratic Gaudin Hamiltonians in the
special case (2.7). We refer to this situation as the _extended Gaudin model_
, and write $\tilde{\mathcal{H}}^{(i)}$ as $\mathcal{H}^{(i)}_{\rm ext}$. Note
that

$\displaystyle\mathcal{H}^{(i)}_{\rm
ext}\in\operatorname{End}\left(M_{\Lambda_{0}}\otimes\bigotimes_{k=0}^{M-1}\bigotimes_{i=1}^{N}M_{\sigma^{k}\Lambda_{i}}\right),\qquad
i=0,1,\dots,NM.$ (3.8)

The following is then a corollary of Theorem 3.1.

######  Corollary 3.4.

To any critical point of the cyclotomic master function, i.e., to any solution
$\bm{t}$ to the equations (2.3), there corresponds a simultaneous eigenvector
of the linear operators $\mathcal{H}^{(i)}_{\rm ext}$, $i=0,1,\dots,nM$, such
that $\mathcal{H}^{(0)}_{\rm ext}$ has eigenvalue zero and, for each
$k=0,\dots,M-1$ and $i=1,\dots,N$, the eigenvalue of
$\mathcal{H}^{(k+Mi)}_{\rm ext}$ is given by $\omega^{-k}E_{i}$ with $E_{i}$
as in (3.6).

###### Proof.

Let $\bm{t}$ be the corresponding (by Lemma 2.9) cyclotomic critical point of
the extended master function $\widehat{\Phi}$. Then the result is a special
case of Theorem 3.1, by substituting (2.7) and (2.11) into (3.1). (To see that
$\mathcal{H}^{(0)}_{\rm ext}$ has eigenvalue zero note that

$\displaystyle\sum_{i=1}^{N}\sum_{s=0}^{M-1}\frac{(\Lambda_{0},\sigma^{s}\Lambda_{i})}{0-\omega^{s}z_{i}}-\sum_{j=1}^{m}\sum_{s=0}^{M-1}\frac{(\Lambda_{0},\sigma^{s}\alpha_{{\mathsf{c}}(j)})}{0-\omega^{s}t_{j}}=0,$

because
$\sum\limits_{s=0}^{M-1}\omega^{-s}\sigma^{-s}\Lambda_{0}=\Lambda_{0}\sum\limits_{s=0}^{M-1}\omega^{-s}=0$
since $\sigma\Lambda_{0}=\Lambda_{0}$ and $M>1$.) ∎

In summary, we have the following observation.

######  Theorem 3.5.

To any critical point of the cyclotomic master function there corresponds both
a simultaneous eigenvector (3.7) of the Hamiltonians $\mathcal{H}^{(i)}$ of
the cyclotomic Gaudin model and a simultaneous eigenvector (3.2) of the
Hamiltonians $\mathcal{H}^{(i)}_{\rm ext}$ of the extended Gaudin model,
$i=1,\dots,n$, with the corresponding eigenvalues equal and in both cases
being given by (3.6).

######  Remark 3.6.

The operators $\mathcal{H}^{(i)}$ and $\mathcal{H}^{(i)}_{\rm ext}$ are acting
in different spaces, (3.4) and (3.8) respectively. It would be interesting to
relate these operators by some means independent of the Bethe ansatz.

##  4 Cyclotomic generation procedure

In [15, 20] a procedure was introduced which generates new critical points of
master functions starting from a given initial critical point. There is an
“elementary generation” step associated to each $i\in I$. The Zariski closure
of the collection of all critical points obtained by recursively applying
elementary generations in all possible ways is called the “population” to
which the initial critical point belongs.

The extended master functions, (2.8) above, are master functions of the
standard form (unlike the cyclotomic master functions (2.2)). Modulo
subtleties coming from the fact that the weight $\Lambda_{0}$ at the origin
need not be dominant integral, that means the generation procedure can be
applied.

In this section we describe this generation procedure and go on to show how,
given a cyclotomic critical point, one can obtain new cyclotomic critical
points by applying the elementary generation steps in certain carefully chosen
combinations. The resulting collections of cyclotomic critical points will be
called “cyclotomic populations”.

###  4.1 Conditions on $\bm{\Lambda_{0}}$

In the remainder of the paper we assume that $\sigma$ is a diagram
automorphism obeying the linking condition (2.1). That means for each $i\in
I$, either $L_{i}=1$ or $L_{i}=2$.

In addition, in this section, Section 4, we place the following conditions on
the weight $\Lambda_{0}\in{\mathfrak{h}}^{\sigma,*}$.

For each $i\in I$ such that $L_{i}=1$, we suppose that

$\displaystyle\langle\Lambda_{0},\alpha_{i}^{\vee}\rangle\in{\mathbb{Z}}_{\geq
0}$ (4.1)

and

$\displaystyle\langle\Lambda_{0},\alpha_{i}^{\vee}\rangle+1\equiv 0\mod
M/M_{i}.$ (4.2)

For each $i\in I$ such that $L_{i}=2$, we suppose that

$\displaystyle
2\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle+1\in{\mathbb{Z}}_{\geq 0}.$ (4.3)

######  Remark 4.1.

One can verify that these conditions are satisfied by the weight $\Lambda_{0}$
of (3.5) in the case of diagram automorphisms of finite-type Dynkin diagrams.
Our assumptions on $\Lambda_{0}$ in the treatment of type $A_{R}$ in Section 5
below are weaker.

###  4.2 Tuples of polynomials

To any pair $(\bm{t};\bm{\mathsf{c}})$ with
$\bm{t}\in{\mathbb{C}}^{\tilde{m}}$ and $\bm{\mathsf{c}}\in I^{\tilde{m}}$, we
may associate a tuple of polynomials ${\bm{y}}=(y_{1}(x),\dots,y_{r}(x))$,
given by

$\displaystyle y_{i}(x):=\prod_{\begin{subarray}{c}j=1\\\
{\mathsf{c}}(j)=i\end{subarray}}^{\tilde{m}}(x-t_{j}),\qquad i\in I.$ (4.4)

We say that this tuple $\bm{y}$ _represents the pair
$(\bm{t};\bm{\mathsf{c}})$_. We consider each coordinate $y_{i}(x)$ only up to
multiplication by a non-zero complex number, since we are only concerned with
their zeros. So the tuple $\bm{y}$ defines a point in the direct product
${\mathbb{P}}({\mathbb{C}}[x])^{|I|}$ of $|I|$ copies of the projective space
${\mathbb{P}}({\mathbb{C}}[x])$, where ${\mathbb{C}}[x]$ is the vector space
of complex polynomials in $x$.

Conversely, given any $\bm{y}\in{\mathbb{P}}({\mathbb{C}}[x])^{|I|}$ we may
extract the pair $(\bm{t};\bm{\mathsf{c}})\in{\mathbb{C}}^{\tilde{m}}\times
I^{\tilde{m}}$ such that (4.4) holds. This pair is unique up to permutation by
an element of $S_{\tilde{m}}$; see (2.10).

Define $T_{i}(x)$, $i\in I$, to be

$\displaystyle
T_{i}(x):=\prod_{s=1}^{N}\prod_{k=0}^{M-1}\big{(}x-\omega^{k}z_{s}\big{)}^{\langle\sigma^{k}\Lambda_{s},\alpha^{\vee}_{i}\rangle}.$
(4.5)

We say that a tuple of polynomials ${\bm{y}}=(y_{i}(x))_{i\in
I}\in{\mathbb{P}}({\mathbb{C}}[x])^{|I|}$ is _generic $($with respect to
$(T_{i}(x))_{i\in I})$_ if for each $i\in I$, $y_{i}(x)$ has no root in common
with $T_{i}(x)$, or with any $y_{j}(x)$, $j\in I{\setminus}\\{i\\}$, such that
$\langle\alpha_{j},\alpha^{\vee}_{i}\rangle\neq 0$.

Note that if $\bm{y}$ represents a critical point of the extended master
function $\widehat{\Phi}(\bm{t};\bm{\mathsf{c}};\bm{z};\bm{\Lambda})$, (2.8),
i.e., its roots obey (2.9), then the tuple $\bm{y}$ must be generic. (Indeed,
if (2.9) holds then in particular each summand on the left hand side of (2.9)
must have non-zero denominator. By definition that implies that the
corresponding tuple is generic.)

###  4.3 Elementary generation: the $\bm{L_{i}=1}$ case

Throughout this subsection, we suppose $i\in I$ is such that $L_{i}=1$. That
means that the simple roots $\alpha_{\sigma^{k}i}$, $i=1,\dots,M_{i}$, are
mutually orthogonal. Equivalently it means that the reflections
${\mathsf{s}}_{\sigma^{k}i}\in W$, $i=1,\dots,M_{i}$, are mutually commuting.

Let $y_{i}^{(i)}(x)$ be of the form

$\displaystyle
y_{i}^{(i)}(x)=y_{i}(x)\int^{x}\xi^{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}T_{i}(\xi)\prod_{j\in
I}y_{j}(\xi)^{-\langle\alpha_{j},\alpha^{\vee}_{i}\rangle}d\xi,$ (4.6)

so that $y_{i}^{(i)}(x)$ is a solution to the equation

$\displaystyle\operatorname{Wr}(y_{i}(x),y_{i}^{(i)}(x))=x^{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}T_{i}(x)\prod_{j\in
I{\setminus}\\{i\\}}y_{j}(x)^{-\langle\alpha_{j},\alpha^{\vee}_{i}\rangle},$
(4.7)

where $\operatorname{Wr}(f(x),g(x)):=f(x)g^{\prime}(x)-f^{\prime}(x)g(x)$
denotes the Wronskian determinant.

######  Proposition 4.2.

If $\bm{y}$ represents a critical point then $y_{i}^{(i)}(x)$ is a polynomial.

###### Proof.

We have $\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle\in{\mathbb{Z}}_{\geq 0}$
as in (4.1), and for each $s\in\\{1,\dots,N\\}$, $\Lambda_{s}$ is integral
dominant so $\langle\Lambda_{s},\alpha^{\vee}_{i}\rangle\in{\mathbb{Z}}_{\geq
0}$. So the integrand is a rational function with poles at most at the points
$t_{p}$, $p\in\\{1,\dots,m\\}$, for which ${\mathsf{c}}(p)=i$. Consider such a
point $t_{p}$. Note that

$\displaystyle\frac{\partial}{\partial x}\log
x^{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}T_{i}(x)(x-t_{p})^{2}\prod_{j\in
I}y_{j}(x)^{-\langle\alpha_{j},\alpha^{\vee}_{i}\rangle}$
$\displaystyle\qquad{}=\sum_{k=0}^{M-1}\sum_{i=1}^{N}\frac{\langle\sigma^{k}\Lambda_{i},\alpha^{\vee}_{{\mathsf{c}}(p)}\rangle}{x-\omega^{k}z_{i}}+\frac{\langle\Lambda_{0},\alpha^{\vee}_{{\mathsf{c}}(p)}\rangle}{x}-\sum_{\begin{subarray}{c}i=1\\\
i\neq
p\end{subarray}}^{\tilde{m}}\frac{\langle\alpha_{{\mathsf{c}}(i)},\alpha_{{\mathsf{c}}(p)}\rangle}{x-t_{i}}.$
(4.8)

This vanishes at $x=t_{p}$ by virtue of the critical point equations (2.9). It
follows that the residue of the integrand at $t_{p}$ vanishes: indeed, this
residue is

$\displaystyle\left(\frac{\partial}{\partial
x}x^{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}T_{i}(x)(x-t_{p})^{2}\prod_{j\in
I}y_{j}(x)^{-\langle\alpha_{j},\alpha^{\vee}_{i}\rangle}\right)_{x=t_{p}},$

which vanishes if (4.8) vanishes. This shows that $y_{i}^{(i)}(x)$ is an
entire function. It is of polynomial growth for large $x$. Therefore it is a
polynomial. ∎

If $y_{i}^{(i)}(x)$ is any solution to (4.7) then so too is
$y_{i}^{(i)}(x)+cy_{i}(x)$ for any $c\in{\mathbb{C}}$.

Thus, given any tuple $\bm{y}$ representing a critical point we have, for each
value of a parameter $c\in{\mathbb{C}}$, a new tuple of polynomials
$\bm{y}^{(i)}$, obtained from the tuple $\bm{y}$ by replacing $y_{i}(x)$ with
$y_{i}^{(i)}(x)+cy_{i}(x)$. We say $\bm{y}^{(i)}$ is obtained from $\bm{y}$ by
_generation in the $i$th direction_, and we call $\bm{y}^{(i)}$ the _immediate
descendant of $\bm{y}$ in the $i$th direction_.

######  Proposition 4.3 ([15]).

The tuple of polynomials $\bm{y}^{(i)}$ is generic for almost all $c$. If
$\bm{y}^{(i)}$ is generic then it represents a critical point.

The tuples $\bm{y}^{(i)}$ describe a projective line in
${\mathbb{P}}({\mathbb{C}}[x])^{|I|}$. It will be useful to have the following
specific parameterization of this line. There exists a unique solution
$y_{i}^{(i)}(x)$ to the equation (4.7), call it $y_{i}^{(i)}(x;0)$, such that
the coefficient of $x^{\deg y_{i}}$ in $y^{(i)}_{i}(x;0)$ is zero. Let us
define

$\displaystyle y_{i}^{(i)}(x;c):=y^{(i)}_{i}(x;0)+cy_{i}(x),$ (4.9)

and define $\bm{y}^{(i)}(c)\in{\mathbb{P}}({\mathbb{C}}[x])^{|I|}$ to be the
tuple obtained from the tuple $\bm{y}$ by replacing $y_{i}(x)$ with
$y^{(i)}_{i}(x;0)+cy_{i}(x)$.

We say generation in the $i$th direction is _degree-increasing_ if $\deg
y_{i}^{(i)}>\deg y_{i}$ for almost all $c$.

Recall that there is a weight at infinity, $\Lambda_{\infty}$, associated to
any critical point. For the critical point represented by $\bm{y}$ this weight
is, cf. (2.4),

$\displaystyle\Lambda_{\infty}(\bm{y})=\Lambda_{0}+\sum_{s=1}^{N}\sum_{k=0}^{M-1}\sigma^{k}\Lambda_{s}-\sum_{j\in
I}\alpha_{j}\deg y_{j}.$ (4.10)

For fixed $\Lambda_{0},\Lambda_{1},\dots,\Lambda_{N}$ we can think of
$\Lambda_{\infty}$ as encoding the degrees of the polynomials $y_{j}$. Note
that $\deg y_{i}^{(i)}(x;0)=\deg
y_{i}+\langle\Lambda_{\infty},\alpha_{i}^{\vee}\rangle+1$. It follows that the
weight at infinity of $\bm{y}^{(i)}(0)$ is

$\displaystyle\Lambda_{\infty}-\alpha_{i}\big{(}\langle\Lambda_{\infty},\alpha^{\vee}_{i}\rangle+1\big{)}=\Lambda_{\infty}-\langle\Lambda_{\infty}+\rho,\alpha^{\vee}_{i}\rangle\alpha_{i}={\mathsf{s}}_{i}\cdot\Lambda_{\infty}.$

This establishes the following lemma.

######  Lemma 4.4.

Generation in the $i$th direction $($with $L_{i}=1)$ is degree-increasing if
and only if $\Lambda_{\infty}$ is $i$-dominant, i.e.,
$\langle\Lambda_{\infty},\alpha^{\vee}_{i}\rangle\in{\mathbb{Z}}_{\geq 0}$.

If generation in the $i$th direction is degree-increasing, then the weight at
infinity associated with the critical point represented by
$\bm{y}_{i}^{(i)}(c)$ is ${\mathsf{s}}_{i}\cdot\Lambda_{\infty}$. Otherwise it
is $\Lambda_{\infty}$ for all $c\neq 0$ $($and
${\mathsf{s}}_{i}\cdot\Lambda_{\infty}$ for $c=0)$.

###  4.4 Cyclotomic generation: the $\bm{L_{i}=1}$ case

We continue to suppose that $i$ is such that $L_{i}=1$.

If $\bm{y}$ represents a cyclotomic point then its immediate descendant
$\bm{y}^{(i)}$ in the $i$th direction generically does not. However if,
starting from a cyclotomic critical point, we successively generate in _each_
of the directions $\sigma^{k}i$, $k=1,\dots,M_{i}$, in turn, in any order,
then we can arrange to arrive at a (new) cyclotomic critical point. This is
the content of Theorem 4.6 below.

Let $\simeq$ denote equality up to a constant (independent of $x$) nonzero
factor. Recall the definition (2.11) of a cyclotomic point.

######  Lemma 4.5.

A tuple of polynomials $\bm{y}$ represents a cyclotomic point if and only if

$\displaystyle y_{\sigma j}(\omega x)\simeq y_{j}(x)$

for all $j\in I$. If $y_{j}(x)$ and $y_{\sigma j}(x)$ share the same leading
coefficient for all $j\in I$, then the tuple $\bm{y}$ represents a cyclotomic
point if and only if

$\displaystyle y_{\sigma j}(\omega x)=\omega^{\deg y_{j}}y_{j}(x)$

for all $j\in I$.

For the rest of this subsection, we suppose $\bm{y}$ represents a cyclotomic
critical point. Hence in particular $\sigma\Lambda_{\infty}=\Lambda_{\infty}$.
Let $y^{(i)}_{i}(x;c)=y^{(i)}_{i,0}(x)+cy_{i}(x)$ be as in (4.9). (So
$y^{(i)}_{i}(x;c)$ is a parameterization of the space of solutions to (4.7).)
Define $\bm{y}^{(i,\sigma)}(c)$ to be the tuple of polynomials given by

$\displaystyle
y^{(i,\sigma)}_{\sigma^{k}i}\big{(}\omega^{k}x;c\big{)}:=\omega^{k\deg
y^{(i)}_{i}}y^{(i)}_{i}(x;c),\qquad k=0,1,\dots,M_{i}-1,$

and $y^{(i,\sigma)}_{j}(x;c)=y_{j}(x)$ for $j\in
I{\setminus}\sigma^{\mathbb{Z}}i$. Recall ${\mathsf{s}}_{i}^{\sigma}$ from
Lemma 2.5.

######  Theorem 4.6.

For almost all $c\in{\mathbb{C}}$, the tuple $\bm{y}^{(i,\sigma)}(c)$
represents a cyclotomic critical point. The exceptional values of $c$ form a
finite subset of ${\mathbb{C}}$.

The weight at infinity of $\bm{y}^{(i,\sigma)}(c)$ is
${\mathsf{s}}^{\sigma}_{i}\cdot\Lambda_{\infty}$ if
$\langle\Lambda_{\infty},\alpha^{\vee,\sigma}_{i}\rangle\in{\mathbb{Z}}_{\geq
0}$. Otherwise it is $\Lambda_{\infty}$ for all $c\neq 0$, and
${\mathsf{s}}^{\sigma}_{i}\cdot\Lambda_{\infty}$ for $c=0$.

###### Proof.

First let us show that $\bm{y}^{(i,\sigma)}$ represents a cyclotomic point for
all $c\in{\mathbb{C}}$. Comparing our definition of $\bm{y}^{(i,\sigma)}$ with
the criterion in Lemma 4.5, one sees that it is enough to check that

$\displaystyle y^{(i)}_{i}\big{(}\omega^{M_{i}}x;c\big{)}=\omega^{M_{i}\deg
y^{(i)}_{i}}y^{(i)}_{i}(x;c).$

Inspecting (4.6), we see that this equality holds for all $c\in{\mathbb{C}}$
if and only if

$\displaystyle\omega^{M_{i}\langle\Lambda_{\infty}+\rho,\alpha_{i}^{\vee}\rangle}=1.$
(4.11)

But now, given (4.10) and the assumption that $\Lambda_{s}$, $s=1,\dots,n$ are
integral, the following lemma implies that (4.11) holds if and only if we
impose the condition (4.2) on $\Lambda_{0}$.

######  Lemma 4.7.

Suppose $\Lambda\in{\mathfrak{h}}^{*}$ is an integral weight. Then, for any
$j\in I$,

$\displaystyle\sum_{k=0}^{M-1}\langle\sigma^{k}\Lambda,\alpha^{\vee}_{j}\rangle
M_{j}\equiv 0\mod M.$

###### Proof.

We have

$\displaystyle\left\langle\sum_{k=0}^{M-1}\sigma^{k}\Lambda,\alpha^{\vee}_{j}\right\rangle
M_{j}=\left\langle\Lambda,\sum_{k=0}^{M-1}\sigma^{-k}\alpha^{\vee}_{j}\right\rangle
M_{j}$
$\displaystyle\hphantom{\left\langle\sum_{k=0}^{M-1}\sigma^{k}\Lambda,\alpha^{\vee}_{j}\right\rangle
M_{j}}{}=\left\langle\Lambda,\frac{M}{M_{j}}\sum_{k=0}^{M_{j}-1}\alpha^{\vee}_{j}\right\rangle
M_{j}=M\left\langle\Lambda,\sum_{k=0}^{M_{j}-1}\alpha^{\vee}_{j}\right\rangle\in
M{\mathbb{Z}}.$ ∎

Now we show that $\bm{y}^{(i,\sigma)}$ represents a critical point for all but
finitely many $c\in{\mathbb{C}}$. Note that from definition (4.5) we have

$\displaystyle T_{\sigma j}(\omega
x)=\omega^{\big{\langle}\sum\limits_{s=1}^{N}\sum\limits_{k=0}^{M-1}\sigma^{k}\Lambda_{s},\alpha^{\vee}_{j}\big{\rangle}}T_{j}(x),\qquad
j\in I.$ (4.12)

Hence, in view of (4.10),

$\displaystyle x^{\langle\Lambda_{0},\alpha^{\vee}_{\sigma i}\rangle}T_{\sigma
i}(\omega x)\prod_{j\in I}y_{j}(\omega
x)^{-\langle\alpha_{j},\alpha^{\vee}_{\sigma
i}\rangle}=x^{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}T_{\sigma i}(\omega
x)\prod_{j\in I}y_{\sigma j}(\omega x)^{-\langle\alpha_{\sigma
j},\alpha^{\vee}_{\sigma i}\rangle}$
$\displaystyle\hphantom{x^{\langle\Lambda_{0},\alpha^{\vee}_{\sigma
i}\rangle}T_{\sigma i}(\omega x)\prod_{j\in I}y_{j}(\omega
x)^{-\langle\alpha_{j},\alpha^{\vee}_{\sigma
i}\rangle}}{}=\omega^{\langle\Lambda_{\infty},\alpha^{\vee}_{i}\rangle}\left(x^{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}T_{i}(x)\prod_{j\in
I}y_{j}(x)^{-\langle\alpha_{j},\alpha^{\vee}_{i}\rangle}\right).$

Note also that since $L_{i}=1$, no node $j$ in the orbit of $i$ is linked by
an edge of the Dynkin diagram to $i$. That is, no $y_{j}$ for $j$ in the orbit
of $i$ appears on the right of (4.7). Hence, for $k=1,\dots,M_{i}-1$,
$y^{(i,\sigma)}_{\sigma^{k}i}(x;c)$ obeys the equation

$\displaystyle\operatorname{Wr}(y_{\sigma^{k}i}(x),y^{(i,\sigma)}_{\sigma^{k}i}(x;c))=x^{\langle\Lambda_{0},\alpha^{\vee}_{\sigma^{k}i}\rangle}T_{\sigma^{k}i}(x)\prod_{j\in
I{\setminus}\\{\sigma^{k}i\\}}y_{j}(x)^{-\langle\alpha_{j},\alpha^{\vee}_{\sigma^{k}i}\rangle}.$

and the tuple $\bm{y}^{(i,\sigma)}$ is indeed the result of generating in each
of the directions $i,\sigma i,\dots,\sigma^{M_{i}-1}i$ (in any order). It
follows from Proposition 4.3 that $\bm{y}^{(i,\sigma)}$ is generic for almost
all $c$, and represents a critical point whenever it is generic.

The statements about the weight at infinity follow from Lemma 4.4 and Section
2.3. This completes the proof of Theorem 4.6. ∎

###  4.5 Elementary generation: the $\bm{L_{i}=2}$ case

For this subsection we suppose that $i\in I$ is such that $L_{i}=2$. That
implies $M_{i}$ is even and the restriction of the Dynkin diagram to the nodes
$\sigma^{\mathbb{Z}}i$ consists of $\frac{M_{i}}{2}\in{\mathbb{Z}}_{\geq 1}$
disconnected copies of the Dynkin diagram of type ${\mathrm{A}}_{2}$, as
sketched below:

$\sigma^{0}i$$\sigma^{0}{\bar{\imath}}$$\sigma^{1}i$$\sigma^{1}{\bar{\imath}}$$\sigma^{2}i$$\sigma^{2}{\bar{\imath}}$$\sigma^{M_{i}/2-1}i$$\sigma^{M_{i}/2-1}{\bar{\imath}}$

Here, for brevity, we write ${\bar{\imath}}:=\sigma^{M_{i}/2}i$.

######  Remark 4.8.

Among finite and affine types, only the case $M_{i}/2=1$ occurs.

We define $y_{i}^{(i)}(x)$ by

$\displaystyle
y_{i}^{(i)}(x):=y_{i}(x)x^{-{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}-1}\int_{0}^{x}\xi^{{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}}T_{i}(\xi)\prod_{j\in
I}y_{j}(\xi)^{-\langle\alpha_{j},\alpha^{\vee}_{i}\rangle}d\xi.$

Here the limits $\int_{0}^{x}$ mean that $y^{(i)}_{i}(x)$ is holomorphic at
$x=0$. This condition defines the integral uniquely, since
$\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle\notin{\mathbb{Z}}$ by our
assumption (4.3).

######  Proposition 4.9.

If $\bm{y}$ represents a critical point then $y^{(i)}_{i}(x)$ is a polynomial.
It has degree $\deg y^{(i)}_{i}=\deg
y_{i}+\langle\Lambda_{\infty}-\Lambda_{0},\alpha^{\vee}_{i}\rangle$.

###### Proof.

The proof is as for Proposition 4.2. ∎

Let $\bm{y}^{(i)}=(y^{(i)}_{j}(x))_{j\in I}$ be the tuple of polynomials whose
$i$th component $y^{(i)}_{i}(x)$ is as above, and whose remaining components
are the same as those of $\bm{y}$, i.e.,

$\displaystyle y^{(i)}_{j}(x)=y_{j}(x)\qquad\text{ for all }\quad j\in
I{\setminus}\\{i\\}.$

Let $(\bm{t}^{(i)};\bm{\mathsf{c}}^{(i)})$ denote the pair represented by this
tuple in the sense of Section 4.2. It turns out that $\bm{t}^{(i)}$ is not in
general a critical point of the extended master function
$\widehat{\Phi}(\bm{t}^{(i)};\bm{\mathsf{c}}^{(i)};\bm{z};\bm{\Lambda})$,
i.e., it does not in general obey the equations (2.9). Instead, the following
result gives the analogous collection of equations that it does obey, provided
$\bm{y}^{(i)}$ is generic.

######  Proposition 4.10.

If $\bm{y}$ represents a critical point and $\bm{y}^{(i)}$ is generic, then

$\displaystyle\frac{\big{\langle}{\mathsf{s}}_{i}\cdot\Lambda_{0},\alpha^{\vee}_{{\mathsf{c}}^{(i)}(p)}\big{\rangle}}{t^{(i)}_{p}}+\sum_{s=1}^{N}\sum_{k=0}^{M-1}\frac{\big{\langle}\sigma^{k}\Lambda_{s},\alpha^{\vee}_{{\mathsf{c}}^{(i)}(p)}\big{\rangle}}{t^{(i)}_{p}-\omega^{k}z_{s}}-\sum_{r\colon
r\neq
p}\frac{\big{\langle}\alpha_{{\mathsf{c}}^{(i)}(r)},\alpha^{\vee}_{{\mathsf{c}}^{(i)}(p)}\big{\rangle}}{t^{(i)}_{p}-t^{(i)}_{r}}=0$

for each $p$.

###### Proof.

By (2.9) for each root $t_{p}$ in the tuple $\bm{t}$ we have

$\displaystyle\frac{\big{\langle}\Lambda_{0},\alpha^{\vee}_{{\mathsf{c}}(p)}\big{\rangle}}{t_{p}}+\sum_{s=1}^{N}\sum_{k=0}^{M-1}\frac{\big{\langle}\sigma^{k}\Lambda_{s},\alpha^{\vee}_{{\mathsf{c}}(p)}\big{\rangle}}{t_{p}-\omega^{k}z_{s}}-\sum_{r\colon
r\neq
p}\frac{\big{\langle}\alpha_{{\mathsf{c}}(r)},\alpha^{\vee}_{{\mathsf{c}}(p)}\big{\rangle}}{t_{p}-t_{r}}=0.$
(4.13)

For all roots of colours $j\in I$ such that
$\langle\alpha_{j},\alpha^{\vee}_{i}\rangle=0$ this is immediately equivalent
to the required equation. So we must consider roots of colour $i$, and roots
of colours $j\in I$ such that $\langle\alpha_{j},\alpha^{\vee}_{i}\rangle<0$.

By definition of $y^{(i)}_{i}(x)$ we have

$\displaystyle\operatorname{Wr}(y_{i}(x),x^{{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}+1}y^{(i)}_{i}(x))=x^{{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}}T_{i}(x)\prod_{j\neq
i}y_{j}(x)^{-\langle\alpha_{j},\alpha^{\vee}_{i}\rangle}$ (4.14)

or equivalently

$\displaystyle\frac{y_{i}^{\prime}(x)}{y_{i}(x)}-\frac{y^{(i)}_{i}{}^{\prime}(x)}{y^{(i)}_{i}(x)}-\frac{1+{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}}{x}=\frac{T_{i}(x)\prod\limits_{j\neq
i}y_{j}(x)^{-\langle\alpha_{j},\alpha^{\vee}_{i}\rangle}}{xy_{i}(x)y^{(i)}_{i}(x)}.$
(4.15)

By definition of $(\bm{t}^{(i)},\bm{\mathsf{c}}^{(i)})$, the left-hand side of
(4.15) is

$\displaystyle\sum_{r\colon{\mathsf{c}}(r)=i}\frac{1}{x-t_{r}}-\sum_{r\colon{\mathsf{c}}^{(i)}(r)=i}\frac{1}{x-t^{(i)}_{r}}-\frac{1+{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}}{x}.$
(4.16)

Now suppose $j\in I$ is such that
$\langle\alpha_{j},\alpha^{\vee}_{i}\rangle\in{\mathbb{Z}}_{<0}$. By
definition $y^{(i)}_{j}(x)=y_{j}(x)$. Suppose $t_{p}$ is a root of $y_{j}(x)$,
i.e., suppose ${\mathsf{c}}(p)=j$. Since $\bm{y}$ represents a critical point,
$\bm{y}$ must be generic, and hence $t_{p}$ is not a root of $y_{i}(x)$. By
our assumption that $\bm{y}^{(i)}$ is generic, $t_{p}$ is not a root of
$y^{(i)}_{i}(x)$ either. Hence the right-hand side of (4.15) is zero at
$x=t_{p}$ and so, in view of (4.16), we have

$\displaystyle\sum_{r\colon{\mathsf{c}}(r)=i}\frac{1}{t_{p}-t_{r}}-\sum_{r\colon{\mathsf{c}}^{(i)}(r)=i}\frac{1}{t_{p}-t^{(i)}_{r}}-\frac{1+{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}}{t_{p}}=0.$

On adding this equation multiplied by
$\langle\alpha_{i},\alpha^{\vee}_{j}\rangle$ to the equation (4.13), we arrive
at

$\displaystyle\frac{\langle\Lambda_{0},\alpha^{\vee}_{j}\rangle-\langle\alpha_{i},\alpha^{\vee}_{j}\rangle\langle\Lambda_{0}+\rho,\alpha^{\vee}_{i}\rangle}{t^{(i)}_{p}}+\sum_{s=1}^{N}\sum_{k=0}^{M-1}\frac{\langle\sigma^{k}\Lambda_{s},\alpha^{\vee}_{j}\rangle}{t^{(i)}_{p}-\omega^{k}z_{s}}-\sum_{r\colon
r\neq
p}\frac{\langle\alpha_{{\mathsf{c}}^{(i)}(r)},\alpha^{\vee}_{j}\rangle}{t^{(i)}_{p}-t^{(i)}_{r}}=0,$

which is the required equality (since
${\mathsf{s}}_{i}\cdot\Lambda_{0}=\Lambda_{0}-\langle\Lambda_{0}+\rho,\alpha^{\vee}_{i}\rangle\alpha_{i}$).

It remains to consider roots of colour $i$. First note that $y_{i}(x)$ and
$y^{(i)}_{i}(x)$ have no common roots. Indeed, if $t$ were a common root of
$y_{i}(x)$ and $y_{i}^{(i)}(x)$ then the right-hand side of (4.14) would have
to vanish at $x=t$. In other words $y_{i}(x)$ would have a root in common with
the right-hand side of (4.14). But by our definition of what it means for
$\bm{y}$ to be generic, Section 4.2, this is impossible.

Suppose $t^{(i)}_{p}$ is any root of $y^{(i)}_{i}(x)$. By our assumption that
$\bm{y}^{(i)}$ is generic, it follows from (4.14) and Lemmas 4.12 and 4.13
below that

$\displaystyle\frac{2(1+{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle})}{t^{(i)}_{p}}-\frac{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}{t^{(i)}_{p}}-\sum_{s=1}^{N}\sum_{k=0}^{M-1}\frac{\langle\sigma^{k}\Lambda_{s},\alpha^{\vee}_{i}\rangle}{t^{(i)}_{p}-\omega^{k}z_{s}}+\sum_{r:r\neq
p}\frac{\langle\alpha_{{\mathsf{c}}^{(i)}(r)},\alpha^{\vee}_{i}\rangle}{t^{(i)}_{p}-t^{(i)}_{r}}=0,$

which is the required equality. ∎

######  Remark 4.11.

Propositions 4.9 and 4.10 also follow from Theorem 3.5 in [17].

######  Lemma 4.12.

For any $\alpha\in{\mathbb{C}}$, if
$g(x)=x^{\alpha}\prod\limits_{j=1}^{J}(x-s_{j})$, where $(s_{j})_{j=1}^{J}$
are all distinct and non-zero, then

$\displaystyle\left.\frac{g^{\prime\prime}(x)}{g^{\prime}(x)}\right|_{x=s_{k}}=\frac{2\alpha}{s_{k}}+\sum_{\begin{subarray}{c}j=1\\\
j\neq k\end{subarray}}^{J}\frac{2}{s_{k}-s_{j}}.$

######  Lemma 4.13.

If $\operatorname{Wr}(f(x),g(x))=W(x)$ then

$\displaystyle\frac{g^{\prime\prime}(x)}{g^{\prime}(x)}-\frac{W^{\prime}(x)}{W(x)}=\frac{g(x)\big{(}W(x)f^{\prime\prime}(x)-W^{\prime}(x)f^{\prime}(x)\big{)}}{f(x)g^{\prime}(x)W(x)}.$

###### Proof.

We have $W\operatorname{Wr}(f,g)^{\prime}=W^{\prime}\operatorname{Wr}(f,g)$.
Hence

$\displaystyle
W(x)f(x)g^{\prime\prime}(x)-W^{\prime}(x)f(x)g^{\prime}(x)=W(x)f^{\prime\prime}(x)g(x)-W^{\prime}(x)f^{\prime}(x)g(x)$

and hence the result. ∎

To deal with the case in which $\bm{y}^{(i)}$ fails to be generic, we shall
also need the following observation, which follows from (4.14).

######  Lemma 4.14.

For any $j\in I$ such that $\langle\alpha_{j},\alpha^{\vee}_{i}\rangle<0$, if
$t$ is a root of both $y_{j}(x)$ and $y^{(i)}_{i}(x)$ then it is a root of
$y^{(i)}_{i}(x)$ with multiplicity $2$. In particular, if $t$ is a root of
both $y_{\bar{\imath}}(x)$ and $y^{(i)}_{i}(x)$ then it is a root of
$y^{(i)}_{i}(x)$ with multiplicity $2$.

Now we define

$\displaystyle
y^{({\bar{\imath}},i)}_{\bar{\imath}}(x):=y_{\bar{\imath}}(x)\int\xi^{{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}}T_{\bar{\imath}}(\xi)\frac{\xi^{1+{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}}y^{(i)}_{i}(\xi)\prod\limits_{j\neq
i,{\bar{\imath}}}y_{j}(\xi)^{-\langle\alpha_{j},\alpha^{\vee}_{\bar{\imath}}\rangle}}{y_{\bar{\imath}}(\xi)^{2}}d\xi$
$\displaystyle\hphantom{y^{({\bar{\imath}},i)}_{\bar{\imath}}(x)}{}=y_{\bar{\imath}}(x)\int\xi^{1+2{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}}T_{\bar{\imath}}(\xi)\frac{y^{(i)}_{i}(\xi)\prod\limits_{j\neq
i,{\bar{\imath}}}y_{j}(\xi)^{-\langle\alpha_{j},\alpha^{\vee}_{\bar{\imath}}\rangle}}{y_{\bar{\imath}}(\xi)^{2}}d\xi.$
(4.17)

######  Proposition 4.15.

If $\bm{y}$ represents a critical point then
$y^{({\bar{\imath}},i)}_{\bar{\imath}}(x)$ is a polynomial.

###### Proof.

By our assumption (4.3) that
$2{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}+1\in{\mathbb{Z}}_{\geq 0}$,
the integrand is regular at $x=0$. Hence, by Lemma 4.14, it is a rational
function with poles at most at those roots of $y_{\bar{\imath}}(x)$ that are
not also roots of $y^{(i)}_{i}(x)$. Let $t_{p}$ be any such root. The residue
of the integrand at $\xi=t_{p}$ is

$\displaystyle\left.\frac{\partial}{\partial
x}(x-t)^{2}x^{{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}}T_{i}(x)\frac{y^{(i)}_{i}(x)\prod\limits_{j\neq
i,{\bar{\imath}}}y^{(i)}_{j}(x)^{-\langle\alpha_{j},\alpha^{\vee}_{\bar{\imath}}\rangle}}{y_{\bar{\imath}}(x)^{2}}\right|_{x=t_{p}},$

which must vanish, because according to Proposition 4.10 the following
vanishes:

$\displaystyle\left.\frac{\partial}{\partial
x}\log(x-t)^{2}x^{{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}}T_{i}(x)\frac{y^{(i)}_{i}(x)\prod\limits_{j\neq
i,{\bar{\imath}}}y^{(i)}_{j}(x)^{-\langle\alpha_{j},\alpha^{\vee}_{\bar{\imath}}\rangle}}{y_{\bar{\imath}}(x)^{2}}\right|_{x=t_{p}}$
$\displaystyle\qquad{}=\frac{1+2{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}}{t_{p}}+\sum_{s=1}^{N}\sum_{k=0}^{M-1}\frac{\langle\sigma^{k}\Lambda_{s},\alpha^{\vee}_{{\bar{\imath}}}\rangle}{t_{p}-\omega^{k}z_{s}}-\sum_{r\colon
r\neq
p}\frac{\langle\alpha_{{\mathsf{c}}^{(i)}(r)},\alpha^{\vee}_{{\bar{\imath}}}\rangle}{t_{p}-t_{r}^{(i)}}.$

(Note
${\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}=\langle\Lambda_{0},\alpha^{\vee}_{\bar{\imath}}\rangle$
since $\sigma\Lambda_{0}=\Lambda_{0}$.) ∎

The polynomial $y^{({\bar{\imath}},i)}_{\bar{\imath}}(x)$ is defined up a to
the addition of a constant multiple of $y_{\bar{\imath}}(x)$, coming from the
constant of integration in (4.17).

We say generation in the $i$th direction from $\bm{y}$ is _degree-increasing_
if $\deg y^{({\bar{\imath}},i)}_{\bar{\imath}}(x)>\deg y_{\bar{\imath}}(x)$.
Generation in the $i$th direction is degree-increasing if and only if

$\displaystyle\langle\Lambda_{\infty}+\rho,\alpha^{\vee}_{i}+\alpha^{\vee}_{\bar{\imath}}\rangle>0.$
(4.18)

Indeed, if (4.18) holds then

$\displaystyle\deg y^{({\bar{\imath}},i)}_{\bar{\imath}}(x)=\deg
y_{{\bar{\imath}}}(x)+\langle\Lambda_{\infty}+\rho,\alpha^{\vee}_{i}+\alpha^{\vee}_{\bar{\imath}}\rangle>\deg
y_{{\bar{\imath}}}(x)$ (4.19)

for all values of the constant of integration. If (4.18) does not hold then
$\deg y^{({\bar{\imath}},i)}_{\bar{\imath}}(x)\leq\deg y_{\bar{\imath}}(x)$,
with equality for all but one value of the constant of integration in (4.17).

Let $y^{({\bar{\imath}},i)}_{{\bar{\imath}}}(x;0)$ be the unique solution to
(4.17) whose coefficient of $x^{\deg y_{{\bar{\imath}}}}$ is zero. The degree
of $y^{({\bar{\imath}},i)}_{{\bar{\imath}}}(x;0)$ is always given by

$\displaystyle\deg y^{({\bar{\imath}},i)}_{{\bar{\imath}}}(x;0)=\deg
y_{{\bar{\imath}}}(x)+\langle\Lambda_{\infty}+\rho,\alpha^{\vee}_{i}+\alpha^{\vee}_{\bar{\imath}}\rangle,$

whether or not generation is degree-increasing. (Note that
$\langle\Lambda_{\infty}+\rho,\alpha^{\vee}_{i}+\alpha^{\vee}_{\bar{\imath}}\rangle$
is odd, by our assumption (4.3), and in particular not zero.)

Let then
$\bm{y}^{({\bar{\imath}},i)}(c)=(y^{({\bar{\imath}},i)}_{j}(x;c))_{j\in I}$ be
the tuple of polynomials whose ${\bar{\imath}}$th component is

$\displaystyle
y_{\bar{\imath}}^{({\bar{\imath}},i)}(x;c):=y^{({\bar{\imath}},i)}_{\bar{\imath}}(x;0)+cy_{\bar{\imath}}(x)$

and whose remaining components are the same as those of $\bm{y}^{(i)}$, i.e.,

$\displaystyle
y^{({\bar{\imath}},i)}_{i}(x;c)=y^{(i)}_{i}(x;c),\qquad\text{and}\qquad
y^{({\bar{\imath}},i)}_{j}(x)=y^{(i)}_{j}(x)=y_{j}(x)\qquad\text{for all}\quad
j\in I{\setminus}\\{i,{\bar{\imath}}\\}.$

Let $(\bm{t}^{({\bar{\imath}}.i)};\bm{\mathsf{c}}^{({\bar{\imath}},i)})$
denote the pair represented by this tuple in the sense of Section 4.2.

The following result says that whenever $\bm{y}^{({\bar{\imath}},i)}(c)$ is
generic, this new pair
$(\bm{t}^{({\bar{\imath}},i)}(c),\bm{\mathsf{c}}^{({\bar{\imath}},i)})$ obeys
the same form of equations as did $(\bm{t}^{(i)},\bm{\mathsf{c}}^{(i)})$.

######  Proposition 4.16.

If $\bm{y}$ represents a critical point then, for all $c\in{\mathbb{C}}$ such
that $\bm{y}^{({\bar{\imath}},i)}(c)$ is generic, we have

$\displaystyle\frac{\big{\langle}{\mathsf{s}}_{i}\cdot\Lambda_{0},\alpha^{\vee}_{{\mathsf{c}}^{({\bar{\imath}},i)}(p)}\big{\rangle}}{t^{({\bar{\imath}},i)}_{p}(c)}+\sum_{s=1}^{N}\sum_{k=0}^{M-1}\frac{\big{\langle}\sigma^{k}\Lambda_{s},\alpha^{\vee}_{{\mathsf{c}}^{({\bar{\imath}},i)}(p)}\big{\rangle}}{t^{({\bar{\imath}},i)}_{p}(c)-\omega^{k}z_{s}}-\sum_{r\colon
r\neq
p}\frac{\big{\langle}\alpha_{{\mathsf{c}}^{({\bar{\imath}},i)}(r)},\alpha^{\vee}_{{\mathsf{c}}^{({\bar{\imath}},i)}(p)}\big{\rangle}}{t^{({\bar{\imath}},i)}_{p}(c)-t^{({\bar{\imath}},i)}_{r}(c)}=0$

for each $p$.

###### Proof.

The proof is analogous to that of Proposition 4.10. ∎

Finally, we define $y_{i}^{(i,{\bar{\imath}},i)}(x;c)$ by

$\displaystyle
y_{i}^{(i,{\bar{\imath}},i)}(x;c)=y^{(i)}_{i}(x)x^{{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}+1}\int_{0}^{x}\xi^{{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}}T_{i}(\xi)\frac{y^{({\bar{\imath}},i)}_{\bar{\imath}}(\xi;c)\prod\limits_{j\in
I{\setminus}\\{i,{\bar{\imath}}\\}}y_{j}(\xi)^{-\langle\alpha_{j},\alpha^{\vee}_{i}\rangle}}{\left(\xi^{{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}+1}y^{(i)}_{i}(\xi)\right)^{2}}d\xi$
$\displaystyle\hphantom{y_{i}^{(i,{\bar{\imath}},i)}(x;c)}{}=y^{(i)}_{i}(x)x^{{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}+1}\int_{0}^{x}\xi^{-{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}-2}T_{i}(\xi)\frac{y^{({\bar{\imath}},i)}_{\bar{\imath}}(\xi;c)\prod\limits_{j\in
I{\setminus}\\{i,{\bar{\imath}}\\}}y_{j}(\xi)^{-\langle\alpha_{j},\alpha^{\vee}_{i}\rangle}}{y^{(i)}_{i}(\xi)^{2}}d\xi.$

Here the limits $\int_{0}^{x}$ mean that $y^{(i,{\bar{\imath}},i)}_{i}(x;c)$
is holomorphic at $x=0$. This condition defines the integral uniquely.

######  Proposition 4.17.

For all $c\in{\mathbb{C}}$, if $\bm{y}$ represents a critical point then
$y^{(i,{\bar{\imath}},i)}_{i}(x;c)$ is a polynomial.

###### Proof.

Pick any root $t^{({\bar{\imath}},i)}_{p}$ of
$y_{i}^{(i)}(x)=y_{i}^{({\bar{\imath}},i)}(x)$. The residue of the integrand
at $\xi=t^{({\bar{\imath}},i)}$ is zero. Indeed, we have

$\displaystyle\left.\frac{\partial}{\partial
x}\log\big{(}x-t^{(i)}_{p}\big{)}^{2}x^{-{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}-2}T_{i}(x)\frac{y^{({\bar{\imath}},i)}_{\bar{\imath}}(x;c)\prod\limits_{j\in
I{\setminus}\\{i,{\bar{\imath}}\\}}y_{j}(x)^{-\langle\alpha_{j},\alpha^{\vee}_{i}\rangle}}{y^{(i)}_{i}(x)^{2}}\right|_{x=t^{({\bar{\imath}},i)}_{p}}$
$\displaystyle\qquad{}=\frac{-{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}-2}{t^{({\bar{\imath}},i)}_{p}}+\sum_{s=1}^{N}\sum_{k=0}^{M-1}\frac{\langle\sigma^{k}\Lambda_{s},\alpha^{\vee}_{{\bar{\imath}}}\rangle}{t^{({\bar{\imath}},i)}_{p}-\omega^{k}z_{s}}-\sum_{r\colon
r\neq
p}\frac{\langle\alpha_{{\mathsf{c}}^{({\bar{\imath}},i)}(r)},\alpha^{\vee}_{i}\rangle}{t^{({\bar{\imath}},i)}_{p}-t_{r}^{({\bar{\imath}},i)}},$

and this vanishes by Proposition 4.16. ∎

Let
$\bm{y}^{(i,{\bar{\imath}},i)}(c)\\!=\\!(y^{(i,{\bar{\imath}},i)}_{j}(x;c))_{j\in
I}$ be the tuple of polynomials whose $i$th component is
$y^{(i,{\bar{\imath}},i)}(x;c)$ as above and whose remaining components are
those of $\bm{y}^{({\bar{\imath}},i)}(c)$, i.e.,

$\displaystyle
y^{(i,{\bar{\imath}},i)}_{\bar{\imath}}(x;c)=y^{({\bar{\imath}},i)}_{\bar{\imath}}(x;c),\qquad\text{and}\qquad
y^{(i,{\bar{\imath}},i)}_{j}(x)=y_{j}(x)\qquad\text{for all}\quad j\in
I{\setminus}\\{i,{\bar{\imath}}\\}.$

Let $(\bm{t}^{(i,{\bar{\imath}}.i)};\bm{\mathsf{c}}^{(i,{\bar{\imath}},i)})$
denote the pair represented by this tuple in the sense of Section 4.2.

######  Proposition 4.18.

If $\bm{y}$ represents a critical point and $\bm{y}^{(i,{\bar{\imath}},i)}(c)$
is generic, then $\bm{y}^{(i,{\bar{\imath}},i)}(c)$ represents a critical
point. That is, the pair
$(\bm{t}^{i,{\bar{\imath}},i}(c),\bm{\mathsf{c}}^{i,{\bar{\imath}},i})$ obeys
the equations

$\displaystyle\frac{\big{\langle}\Lambda_{0},\alpha^{\vee}_{{\mathsf{c}}(p)}\big{\rangle}}{t^{(i,{\bar{\imath}},i)}_{p}(c)}+\sum_{s=1}^{N}\sum_{k=0}^{M-1}\frac{\big{\langle}\sigma^{k}\Lambda_{s},\alpha^{\vee}_{{\mathsf{c}}(p)}\big{\rangle}}{t^{(i,{\bar{\imath}},i)}_{p}(c)-\omega^{k}z_{s}}-\sum_{r\colon
r\neq
p}\frac{\big{\langle}\alpha_{{\mathsf{c}}(r)},\alpha^{\vee}_{{\mathsf{c}}(p)}\big{\rangle}}{t^{(i,{\bar{\imath}},i)}_{p}(c)-t^{(i,{\bar{\imath}},i)}_{r}(c)}=0$

for each $p$.

###### Proof.

The proof is analogous to that of Proposition 4.10. ∎

We say $\bm{y}^{(i,{\bar{\imath}},i)}(c)$ is obtained from $\bm{y}$ by
_generation in the $i$th direction_, and we call
$\bm{y}^{(i,{\bar{\imath}},i)}(c)$ the _immediate descendant of $\bm{y}$ in
the $i$th direction_. We have the following; cf. Lemma 4.4.

######  Lemma 4.19.

Generation in the $i$th direction $($with $L_{i}=2)$ is degree-increasing if
and only if
$\langle\Lambda_{\infty}+\rho,\alpha^{\vee}_{i}+\alpha^{\vee}_{\bar{\imath}}\rangle\in{\mathbb{Z}}_{>0}$.

If generation in the $i$th direction is degree-increasing, then the weight at
infinity associated with the critical point represented by
$\bm{y}_{i}^{(i,{\bar{\imath}},i)}(c)$ is
$({\mathsf{s}}_{i}{\mathsf{s}}_{\bar{\imath}}{\mathsf{s}}_{i})\cdot\Lambda_{\infty}$.
Otherwise it is $\Lambda_{\infty}$ for all $c\neq 0$ $($and
$({\mathsf{s}}_{i}{\mathsf{s}}_{\bar{\imath}}{\mathsf{s}}_{i})\cdot\Lambda_{\infty}$
for $c=0)$.

###### Proof.

Recall that (4.19) holds if and only if (4.18) holds. Note also that

$\displaystyle\deg y^{(i,{\bar{\imath}},i)}_{i}=\deg
y_{i}+\langle\Lambda_{\infty}+\rho,\alpha^{\vee}_{i}+\alpha^{\vee}_{\bar{\imath}}\rangle.$

By direct calculation, one verifies that

$\displaystyle({\mathsf{s}}_{i}{\mathsf{s}}_{\bar{\imath}}{\mathsf{s}}_{i})\cdot\Lambda_{\infty}=\Lambda_{\infty}-(\alpha_{\bar{\imath}}+\alpha_{i})\langle\Lambda_{\infty}+\rho,\alpha^{\vee}_{i}+\alpha^{\vee}_{\bar{\imath}}\rangle,$

so we have the result. ∎

###  4.6 Cyclotomic generation: the $\bm{L_{i}=2}$ case

We continue to suppose that $i\in I$ is such that $L_{i}=2$.

Suppose for the rest of this subsection that $\bm{y}$ represents a cyclotomic
critical point. Define $\bm{y}^{(i,\sigma)}(c)$ to be the tuple of polynomials
given by

$\displaystyle
y^{(i,\sigma)}_{\sigma^{k}i}(\omega^{k}x;c):=y^{(i,{\bar{\imath}},i)}_{i}(x;c),$
$\displaystyle
y^{(i,\sigma)}_{\sigma^{k}{\bar{\imath}}}(\omega^{k}x;c):=y^{(i,{\bar{\imath}},i)}_{{\bar{\imath}}}(x;c),\qquad
k=0,1,\dots,M_{i}/2-1,$ (4.20)

and $y^{(i,\sigma)}_{j}(x;c)=y_{j}(x)$ for $j\in
I{\setminus}\sigma^{\mathbb{Z}}i$.

######  Theorem 4.20.

For almost all $c\in{\mathbb{C}}$, the tuple $\bm{y}^{(i,\sigma)}(x;c)$
represents a cyclotomic critical point. The exceptional values of $c$ form a
finite subset of ${\mathbb{C}}$.

The weight at infinity of $\bm{y}^{(i,\sigma)}(x;c)$ is
${\mathsf{s}}^{\sigma}_{i}\cdot\Lambda_{\infty}$ if
$\langle\Lambda_{\infty}+\rho,\alpha^{\sigma}_{i}+\alpha^{\sigma}_{\bar{\imath}}\rangle\in{\mathbb{Z}}_{\geq
1}$. Otherwise it is $\Lambda_{\infty}$ for all $c\neq 0$, and
${\mathsf{s}}^{\sigma}_{i}\cdot\Lambda_{\infty}$ for $c=0$.

###### Proof.

First let us show that $\bm{y}^{(i,\sigma)}(x;c)$ represents a critical point
for all but finitely many $c\in{\mathbb{C}}$. As in the proof of Theorem 4.6,
we first observe that $\bm{y}^{(i,\sigma)}$ is indeed the result of generating
in each of the directions $i,\sigma i,\dots,\sigma^{M_{i}/2-1}i$ (in any
order). By Proposition 4.18 it is enough to check that
$\bm{y}^{(i,{\bar{\imath}},i)}(c)$ is generic for all but finitely many
$c\in{\mathbb{C}}$. This follows from (4.21) and Lemma 4.22, below.

The statements about the weight at infinity follow from Lemma 4.19 and Section
2.3.

Finally we must check that $\bm{y}^{(i,\sigma)}(x;c)$ represents a cyclotomic
point. Given Lemma 4.5 and the definition (4.20), it is enough to check that

$\displaystyle y^{(i,{\bar{\imath}},i)}_{\bar{\imath}}(-x;c)=(-1)^{\deg
y^{(i,{\bar{\imath}},i)}_{i}}y^{(i,{\bar{\imath}},i)}_{i}(x;c).$ (4.21)

This is effectively a statement about the case of type $A_{2}$ and we are in
the setting of Section 5 below, with $R=2n$, $n=1$, $p=1$. The statement
(4.21) follows from Theorem 5.34 and Lemma 5.36. ∎

######  Lemma 4.21.

We have

$\displaystyle y^{({\bar{\imath}},i,{\bar{\imath}})}_{j}(-x;c)=(-1)^{\deg
y^{(i,{\bar{\imath}},i)}_{\bar{\jmath}}}y^{(i,{\bar{\imath}},i)}_{\bar{\jmath}}(x;-c)$

for all $j\in I$.

###### Proof.

Note first that from (4.12) we have

$\displaystyle
T_{\bar{\jmath}}(-x)=(-1)^{\big{\langle}\sum\limits_{k=0}^{M-1}\sum\limits_{s=1}^{N}\omega^{k}\Lambda_{s},\alpha^{\vee}_{j}\big{\rangle}}T_{j}(x)$

for all $j\in I$. It follows that

$\displaystyle y^{({\bar{\imath}})}_{\bar{\imath}}(-x)=(-1)^{\deg
y_{i}+\langle\Lambda_{\infty}-\Lambda_{0},\alpha^{\vee}_{i}\rangle}y^{(i)}_{i}(x).$

Then, from the definition of $y^{({\bar{\imath}},i)}_{\bar{\imath}}(x;c)$ and
(4.19) we have that

$\displaystyle y^{(i,{\bar{\imath}})}_{i}(-x;c)=(-1)^{\deg
y^{(i,{\bar{\imath}})}_{\bar{\imath}}}y^{({\bar{\imath}},i)}_{\bar{\imath}}(x;\bar{c})$

if and only if $c$ and $\bar{c}$ are related by
$c=(-1)^{2+\langle\Lambda_{\infty},\alpha^{\vee}_{\bar{\imath}}+\alpha^{\vee}_{i}\rangle}\bar{c}$.
Since the $\Lambda_{s}$, $s=1,\dots,N$, are integral, we have

$\displaystyle(-1)^{\langle\Lambda_{\infty},\alpha^{\vee}_{i}+\alpha^{\vee}_{\bar{\imath}}\rangle}=(-1)^{\langle\Lambda_{0},\alpha^{\vee}_{i}+\alpha^{\vee}_{\bar{\imath}}\rangle}=(-1)^{2\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}=-1,$

using $\sigma\Lambda_{0}=\Lambda_{0}$ and the property (4.1). ∎

######  Lemma 4.22.

For all but finitely many $c\in{\mathbb{C}}$,
$y^{(i,{\bar{\imath}},i)}_{\bar{\imath}}(x;c)$ and
$y^{(i,{\bar{\imath}},i)}_{\bar{\imath}}(-x;c)$ have no root in common.

###### Proof.

Recall
$y^{(i,{\bar{\imath}},i)}_{\bar{\imath}}(x;c)=y^{({\bar{\imath}},i)}_{\bar{\imath}}(x;c)$.
Consider the leading behaviour in small $c$. As $c\to 0$, $\deg
y_{\bar{\imath}}$ of the roots of $y^{({\bar{\imath}},i)}_{\bar{\imath}}(x;c)$
tend to the $\deg y_{\bar{\imath}}$ roots of $y_{\bar{\imath}}(x)$. By the
assumption that $\bm{y}$ was generic and cyclotomic, none of these are roots
of $y_{\bar{\imath}}(-x)\simeq y_{i}(x)$.

Recall (4.19) and the fact that
$\langle\Lambda_{\infty}+\rho,\alpha^{\vee}_{i}+\alpha^{\vee}_{\bar{\imath}}\rangle$
is odd, by the assumption (4.3).

If
$\langle\Lambda_{\infty}+\rho,\alpha^{\vee}_{i}+\alpha^{\vee}_{\bar{\imath}}\rangle<0$,
then these are all the roots of $y^{({\bar{\imath}},i)}_{\bar{\imath}}(x;c)$.

If
$\langle\Lambda_{\infty}+\rho,\alpha^{\vee}_{i}+\alpha^{\vee}_{\bar{\imath}}\rangle\not<0$
then the remaining
$\langle\Lambda_{\infty}+\rho,\alpha^{\vee}_{i}+\alpha^{\vee}_{\bar{\imath}}\rangle>0$
roots of $y^{({\bar{\imath}},i)}_{\bar{\imath}}(x;c)$ tend to the roots of the
equation
$cx^{\langle\Lambda_{\infty}+\rho,\alpha^{\vee}_{i}+\alpha^{\vee}_{\bar{\imath}}\rangle}+1=0$.
This limiting set of roots multiplied by $-1$ does not intersect itself. This
implies the lemma. ∎

###  4.7 Definition of the cyclotomic population

Suppose $\bm{y}\in{\mathbb{P}}({\mathbb{C}}[x])^{|I|}$ is a tuple of
polynomials representing a cyclotomic critical point.

Recall the definition of $\bm{y}^{(i,\sigma)}(c)$, from Section 4.5 when
$L_{i}=1$ and from Section 4.6 when $L_{i}=2$. We say $\bm{y}^{(i,\sigma)}(c)$
is obtained from $\bm{y}$ by _cyclotomic generation_ in the direction $i$.

Let us define the _cyclotomic population_ originated at $\bm{y}$ to be the
Zariski closure of the set of all tuples of polynomials obtained from $\bm{y}$
by repeated cyclotomic generation, in all directions $i\in I$.

##  5 The case of type $\bm{A_{R}}$: vector spaces of quasi-polynomials

###  5.1 Type $\bm{A}$ data

Throughout this section we specialise to
${\mathfrak{g}}={\mathfrak{sl}}_{R+1}$. We shall treat in parallel the cases
where $R=2n-1$ and $R=2n$, $n\in{\mathbb{Z}}_{\geq 0}$. We have the usual
identification of ${\mathfrak{h}}\cong{\mathfrak{h}}^{*}$ with a subspace of
$(R+1)$-dimensional Euclidean space, given by
$\alpha_{i}=\alpha^{\vee}_{i}=\epsilon_{i+1}-\epsilon_{i}$, $i=1,\dots,R$,
where $(\epsilon_{i})_{i=1}^{R+1}$ is the standard orthonormal basis.

Let $\sigma\colon{\mathfrak{g}}\to{\mathfrak{g}}$ be the unique non-trivial
diagram automorphism, whose order is 2. The nodes $I$ of the Dynkin diagram,
and the action of $\sigma$ on these nodes, are as shown below:

$n-1$$\phantom{1}n\phantom{1}$$n+1$$2n-1$$1$
$n-1$$\phantom{1}n\phantom{1}$$n+1$$n+2$$\,\,2n\,\,$$1$

When $R=2n-1$, then $L_{i}=1$ for all $i\in I$, and
$M_{i}=\begin{cases}1,&i=n,\\\ 2,&i\neq n.\end{cases}$ When $R=2n$ then
$L_{i}=\begin{cases}2,&i=n,n+1\\\ 1,&\text{otherwise}\end{cases}$ and
$M_{i}=2$ for all $i\in I$.

Let $(z_{i})_{i=1}^{N}$ be nonzero points $z_{i}\in{\mathbb{C}}^{\times}$ such
that $z_{i}\pm z_{j}\neq 0$ whenever $i\neq j$. Let
$\Lambda_{1},\dots,\Lambda_{N}$ be dominant integral weights.

We suppose the weight at the origin, $\Lambda_{0}\in{\mathfrak{h}}^{*}$, obeys
$\sigma\Lambda_{0}=\Lambda_{0}$ (as always). That is,

$\displaystyle\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle=\langle\Lambda_{0},\alpha^{\vee}_{R+1-i}\rangle,\qquad
i=1,\dots,R.$

In addition, we pick and fix an integer $p\in\\{0,1,\dots,n\\}$, and suppose
that

$\displaystyle\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle\in
2{\mathbb{Z}}_{\geq 0}/M_{i}\qquad\text{if $i\notin\\{p,R+1-p\\}$}$ (5.1a) and
$\displaystyle\langle\Lambda_{0},\alpha^{\vee}_{p}\rangle\in\begin{cases}\frac{1}{2}(2{\mathbb{Z}}_{\geq
0}-1)=\\{-\frac{1}{2},\frac{1}{2},\frac{3}{2},\dots\\}&\text{if $p\leq
R/2$},\\\ 2{\mathbb{Z}}_{\geq 0}+1=\\{1,3,\dots\\}&\text{if $p=n$ and
$R=2n-1$.}\end{cases}$ (5.1b)

Note the following particular cases:

  * •

If $R=2n$ is even and $p=0$ then (5.1) just says that $\Lambda_{0}$ is
dominant integral.

  * •

If $R=2n-1$ is odd and $p=0$ then $\Lambda_{0}$ is dominant integral and
$\langle\Lambda_{0},\alpha^{\vee}_{n}\rangle$ is even.

  * •

If $R=2n-1$ is odd and $p=n$ then $\Lambda_{0}$ is dominant integral and
$\langle\Lambda_{0},\alpha^{\vee}_{n}\rangle$ is odd.

In the case $p=n$ (and any $R$) our choice of $\Lambda_{0}$ obeys the
assumptions set out in Section 4.1.

###  5.2 Vector spaces of quasi-polynomials

Let

$\displaystyle{\tilde{T}}_{i}(x)=x^{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}\prod_{s=1}^{N}(x-z_{s})^{\langle\Lambda_{s},\alpha^{\vee}_{i}\rangle}(x+z_{s})^{\langle\Lambda_{R+1-s},\alpha^{\vee}_{i}\rangle},\qquad
i\in I.$

Thus
${\tilde{T}}_{i}(x)=x^{\langle\Lambda_{0},\alpha^{\vee}_{i}\rangle}T_{i}(x)$
with $T_{i}(x)$ as in (4.5).

In view of (5.1), ${\tilde{T}}_{i}(x)\in{\mathbb{C}}[x]$ for all
$i\notin\\{p,R+1-p\\}$. If $0<p<R+1-p<R$ then ${\tilde{T}}_{p}(x)$ and
${\tilde{T}}_{R+1-p}(x)$ belong to $x^{-\frac{1}{2}}{\mathbb{C}}[x]$. If
$p=R+1-p$ then ${\tilde{T}}_{p}(x)\in{\mathbb{C}}[x]$.

We define the _degree_ , $\deg p$, of a Laurent polynomial
$p(x)\in{\mathbb{C}}[x^{\pm\frac{1}{2}}]$ to be the leading power of $x$ (for
large $x$) that appears in $p(x)$ with non-zero coefficient.

We will call any polynomial in $x^{\frac{1}{2}}$ a _quasi-polynomial_.

A vector space $V\subset{\mathbb{C}}\big{[}x^{\frac{1}{2}}\big{]}$ of quasi-
polynomials is _decomposable_ if

$\displaystyle V=V\cap{\mathbb{C}}[x]\oplus V\cap
x^{\frac{1}{2}}{\mathbb{C}}[x].$

A tuple of quasi-polynomials is _decomposable_ if each element lies in either
${\mathbb{C}}[x]$ or $x^{\frac{1}{2}}{\mathbb{C}}[x]$. In particular, a
_decomposable basis_ of a decomposable vector space
$V\subset{\mathbb{C}}\big{[}x^{\frac{1}{2}}\big{]}$ is one in which each basis
vector lies in either ${\mathbb{C}}[x]$ or $x^{\frac{1}{2}}{\mathbb{C}}[x]$.

Define the _divided Wronksian determinant_ of quasi-polynomials
$u_{1},\dots,u_{k}\in{\mathbb{C}}\big{[}x^{\frac{1}{2}}\big{]}$ by

$\displaystyle\operatorname{Wr}^{\dagger}(u_{1},\dots,u_{k}):=\frac{\operatorname{Wr}(u_{1},\dots,u_{k})}{{\tilde{T}}_{1}^{k-1}{\tilde{T}}_{2}^{k-2}\cdots{\tilde{T}}_{k-1}},\qquad\operatorname{Wr}(u_{1},\dots,u_{k}):=\det\left(\frac{d^{j-1}u_{i}}{dx^{j-1}}\right)_{i,j=1}^{k},$

for $k=1,\dots,R+1$.

Define

$\displaystyle\Lambda:=\Lambda_{0}+\sum_{s=1}^{N}(\Lambda_{s}+\sigma\Lambda_{s}),$
(5.2)

and suppose $\tilde{\Lambda}_{\infty}\in{\mathfrak{h}}^{*}$ is a dominant
weight such that $\Lambda-\tilde{\Lambda}_{\infty}=\sum\limits_{i\in
I}k_{i}\alpha_{i}$ for some $k_{i}\in{\mathbb{Z}}_{\geq 0}$. Such a weight
defines numbers $d_{1},\dots,d_{R+1}\in{\mathbb{Z}}/2$, $0\leq
d_{1}<\dots<d_{R+1}$, by

$\displaystyle
d_{1}:=\langle\Lambda-\tilde{\Lambda}_{\infty},\epsilon_{1}\rangle,\qquad
d_{k}:=\langle\Lambda-({\mathsf{s}}_{1}\cdots{\mathsf{s}}_{k-1})\cdot\tilde{\Lambda}_{\infty},\epsilon_{1}\rangle,\qquad
k=2,\dots,R+1.$ (5.3)

######  Lemma 5.1.

We have

$\displaystyle
d_{k}=d_{1}+\langle\tilde{\Lambda}_{\infty}+\rho,\alpha^{\vee}_{1}+\dots+\alpha^{\vee}_{k-1}\rangle,\qquad
k=2,\dots,R+1.$ (5.4)

Hence, for all $p>0$, $d_{1},\dots,d_{p}$ and $d_{R+2-p},\dots,d_{R+1}$ are
integers while $d_{p+1},\dots,d_{R+1-p}$ are half odd integers, i.e., have the
form $m+\frac{1}{2}$ for $m\in{\mathbb{Z}}$. If $p=0$ then
$d_{1},\dots,d_{R+1}$ are all integers.

###### Proof.

We have
$d_{k}-d_{1}=\langle\tilde{\Lambda}_{\infty}+\rho-({\mathsf{s}}_{1}\cdots{\mathsf{s}}_{k-1})(\tilde{\Lambda}_{\infty}+\rho),\epsilon_{1}\rangle=\langle\tilde{\Lambda}_{\infty}+\rho,\epsilon_{1}-({\mathsf{s}}_{k-1}\cdots{\mathsf{s}}_{1})\epsilon_{1}\rangle=\langle\tilde{\Lambda}_{\infty}+\rho,\epsilon_{1}-\epsilon_{k}\rangle$
and hence (5.4). ∎

######  Definition 5.2.

We say a vector space of quasi-polynomials
${\mathcal{K}}\\!\subset\\!{\mathbb{C}}\big{[}x^{\frac{1}{2}}\big{]}$ has
_frame_ $\tilde{T}_{1},\dots,\tilde{T}_{R};\tilde{\Lambda}_{\infty}$ if the
following conditions hold:

  1. (i)

There is a basis $(u_{k}(x))_{k=1}^{R+1}$ of ${\mathcal{K}}$ such that $\deg
u_{k}=d_{k}$ for each $k=1,\dots,R+1$.

  2. (ii)

For any $z\in{\mathbb{C}}{\setminus}\\{0\\}$ and
$v_{1},\dots,v_{k}\in{\mathcal{K}}$, $k=1,\dots,R+1$, the divided Wronskian
$\operatorname{Wr}^{\dagger}(v_{1}$, $\dots,v_{k})$ is regular at $z$, and
moreover, $\operatorname{Wr}^{\dagger}(v_{1},\dots,v_{k})$ is nonzero at $z$
for suitable $v_{1},\dots,v_{k}$.

  3. (iii)

For all $v_{1},\dots,v_{k}\in{\mathcal{K}}$, $k=1,\dots,R+1$, the divided
Wronskian $\operatorname{Wr}^{\dagger}(v_{1},\dots,v_{k})$ has at $x=0$ an
expansion of the form $\sum\limits_{m\in{\mathbb{Z}}_{\geq 0}/2}a_{m}x^{m}$
and moreover this expansion has nonzero $a_{0}$ for suitable
$v_{1},\dots,v_{k}$.

_In the remainder of this section, ${\mathcal{K}}$ will denote a decomposable
vector space of quasi-polynomials with frame
$\tilde{T}_{1},\dots,\tilde{T}_{R};\tilde{\Lambda}_{\infty}$._

Conditions (ii) and (iii) specify the _ramification conditions_ of
${\mathcal{K}}$ at every point $z\in{\mathbb{C}}$. Condition (i) specifies the
ramification conditions at $\infty$. See [15, Section 5.5]. The degrees $0\leq
d_{1}<d_{2}<\dots<d_{R+1}$ will be called the _exponents of ${\mathcal{K}}$ at
infinity_.

Note that conditions (ii) and (iii) together imply in particular that
${\mathcal{K}}$ has no  _base points_. That is, there is no $z\in{\mathbb{C}}$
such that $u(z)=0$ for all $u\in{\mathcal{K}}$. They also imply the following
important lemma.

######  Lemma 5.3.

For all $v_{1},\dots,v_{k}\in{\mathcal{K}}$, $k=1,\dots,R+1$, the divided
Wronskian $\operatorname{Wr}^{\dagger}(v_{1},\dots,v_{k})$ is a quasi-
polynomial.

Since ${\mathcal{K}}$ is decomposable it follows from condition (i) that
${\mathcal{K}}$ admits a decomposable basis $(u_{k})_{k=1}^{R+1}$ such that
$\deg u_{k}=d_{k}$ for each $k$. We call any such basis a _special_ basis.

######  Lemma 5.4.

Any two special bases $(u_{k})_{k=1}^{R+1}$ and $(u^{\prime}_{k})_{k=1}^{R+1}$
are related by a triangular change of basis,
$u^{\prime}_{k}=\sum\limits_{j\leq k}a_{kj}u_{j}$, such that $a_{kj}=0$
whenever $d_{k}-d_{j}\notin{\mathbb{Z}}$.

######  Lemma 5.5.

Let $m\in{\mathbb{Z}}_{\geq 1}$. Let $n_{1},\dots,n_{m}$ be non-negative
integers. Then

$\displaystyle\operatorname{Wr}(x^{n_{1}},\dots,x^{n_{m}})=x^{\sum\limits_{i=1}^{m}n_{i}-\frac{m(m-1)}{2}}\prod_{1\leq
j<i\leq m}(n_{i}-n_{j}).$

######  Lemma 5.6.

Let $(u_{i}(x))_{i=1}^{R+1}$ be a special basis of ${\mathcal{K}}$. Then

$\displaystyle\operatorname{Wr}^{\dagger}(u_{1},\dots,u_{R+1})=\prod_{1\leq
j<i\leq R+1}(d_{i}-d_{j}).$

###### Proof.

By Lemma 5.3,
$\operatorname{Wr}^{\dagger}(u_{1},\dots,u_{R+1})\in{\mathbb{C}}\big{[}x^{\frac{1}{2}}\big{]}$.
We must show that it has degree zero and compute the constant term. From the
condition that $\Lambda-\tilde{\Lambda}_{\infty}\in{\mathbb{Z}}_{\geq
0}[\alpha_{i}]_{i\in I}$ it follows that
$0=\langle\Lambda-\tilde{\Lambda}_{\infty},-(R+1)\epsilon_{1}+R\alpha^{\vee}_{1}+(R-1)\alpha^{\vee}_{2}+\dots+2\alpha^{\vee}_{R-1}+\alpha^{\vee}_{R}\rangle$
and therefore

$\displaystyle(R+1)d_{1}=\langle\Lambda-\tilde{\Lambda}_{\infty},R\alpha^{\vee}_{1}+(R-1)\alpha^{\vee}_{2}+\dots+2\alpha^{\vee}_{R-1}+\alpha^{\vee}_{R}\rangle.$
(5.5)

Then (5.4) implies

$\displaystyle\sum_{i=1}^{R+1}d_{i}-\frac{(R+1)R}{2}=\langle\Lambda,R\alpha^{\vee}_{1}+(R-1)\alpha^{\vee}_{2}+\dots+2\alpha^{\vee}_{R-1}+\alpha^{\vee}_{R}\rangle.$
(5.6)

The result follows by Lemma 5.5. ∎

######  Corollary 5.7.

$\operatorname{Wr}^{\dagger}(v_{1},\dots,v_{R+1})$ is a constant
$($independent of $x)$ for all $v_{1},\dots,v_{R+1}\in{\mathcal{K}}$.

Let $(u_{k})_{k=1}^{R+1}$ be a special basis of ${\mathcal{K}}$. Introduce the
subspaces

$\displaystyle{\mathcal{K}}_{\textup{Sp}}:=\operatorname{span}_{\mathbb{C}}(u_{1},\dots,u_{p})\oplus\operatorname{span}_{\mathbb{C}}(u_{R+2-p},\dots,u_{R+1}),$
$\displaystyle{\mathcal{K}}_{\textup{O}}:=\operatorname{span}_{\mathbb{C}}(u_{p+1},\dots,u_{R+1-p}),$

so that

$\displaystyle{\mathcal{K}}={\mathcal{K}}_{\textup{Sp}}\oplus{\mathcal{K}}_{\textup{O}}.$

By Lemma 5.4, these definitions of do not depend on the choice of special
basis $(u_{k})_{k=1}^{R+1}$. By Lemma 5.1 we have that, whenever $p>0$, then

$\displaystyle{\mathcal{K}}_{\textup{Sp}}={\mathcal{K}}\cap{\mathbb{C}}[x],\qquad{\mathcal{K}}_{\textup{O}}={\mathcal{K}}\cap
x^{\frac{1}{2}}{\mathbb{C}}[x].$ (5.7)

Exceptionally, when $p=0$, we have ${\mathcal{K}}_{\textup{Sp}}=\\{0\\}$,
${\mathcal{K}}_{\textup{O}}={\mathcal{K}}\subset{\mathbb{C}}[x]$.

Given a decomposable subspace $V$, we write $\operatorname{sdim}V$ for the
pair of numbers

$\displaystyle\operatorname{sdim}V:=(\dim
V\cap{\mathcal{K}}_{\textup{Sp}}|\dim V\cap{\mathcal{K}}_{\textup{O}}).$

###  5.3 Flags in $\bm{{\mathcal{K}}}$

Let $F\\!L({\mathcal{K}})$ denote the space of full (i.e., $R+1$-step) flags
in ${\mathcal{K}}$.

We say an $r$-step flag ${\mathcal{F}}=\\{0=F_{0}\subset F_{1}\subset
F_{2}\subset\dots\subset F_{r}={\mathcal{K}}\\}$ in ${\mathcal{K}}$ is
_decomposable_ if each $F_{k}$ is decomposable.

The space of decomposable full flags in ${\mathcal{K}}$ has $\binom{R+1}{2p}$
connected components. These connected components are labeled by $2p$-element
subsets $Q\subset\\{1,\dots,R+1\\}$. Define $F\\!L_{Q}({\mathcal{K}})$ to be
the subset consisting of the flags ${\mathcal{F}}=\\{0=F_{0}\subset
F_{1}\subset\dots\subset F_{R+1}={\mathcal{K}}\\}$ such that for each $k$,

$\displaystyle\operatorname{sdim}F_{k}-\operatorname{sdim}F_{k-1}=\begin{cases}(1|0)&\text{if}\quad
k\in Q,\\\ (0|1)&\text{if}\quad k\notin Q.\end{cases}$

We call elements of $F\\!L_{Q}({\mathcal{K}})$ _flags of type $Q$_.

For each $Q$ the variety $F\\!L_{Q}({\mathcal{K}})$ is isomorphic to the
direct product of full flag spaces $F\\!L({\mathcal{K}}_{\textup{Sp}})\times
F\\!L({\mathcal{K}}_{\textup{O}})$. The isomorphism

$\displaystyle\eta_{Q}\colon\ F\\!L({\mathcal{K}}_{\textup{Sp}})\times
F\\!L({\mathcal{K}}_{\textup{O}})\to F\\!L_{Q}({\mathcal{K}})$ (5.8)

sends a pair of flags $F_{1,+}\subset\dots\subset F_{2p,+}$,
$F_{1,-}\subset\dots\subset F_{R+1-2p,-}$ to the flag
$F_{1}\subset\dots\subset F_{R}$, where $F_{k}=F_{k_{1},+}\oplus F_{k_{2},-}$,
$k_{1}=|Q\cap\\{1,\dots,k\\}|$, $k_{2}=k-k_{1}$.

Call a $2p$-element subset $Q\subset\\{1,\dots,R+1\\}$ _symmetric_ if $Q$ is
invariant with respect to the involution $k\mapsto R+2-k$. In particular, the
following subset $S$ is symmetric

$\displaystyle S:=\\{1,\dots,p,R+2-p,\dots,R+1\\}.$ (5.9)

If $(u_{k})_{k=1}^{R+1}$ is a special basis of ${\mathcal{K}}$ then the full
flag ${\mathcal{F}}=\\{0=F_{0}\subset F_{1}\subset F_{2}\subset\dots\subset
F_{R+1}={\mathcal{K}}\\}$ defined by

$\displaystyle
F_{k}=\operatorname{span}_{\mathbb{C}}(u_{1},\dots,u_{k}),\qquad
k=1,\dots,R+1,$ (5.10)

belongs to $F\\!L_{S}({\mathcal{K}})$. By Lemma 5.4 this flag is independent
of the choice of special basis.

To any full flag ${\mathcal{F}}=\\{0\subset F_{1}\subset
F_{2}\subset\dots\subset F_{R+1}={\mathcal{K}}\\}$ in $F\\!L({\mathcal{K}})$
one can associate a tuple
$\bm{y}^{\mathcal{F}}=(y_{i}(x))_{i=1}^{R}\in{\mathbb{P}}\big{(}{\mathbb{C}}\big{[}x^{\frac{1}{2}}\big{]}\big{)}^{R}$.
Namely, let $(u^{\mathcal{F}}_{k}(x))_{k=1}^{R+1}$ be any basis of
${\mathcal{K}}$ such that

$\displaystyle
F_{k}=\operatorname{span}_{\mathbb{C}}\big{(}u^{\mathcal{F}}_{1},\dots,u^{\mathcal{F}}_{k}\big{)},\qquad
k=1,\dots,R+1.$

(we say such a basis is _adjusted to_ ${\mathcal{F}}$) and then let

$\displaystyle
y^{\mathcal{F}}_{k}:=\operatorname{Wr}^{\dagger}\big{(}u^{\mathcal{F}}_{1},\dots,u^{\mathcal{F}}_{k}\big{)},\qquad
k=1,\dots,R.$ (5.11)

By Lemma 5.3, these are quasi-polynomials.

We have the shifted action of the Weyl group of type $A_{R}$ on weights as in
Section 2.1. The weight at infinity $\Lambda_{\infty}(\bm{y}^{\mathcal{F}})$,
as in (4.10), belongs to the shifted Weyl orbit of $\tilde{\Lambda}_{\infty}$
[15, Section 3.6]. It is equal to $\tilde{\Lambda}_{\infty}$ if and only if
${\mathcal{F}}$ is the flag given in (5.10).

The map ${\mathcal{F}}\mapsto\bm{y}^{\mathcal{F}}$ defines a morphism of
varieties,

$\displaystyle\beta\colon\
F\\!L({\mathcal{K}})\to{\mathbb{P}}\big{(}{\mathbb{C}}\big{[}x^{1/2}\big{]}\big{)}^{R}.$

This morphism $\beta$ defines an isomorphism of $F\\!L({\mathcal{K}})$ onto
its image, as in Lemmas 5.14–5.16 of [15].

######  Lemma 5.8.

The image $\beta(F\\!L_{S}({\mathcal{K}}))$ of the variety of flags of type
$S$ lies in ${\mathbb{P}}({\mathbb{C}}[x])^{R}$, i.e., consists of tuples of
polynomials.

###### Proof.

In the exceptional case $p=0$ no fractional powers are present at all and the
result is clear. Suppose $p>0$. Let ${\mathcal{F}}\in
F\\!L_{S}({\mathcal{K}})$ and let $(u_{k}^{\mathcal{F}})_{k=1}^{R+1}$ be a
basis of ${\mathcal{K}}$ adjusted to ${\mathcal{F}}$. By inspection one sees
that because ${\mathcal{F}}\in F\\!L_{S}({\mathcal{K}})$,
$\operatorname{Wr}(u^{\mathcal{F}}_{1},\dots,u^{\mathcal{F}}_{k})$ lies in
${\mathbb{C}}[x]$ (resp. $x^{\frac{1}{2}}{\mathbb{C}}[x]$) for precisely those
$k$ such that the product ${\tilde{T}}_{1}^{k-1}\cdots{\tilde{T}}_{k-1}$ lies
in ${\mathbb{C}}[x^{\pm 1}]$ (resp. $x^{\frac{1}{2}}{\mathbb{C}}[x^{\pm 1}]$).
For each $k$, Lemma 5.3 guarantees that
$y^{\mathcal{F}}_{k}\in{\mathbb{C}}\big{[}x^{\frac{1}{2}}\big{]}$. Hence in
fact $y^{\mathcal{F}}_{k}\in{\mathbb{C}}[x]$. ∎

######  Lemma 5.9.

The tuple $\beta({\mathcal{F}})=\bm{y}^{\mathcal{F}}$ is decomposable if and
only if ${\mathcal{F}}$ is a decomposable flag. If ${\mathcal{F}}$ is a
decomposable flag of type $Q$ then

$\displaystyle
y_{k}^{\mathcal{F}}\in\begin{cases}\phantom{x^{\frac{1}{2}}}{\mathbb{C}}[x]&\text{if}\quad\left|S\triangle
Q\cap\\{1,\dots,k\\}\right|\in 2{\mathbb{Z}},\\\
x^{\frac{1}{2}}{\mathbb{C}}[x]&\text{if}\quad\left|S\triangle
Q\cap\\{1,\dots,k\\}\right|\in 2{\mathbb{Z}}+1,\end{cases}$

where $S\triangle Q:=(S{\setminus}Q)\cup(Q{\setminus}S)$ denotes the symmetric
difference of $S$ and $Q$. In particular $\bm{y}^{\mathcal{F}}$ is a tuple of
polynomials if and only if $Q=S$.

###  5.4 Fundamental differential operator and the recovery theorem

To any given a tuple
$\bm{y}=(y_{i}(x))_{i=1}^{R}\in{\mathbb{P}}\big{(}{\mathbb{C}}\big{[}x^{\frac{1}{2}}\big{]}\big{)}^{R}$
of quasi-polynomials, we may associate a differential operator
${\mathcal{D}}(\bm{y})$, defined by

$\displaystyle{\mathcal{D}}(\bm{y}):=\left(\partial-\log^{\prime}\frac{{\tilde{T}}_{1}{\tilde{T}}_{2}\cdots{\tilde{T}}_{R}}{y_{R}}\right)\left(\partial-\log^{\prime}\frac{y_{R}{\tilde{T}}_{1}{\tilde{T}}_{2}\cdots{\tilde{T}}_{R-1}}{y_{R-1}}\right)\cdots$
$\displaystyle\hphantom{{\mathcal{D}}(\bm{y})=}{}\times\left(\partial-\log^{\prime}\frac{y_{2}{\tilde{T}}_{1}}{y_{1}}\right)\big{(}\partial-\log^{\prime}y_{1}\big{)}=\overset{\longrightarrow}{\prod_{i=0}^{R}}\left(\partial-\log^{\prime}\frac{y_{R+1-i}\prod\limits_{j=1}^{R-i}{\tilde{T}}_{j}}{y_{R-i}}\right),$

with the understanding that $y_{0}=y_{R+1}=1$. Here
$\partial:=\partial/\partial x$ and $\log^{\prime}f:=f^{\prime}(x)/f(x)$.

######  Theorem 5.10 ([15, Lemma 5.6]).

Let $\bm{y}\in\beta(F\\!L({\mathcal{K}}))$. Then
${\mathcal{K}}=\ker{\mathcal{D}}$.

###  5.5 The dual space $\bm{{\mathcal{K}}^{\dagger}}$

Let ${\mathcal{K}}^{\dagger}$ be the complex vector space

$\displaystyle{\mathcal{K}}^{\dagger}:=\operatorname{span}_{\mathbb{C}}\big{\\{}{\operatorname{Wr}}^{\dagger}(v_{1},\dots,v_{R})\colon
v_{1},\dots,v_{R}\in{\mathcal{K}}\big{\\}}\subset{\mathbb{C}}\big{[}x^{\frac{1}{2}}\big{]}.$

The space ${\mathcal{K}}^{\dagger}$ is a space of quasi-polynomials by Lemma
5.3. The spaces ${\mathcal{K}}^{\dagger}$ and ${\mathcal{K}}$ are dual with
respect to the pairing

$\displaystyle(\cdot,\cdot)\colon\
{\mathcal{K}}^{\dagger}\times{\mathcal{K}}\to{\mathbb{C}}$

defined by

$\displaystyle\big{(}v_{1},\operatorname{Wr}^{\dagger}(v_{2},\dots,v_{R+1})\big{)}:=\operatorname{Wr}^{\dagger}(v_{1},v_{2},\dots,v_{R+1}).$

Given any basis $(u_{i}(x))_{i=1}^{R+1}$ of ${\mathcal{K}}$ there is a basis
$(W_{i}(x))_{i=1}^{R+1}$ of ${\mathcal{K}}^{\dagger}$ defined by

$\displaystyle
W_{i}:=\operatorname{Wr}^{\dagger}(u_{1},\dots,\widehat{u}_{i},\dots,u_{R+1})\in{\mathcal{K}}^{\dagger},\qquad
i=1,\dots,R+1,$

where $\widehat{u}_{i}$ denotes omission. We have

$\displaystyle(u_{i},W_{j})=0\qquad\text{if}\quad i\neq
j,\qquad(u_{i},W_{i})\neq 0.$

Let $d^{\dagger}_{1}>\dots>d^{\dagger}_{R+1}$ be the numbers given by

$\displaystyle
d^{\dagger}_{R+1}:=-\langle\Lambda-\tilde{\Lambda}_{\infty},\epsilon_{R+1}\rangle,\qquad
d^{\dagger}_{k}:=-\langle\Lambda-({\mathsf{s}}_{R}\cdots{\mathsf{s}}_{k})\cdot\tilde{\Lambda}_{\infty},\epsilon_{R+1}\rangle,\qquad
k=1,\dots,R,$

cf. (5.3). We have

$\displaystyle
d^{\dagger}_{k}=d^{\dagger}_{R+1}+\langle\tilde{\Lambda}_{\infty}+\rho,\alpha^{\vee}_{k}+\dots+\alpha^{\vee}_{R}\rangle,\qquad
k=1,\dots,R,$ (5.12)

by an argument as for Lemma 5.1.

######  Lemma 5.11.

Let $(u_{i}(x))_{i=1}^{R+1}$ be a special basis of ${\mathcal{K}}$. Then $\deg
W_{k}=d^{\dagger}_{k}$, $k=1,\dots,R+1$, and the basis $(W_{k})_{k=1}^{R+1}$
is decomposable.

###### Proof.

From $\Lambda-\tilde{\Lambda}_{\infty}\in{\mathbb{Z}}_{\geq
0}[\alpha_{i}]_{i\in I}$ we have
$0=\langle\Lambda-\tilde{\Lambda}_{\infty},(R+1)\epsilon_{R+1}+\alpha^{\vee}_{1}+2\alpha^{\vee}_{2}+\dots+(R-1)\alpha^{\vee}_{R-1}+R\alpha^{\vee}_{R}\rangle$,
and hence

$\displaystyle(R+1)d^{\dagger}_{R+1}=\langle\Lambda-\tilde{\Lambda}_{\infty},\alpha^{\vee}_{1}+2\alpha^{\vee}_{2}+\dots+R\alpha^{\vee}_{R}\rangle.$
(5.13)

Now

$\displaystyle\deg
W_{R+1}=\deg\operatorname{Wr}^{\dagger}(u_{1},\dots,u_{R})=\sum_{i=1}^{R}d_{i}-\frac{R(R-1)}{2}-\langle\Lambda,(R-1)\alpha^{\vee}_{1}+\dots+\alpha^{\vee}_{R-1}\rangle$
$\displaystyle\hphantom{\deg
W_{R+1}}{}=Rd_{1}+\langle\tilde{\Lambda}_{\infty}+\rho,(R-1)\alpha^{\vee}_{1}+\dots+\alpha^{\vee}_{R-1}\rangle-\frac{R(R-1)}{2}$
$\displaystyle\hphantom{\deg
W_{R+1}=}{}-\langle\Lambda,(R-1)\alpha^{\vee}_{1}+\dots+\alpha^{\vee}_{R-1}\rangle$
$\displaystyle\hphantom{\deg
W_{R+1}}{}=Rd_{1}+\langle\tilde{\Lambda}_{\infty}-\Lambda,(R-1)\alpha^{\vee}_{1}+\dots+\alpha^{\vee}_{R-1}\rangle,$

where we used (5.4). Hence, using (5.5), we have

$\displaystyle\begin{split}&(R+1)\deg
W_{R+1}=R\langle\Lambda-\tilde{\Lambda}_{\infty},R\alpha^{\vee}_{1}+(R-1)\alpha^{\vee}_{2}+\dots+2\alpha^{\vee}_{R-1}+\alpha^{\vee}_{R}\rangle\\\
&\hphantom{(R+1)\deg
W_{R+1}=}{}-(R+1)\langle\Lambda-\tilde{\Lambda}_{\infty},(R-1)\alpha^{\vee}_{1}+\dots+\alpha^{\vee}_{R-1}\rangle\\\
&\hphantom{(R+1)\deg
W_{R+1}}{}=\langle\Lambda-\tilde{\Lambda}_{\infty},\alpha^{\vee}_{1}+2\alpha^{\vee}_{2}+\dots+R\alpha^{\vee}_{R}\rangle\end{split}$

since $R(R+1-k)-(R+1)(R-k)=k$. Comparing this with (5.13) we see that
$d^{\dagger}_{R+1}=\deg W_{R+1}$. Then for the remaining $W_{k}$, we note that
$\deg W_{R+1}-\deg W_{k}=d_{k}-d_{R+1}$ for $k=1,\dots,R$. And by (5.4) and
(5.12),

$\displaystyle
d_{k}-d_{R+1}=-\langle\tilde{\Lambda}_{\infty}+\rho,\alpha^{\vee}_{k}+\dots+\alpha^{\vee}_{R}\rangle=d^{\dagger}_{R+1}-d^{\dagger}_{k}.$

Thus $d^{\dagger}_{k}=\deg W_{k}$ for $k=1,\dots,R+1$. Finally, since the
basis $(u_{k})_{k=1}^{R+1}$ is decomposable and each ${\tilde{T}}_{k}$ lies in
either ${\mathbb{C}}[x^{\pm 1}]$ or $x^{\frac{1}{2}}{\mathbb{C}}[x^{\pm 1}]$,
it follows that $(W_{k})_{k=1}^{R+1}$ is decomposable. ∎

###  5.6 Cyclotomic points and cyclotomic self-duality

Let us fix $(-1)^{m}:=e^{m\pi i}$ for $m\in{\mathbb{Z}}/2$. Then given a
monomial $q(x)=x^{m}$, $m\in{\mathbb{Z}}/2$, we define $q(-x):=(-1)^{m}x^{m}$.
We extend the transformation $q(x)\mapsto q(-x)$ to Laurent polynomials in
$x^{\frac{1}{2}}$ by linearity.

We say that ${\mathcal{K}}$ is _cyclotomically self-dual_ if

$\displaystyle u(x)\in{\mathcal{K}}\ \Leftrightarrow\
u(-x)\in{\mathcal{K}}^{\dagger}.$

######  Lemma 5.12.

If ${\mathcal{K}}$ is cyclotomically self-dual then

$\displaystyle
d_{k}+d_{R+2-k}=R+\langle\Lambda,\alpha^{\vee}_{1}+\dots+\alpha^{\vee}_{R}\rangle,\qquad
k=1,\dots,R+1.$

###### Proof.

If ${\mathcal{K}}$ is cyclotomically self-dual then we must have
$d_{k}=d^{\dagger}_{R+2-k}$, $k=1,\dots,R+1$. Comparing (5.4) and (5.12) we
see that this implies that

$\displaystyle\langle\tilde{\Lambda}_{\infty}+\rho,\alpha^{\vee}_{1}+\dots\alpha^{\vee}_{k}\rangle=d_{k+1}-d_{1}=d^{\dagger}_{R+1-k}-d^{\dagger}_{R+1}$
$\displaystyle\hphantom{\langle\tilde{\Lambda}_{\infty}+\rho,\alpha^{\vee}_{1}+\dots\alpha^{\vee}_{k}\rangle}{}=\langle\tilde{\Lambda}_{\infty}+\rho,\alpha^{\vee}_{R+1-k}+\dots+\alpha^{\vee}_{R}\rangle,\qquad
k=1,\dots,R,$

and hence

$\displaystyle\langle\tilde{\Lambda}_{\infty},\alpha^{\vee}_{k}\rangle=\langle\tilde{\Lambda}_{\infty},\alpha^{\vee}_{R+1-k}\rangle,\qquad
k=1,\dots,R.$

Therefore

$\displaystyle
d_{k}+d_{R+2-k}=2d_{1}+\langle\tilde{\Lambda}_{\infty}+\rho,\alpha^{\vee}_{1}+\dots+\alpha^{\vee}_{R}\rangle,\qquad
k=1,\dots,R+1,$

and so, because the right-hand side here does not depend on $k$,

$\displaystyle d_{k}+d_{R+2-k}=\frac{2}{R+1}\sum_{j=1}^{R+1}d_{j},\qquad
k=1,\dots,R+1.$ (5.14)

Recall (5.6) and the definition (5.2) of $\Lambda$. Using now the fact that
$\langle\Lambda,\alpha^{\vee}_{i}\rangle=\langle\Lambda,\alpha^{\vee}_{R+1-i}\rangle$,
$i=1,\dots,R$, we have

$\displaystyle\sum_{j=1}^{R+1}d_{j}-\frac{(R+1)R}{2}=\frac{R+1}{2}\langle\Lambda,\alpha^{\vee}_{1}+\dots+\alpha^{\vee}_{R}\rangle.$

Thus, given (5.14), we have the result. ∎

If ${\mathcal{K}}$ is cyclotomically self-dual then there is a non-degenerate
bilinear form $B$ on ${\mathcal{K}}$ defined by

$\displaystyle B(u(x),v(x)):=(u(x),v(-x)),$

i.e.,

$\displaystyle
B(u,v)=\operatorname{Wr}^{\dagger}(u,v_{1},\dots,v_{R}),\qquad\text{where}\quad
v(-x)=\operatorname{Wr}^{\dagger}(v_{1},\dots,v_{R}).$

Let us call a tuple of quasi-polynomials
$\bm{y}\in{\mathbb{P}}\big{(}{\mathbb{C}}\big{[}x^{\frac{1}{2}}\big{]}\big{)}^{R}$
_cyclotomic_ if

$\displaystyle y_{k}(-x)\simeq y_{R+1-k}(x),\qquad k=1,\dots,R.$

######  Proposition 5.13.

Let ${\mathcal{F}}\in F\\!L({\mathcal{K}})$. If the tuple
$\beta({\mathcal{F}})\in{\mathbb{P}}\big{(}{\mathbb{C}}\big{[}x^{\frac{1}{2}}\big{]}\big{)}^{R}$
is cyclotomic then ${\mathcal{F}}$ is a decomposable flag.

###### Proof.

Let $\bm{y}^{\mathcal{F}}=\beta({\mathcal{F}})$. To prove that ${\mathcal{F}}$
is decomposable it is enough to show that each entry $y^{\mathcal{F}}_{k}$ of
this tuple lies in ${\mathbb{C}}[x]$ or in $x^{1/2}{\mathbb{C}}[x]$. For each
$k=1,\dots,R$ we have
$y^{\mathcal{F}}_{k}(x)=x^{\frac{1}{2}}a_{k}(x)+b_{k}(x)$ for some polynomials
$a_{k}(x)$ and $b_{k}(x)$ in $x$. If $\bm{y}^{\mathcal{F}}$ is cyclotomic then
$y_{R+1-k}(x)\simeq
y^{\mathcal{F}}_{k}(-x)=(-1)^{\frac{1}{2}}x^{\frac{1}{2}}a_{k}(-x)+b_{k}(-x)$
for each $k$. That is, $a_{R+1-k}(x)=(-1)^{\frac{1}{2}}c_{k}a_{k}(-x)$ and
$b_{R+1-k}(x)=c_{k}b_{k}(-x)$ for some non-zero constants $c_{k}$. But that
means

$\displaystyle
a_{k}(x)=(-1)^{\frac{1}{2}}c_{R+1-k}a_{R+1-k}(-x)=-c_{R+1-k}c_{k}a_{k}(x),$
$\displaystyle b_{k}(x)=c_{R+1-k}b_{R+1-k}(-x)=+c_{R+1-k}c_{k}b_{k}(x),$

from which we conclude that for each $k$ at least one of $a_{k}(x)$ and
$b_{k}(x)$ must vanish. ∎

######  Theorem 5.14.

Suppose $\beta(F\\!L({\mathcal{K}}))$ contains a cyclotomic tuple. Then
${\mathcal{K}}$ is cyclotomically self-dual.

###### Proof.

We shall need the following identity among Wronskian determinants.

######  Lemma 5.15 ([15]).

Given integers $0\leq k\leq s$ and functions $f_{1},\dots,f_{s+1}$, we have

$\displaystyle\operatorname{Wr}\big{(}\operatorname{Wr}(f_{1},\dots,f_{s-k},\dots,f_{s},\widehat{f}_{s+1}),\operatorname{Wr}(f_{1},\dots,f_{s-k},\dots,\widehat{f}_{s},f_{s+1}),\dots,$
$\displaystyle\qquad\operatorname{Wr}(f_{1},\dots,f_{s-k},\widehat{f}_{s-k+1},\dots,f_{s+1})\big{)}=\operatorname{Wr}(f_{1},\dots,f_{s-k})\big{(}\operatorname{Wr}(f_{1},\dots,f_{s+1})\big{)}^{k},$

where $\widehat{f}$ denotes omission.

To prove Theorem 5.14 we argue as for Theorem 6.8 in [15]. Let
${\mathcal{F}}\in F\\!L({\mathcal{K}})$ be a full flag in ${\mathcal{K}}$ and
$(u_{i}(x))_{i=1}^{R+1}$ a basis of ${\mathcal{K}}$ adjusted to this flag. Let
$\bm{y}=\bm{y}^{\mathcal{F}}$ be the corresponding tuple of quasi-polynomials
as in (5.11), and $(W_{i}(x))_{i=1}^{R+1}$ the corresponding basis of
${\mathcal{K}}^{\dagger}$ as in (5.5). Then Theorem 5.14 follows from the case
$k=R+1$ of the following lemma.

######  Lemma 5.16.

If $\bm{y}$ is cyclotomic then

$\displaystyle\operatorname{span}_{\mathbb{C}}(u_{1}(-x),\dots,u_{k}(-x))=\operatorname{span}_{\mathbb{C}}(W_{R+1},W_{R},\dots,W_{R+2-k}),\qquad
k=1,\dots,R+1.$

###### Proof.

Let us prove the lemma by induction on $k$. For $k=1$ we have

$\displaystyle u_{1}(-x)=y_{1}(-x)\simeq
y_{R}(x)=\operatorname{Wr}^{\dagger}(u_{1},\dots,u_{R})=W_{R+1}$

as required. Assume the statement holds for all values up to some $k$. For the
inductive step it is enough to show that

$\displaystyle\operatorname{Wr}(u_{1}(-x),\dots,u_{k}(-x),W_{R+1-k})\simeq\operatorname{Wr}(u_{1}(-x),\dots,u_{k}(-x),u_{k+1}(-x)).$
(5.15)

Indeed, (5.15) is an inhomogeneous differential equation in $W_{R+1-k}(x)$ and
if it holds then it must be that $W_{R+1-k}(x)$ is proportional to
$u_{k+1}(-x)$ modulo
$\operatorname{span}_{\mathbb{C}}(u_{1}(-x),\dots,u_{k}(-x))$, which is
sufficient given the inductive assumption.

By the inductive assumption, we have

$\displaystyle\operatorname{Wr}(u_{1}(-x),\dots,u_{k}(-x),W_{R+1-k})$
$\displaystyle\qquad{}\simeq\operatorname{Wr}(W_{R+1},W_{R},\dots,W_{R+1-k+1},W_{R+1-k})$
$\displaystyle\qquad{}=\operatorname{Wr}\big{(}\operatorname{Wr}(u_{1},\dots,u_{R+1-k-1},\dots,u_{R},\widehat{u}_{R+1}),$
$\displaystyle\qquad\quad{}\
\operatorname{Wr}(u_{1},\dots,u_{R+1-k-1},\dots,\widehat{u}_{R},u_{R+1}),\dots,$
$\displaystyle\qquad\quad{}\
\operatorname{Wr}(u_{1},\dots,u_{R+1-k-1},\widehat{u}_{R+1-k},\dots,u_{R+1})\big{)}\big{/}\big{(}{\tilde{T}}_{1}^{R-1}{\tilde{T}}_{2}^{R+1-3}\cdots{\tilde{T}}_{R-1}^{1}\big{)}^{k+1}$
$\displaystyle\qquad{}=\operatorname{Wr}(u_{1},\dots,u_{R-k})(\operatorname{Wr}(u_{1},\dots,u_{R+1}))^{k}\big{/}\big{(}{\tilde{T}}_{1}^{R-1}{\tilde{T}}_{2}^{R+1-3}\cdots{\tilde{T}}_{R-1}^{1}\big{)}^{k+1},$

the final equality by Lemma 5.15. Since
$\operatorname{Wr}^{\dagger}(u_{1},\dots,u_{R+1})=\operatorname{Wr}(u_{1},\dots,u_{R+1})/{\tilde{T}}_{1}^{R}{\tilde{T}}_{2}^{R-1}\cdots{\tilde{T}}_{R}^{1}$
is a nonzero constant by Lemma 5.6 we therefore have

$\displaystyle\operatorname{Wr}(u_{1}(-x),\dots,u_{k}(-x),W_{R+1-k})\simeq\operatorname{Wr}(u_{1},\dots,u_{R-k})\frac{\big{(}{\tilde{T}}_{1}^{R}\cdots{\tilde{T}}_{R}^{1}\big{)}^{k}}{\big{(}{\tilde{T}}_{1}^{R-1}\cdots{\tilde{T}}_{R-1}\big{)}^{k+1}}$
$\displaystyle\hphantom{\operatorname{Wr}(u_{1}(-x),\dots,u_{k}(-x),W_{R+1-k})}{}=\frac{\operatorname{Wr}(u_{1},\dots,u_{R-k})}{{\tilde{T}}_{1}^{R+1-k-2}\cdots{\tilde{T}}_{R+1-k-2}^{1}}{\tilde{T}}_{R}^{k}\cdots{\tilde{T}}_{R+1-k}^{1}.$

Now we may use again the fact that $\bm{y}$ is cyclotomic, so $y_{k}(-x)\simeq
y_{R+1-k}(x)$. In view of (5.11), that implies

$\displaystyle\frac{\operatorname{Wr}(u_{1},\dots,u_{R-k})}{{\tilde{T}}_{1}^{R+1-k-2}\cdots{\tilde{T}}_{R+1-k-2}^{1}}\simeq\frac{\operatorname{Wr}(u_{1}(-x),\dots,u_{k+1}(-x))}{{\tilde{T}}_{1}^{k}(-x)\cdots{\tilde{T}}_{k}^{1}(-x)}.$
(5.16)

Recall that ${\tilde{T}}_{R+1-k}(x)\simeq{\tilde{T}}_{k}(-x)$. Hence we have
indeed that

$\displaystyle\operatorname{Wr}(u_{1}(-x),\dots,u_{k}(-x),W_{R+1-k})\simeq\operatorname{Wr}(u_{1}(-x),\dots,u_{k+1}(-x)),$

as required. ∎

This completes the proof of Theorem 5.14. ∎

Given a subspace $U\subset{\mathcal{K}}$, let

$\displaystyle U^{\perp}:=\\{v\in{\mathcal{K}}\colon B(u,v)=0\text{ for all
}u\in U\\}$

denote its orthogonal complement in ${\mathcal{K}}$ with respect to the
bilinear form $B$. Recall that a full flag ${\mathcal{F}}=\\{0=F_{0}\subset
F_{1}\subset F_{2}\subset\dots\subset F_{R}\subset F_{R+1}={\mathcal{K}}\\}\in
F\\!L({\mathcal{K}})$ is called _isotropic_ with respect to $B$ if
$F_{k}=F^{\perp}_{R+1-k}$ for $k=1,\dots,R$.

######  Theorem 5.17.

Suppose ${\mathcal{K}}$ is cyclotomically self-dual. A full flag
${\mathcal{F}}\in F\\!L({\mathcal{K}})$ is isotropic if and only if the
associated tuple $\bm{y}^{\mathcal{F}}$ is cyclotomic.

###### Proof.

Let $(u_{i}(x))_{i=1}^{R+1}$ be a basis of ${\mathcal{K}}$ adjusted to
${\mathcal{F}}$, so that we have (5.11).

For the “only if” direction, suppose $\bm{y}^{\mathcal{F}}$ is cyclotomic. By
Lemma 5.16,

$\displaystyle
F_{k}=\operatorname{span}_{\mathbb{C}}(u_{1},\dots,u_{k})=\operatorname{span}_{\mathbb{C}}(W_{R+1}(-x),\dots,W_{R+2-k}(-x)).$

We also have
$F_{R+1-k}^{\perp}=\operatorname{span}_{\mathbb{C}}(u_{1},\dots,u_{R+1-k})^{\perp}=\operatorname{span}_{\mathbb{C}}(W_{R+1}(-x),\dots,W_{R+2-k}(-x))$
by (5.5). Therefore $F_{k}=F_{R+1-k}^{\perp}$.

For the “if” direction, suppose ${\mathcal{F}}=\\{F_{k}\\}$ is isotropic.
Since $F_{k}=F_{R+1-k}^{\perp}$, and given (5.5), we have two bases for
$F_{k}$, namely $(u_{1},\dots,u_{k})$ and $(W_{R+1}(-x),\dots,W_{R+2-k}(-x))$.
So to prove that $\bm{y}$ is cyclotomic it suffices to establish the following
lemma, which is the converse of Lemma 5.16.

######  Lemma 5.18.

If

$\displaystyle\operatorname{span}_{\mathbb{C}}(u_{1}(-x),\dots,u_{k}(-x))=\operatorname{span}_{\mathbb{C}}(W_{R+1},W_{R},\dots,W_{R+2-k}),\qquad
k=1,\dots,R+1,$

then $\bm{y}$ is cyclotomic.

###### Proof.

Examining the induction in the proof of Lemma 5.16, one sees that we also
have, by a similar induction, that if
$\operatorname{span}_{\mathbb{C}}(u_{1}(-x),\dots,u_{k}(-x))=\operatorname{span}_{\mathbb{C}}(W_{R+1},W_{R},\dots,W_{R+2-k})$
for each $k$ then (5.16) must hold for each $k$, which says that $\bm{y}$ is
cyclotomic. ∎

This completes the proof of Theorem 5.17. ∎

In view of Proposition 5.13 we have the following corollary.

######  Corollary 5.19.

If ${\mathcal{F}}\in F\\!L({\mathcal{K}})$ is isotropic then ${\mathcal{F}}$
is decomposable.

###  5.7 Witt bases and the symmetries of the bilinear form $\bm{B}$

We say that $(r_{k})_{k=1}^{R+1}$ is a _Witt basis_ of the cyclotomically
self-dual space ${\mathcal{K}}$ if

$\displaystyle\operatorname{Wr}^{\dagger}(r_{1},\dots,\widehat{r}_{k},\dots,r_{R+1})\simeq
r_{R+2-k}(-x),\qquad k=1,\dots,R+1.$ (5.17)

The following lemma gives a useful alternative characterization of Witt bases.

######  Lemma 5.20.

The basis $(r_{k})_{k=1}^{R+1}$ is a Witt basis if and only if

$\displaystyle B(r_{i},r_{j})=0\qquad\text{whenever}\quad i+j\neq R+2.$ (5.18)

###### Proof.

Suppose $(u_{k})_{k=1}^{R+1}$ is a basis of ${\mathcal{K}}$ and let
$(W_{k})_{k=1}^{R+1}$ be as in (5.5). Then $(W_{i}(x))_{i=1}^{R+1}$ and
$(u_{i}(-x))_{i=1}^{R+1}$ are two bases of ${\mathcal{K}}^{\dagger}$ and so
$u_{i}(-x)=\sum\limits_{j=1}^{R+1}C_{ij}W_{j}(x)$, for some invertible matrix
$C_{ij}$. We have
$B(u_{i},u_{j})=\sum\limits_{k=1}^{R+1}C_{jk}\operatorname{Wr}^{\dagger}(u_{i},u_{1},u_{2},\dots,\widehat{u}_{k},\dots,u_{R+1})=(-1)^{i-1}C_{ji}\operatorname{Wr}^{\dagger}(u_{1},\dots,u_{R+1})$.
Hence (5.17) is equivalent to (5.18). ∎

######  Theorem 5.21.

Every cyclotomically self-dual space ${\mathcal{K}}$ has a special basis
$(r_{k})_{k=1}^{R+1}$ which is also Witt basis, and in which in fact

$\displaystyle\operatorname{Wr}^{\dagger}(r_{1},\dots,\widehat{r}_{k},\dots,r_{R+1})=(-1)^{-\deg
r_{R+2-k}}r_{R+2-k}(-x),\qquad k=1,\dots,R+1.$ (5.19)

###### Proof.

Let $(u_{k}(x))_{k=1}^{R+1}$ be a special basis of ${\mathcal{K}}$. We may
suppose that the $u_{k}(x)$ all have leading coefficient 1. Let
$(W_{k}(x))_{k=1}^{R+1}$ be the basis of ${\mathcal{K}}^{\dagger}$ as in
(5.5). By Lemma 5.11, $\deg W_{k}=d^{\dagger}_{k}$. By Lemma 5.5, we have

$\displaystyle
W_{k}=\operatorname{Wr}^{\dagger}(u_{1},\dots,\widehat{u}_{k},\dots,u_{R+1})=D_{k}x^{d^{\dagger}_{k}}+\cdots,$

where the ellipsis indicates terms of lower degree in $x$ and where

$\displaystyle D_{k}:=\prod_{\begin{subarray}{c}1\leq j<i\leq R+1\\\ i\neq
k,\,j\neq k\end{subarray}}(d_{i}-d_{j}),\qquad k=1,\dots,R+1.$

Since ${\mathcal{K}}$ is cyclotomically self-dual we must have

$\displaystyle d_{k}=d^{\dagger}_{R+2-k},\qquad k=1,\dots,R+1,$

and

$\displaystyle
W_{k}=\operatorname{Wr}^{\dagger}(u_{1},\dots,\hat{u}_{k},\dots,u_{R+1})=D_{k}(-1)^{-d_{R+2-k}}u_{R+2-k}(-x)+\cdots.$

Now from (5.4) we have

$\displaystyle d_{k}-d_{l}=d_{R+2-l}-d_{R+2-k},\qquad 1\leq k<l\leq R+1,$

using which one verifies that

$\displaystyle D_{k}=D_{R+2-k},\qquad k=1,\dots,R+1.$

Given this equality, if we set

$\displaystyle
q_{k}:=u_{k}D_{k}^{\frac{1}{2}}\prod_{j=1}^{R+1}D_{j}^{-\frac{1}{2R-2}}$

then we have

$\displaystyle\operatorname{Wr}^{\dagger}(q_{1},\dots,\widehat{q}_{k},\dots,q_{R+1})=(-1)^{-d_{R+2-k}}q_{R+2-k}(-x)+\cdots.$

In this way, we arrive at

$\displaystyle\operatorname{Wr}^{\dagger}(q_{1},\dots,\widehat{q}_{k},\dots,q_{R+1})$
$\displaystyle\qquad{}=(-1)^{-d_{R+2-k}}q_{R+2-k}(-x)+\sum_{j=2}^{R+1}q_{R+2-j}(-x)c^{j}_{k},\qquad
k=1,\dots,R+1,$ (5.20)

for some constants $c^{j}_{k}$. That is, we have

$\displaystyle\operatorname{Wr}^{\dagger}(q_{1},\dots,q_{R-1},q_{R},\widehat{q}_{R+1})=(-1)^{-d_{1}}q_{1}(-x),$
$\displaystyle\operatorname{Wr}^{\dagger}(q_{1},\dots,q_{R-1},\widehat{q}_{R},q_{R+1})=(-1)^{-d_{2}}q_{2}(-x)+c_{R}^{1}q_{1}(-x),$
$\displaystyle\operatorname{Wr}^{\dagger}(q_{1},\dots,\widehat{q}_{R-1},q_{R},q_{R+1})=(-1)^{-d_{3}}q_{3}(-x)+c_{R-1}^{2}q_{2}(-x)+c_{R-1}^{1}q_{1}(-x),$
$\displaystyle\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots.$

We define

$\displaystyle(-1)^{-d_{1}}r_{1}:=(-1)^{-d_{1}}q_{1},\qquad(-1)^{-d_{2}}r_{2}:=(-1)^{-d_{2}}q_{2}+c_{R}^{1}r_{1},$

so that

$\displaystyle\operatorname{Wr}^{\dagger}(r_{1},r_{2},q_{3},\dots,q_{R-1},q_{R},\widehat{q}_{R+1})=(-1)^{-d_{1}}r_{1}(-x),$
$\displaystyle\operatorname{Wr}^{\dagger}(r_{1},r_{2},q_{3},\dots,q_{R-1},\widehat{q}_{R},q_{R+1})=(-1)^{-d_{2}}r_{2}(-x),$
$\displaystyle\operatorname{Wr}^{\dagger}(r_{1},r_{2},q_{3},\dots,\widehat{q}_{R-1},q_{R},q_{R+1})=(-1)^{-d_{3}}q_{3}(-x)+\tilde{c}_{R-1}^{2}r_{2}(-x)+\tilde{c}_{R-1}^{1}r_{1}(-x),$
$\displaystyle\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots$

for some new constants $\tilde{c}^{j}_{k}$, and we then define

$\displaystyle(-1)^{-d_{3}}r_{3}:=(-1)^{-d_{3}}q_{3}+\tilde{c}_{R-1}^{1}r_{2}+\tilde{c}_{R-1}^{1}r_{1},$

and so on. By an obvious induction, we arrive at a Witt basis
$(r_{k})_{k=1}^{R+1}$. By construction $\deg r_{k}=d_{k}$. Finally, note in
(5.20) that $c^{j}_{k}$ can be non-zero only when $d_{k}-d_{j}\in{\mathbb{Z}}$
since both sides lie in either ${\mathbb{C}}[x]$ or
$x^{\frac{1}{2}}{\mathbb{C}}[x]$. Therefore this Witt basis
$(r_{k})_{k=1}^{R+1}$ is special. ∎

######  Lemma 5.22.

Let $(r_{k})_{k=1}^{R+1}$ be the Witt basis of Theorem 5.21. Then
$\operatorname{Wr}^{\dagger}(r_{1},\dots,r_{R+1})=1$.

###### Proof.

We have

$\displaystyle\operatorname{Wr}^{\dagger}(q_{1},\dots,q_{R+1})=\operatorname{Wr}^{\dagger}(u_{1},\dots,u_{R+1})\prod_{k=1}^{R+1}\left(D_{k}^{\frac{1}{2}}\prod_{j=1}^{R+1}D_{j}^{-\frac{1}{2R-2}}\right)$
$\displaystyle\hphantom{\operatorname{Wr}^{\dagger}(q_{1},\dots,q_{R+1})}{}=\operatorname{Wr}^{\dagger}(u_{1},\dots,u_{R+1})\prod_{k=1}^{R+1}D_{k}^{-\frac{1}{R-1}}.$

But then noting that

$\displaystyle\prod_{k=1}^{R+1}D_{k}=\prod_{1\leq j<i\leq
R+1}(d_{i}-d_{j})^{R-1}$

and recalling Lemma 5.6, one finds

$\displaystyle\operatorname{Wr}^{\dagger}(q_{1},\dots,q_{R+1})=1$

and hence the result. ∎

######  Theorem 5.23.

The subspaces ${\mathcal{K}}_{\textup{Sp}}$ and ${\mathcal{K}}_{\textup{O}}$
are mutually orthogonal with respect to $B$. The bilinear form $B$ is skew-
symmetric on ${\mathcal{K}}_{\textup{Sp}}$ and symmetric on
${\mathcal{K}}_{\textup{O}}$.

###### Proof.

Let $(r_{k})_{k=1}^{R+1}$ be the special Witt basis constructed in Theorem
5.21. From (5.19) and Lemma 5.22, we have

$\displaystyle B(r_{k},r_{R+2-k})=(-1)^{d_{R+2-k}+k+1},\qquad k=1,\dots,R+1,$
(5.21)

and $B(r_{i},r_{j})=0$ if $i+j\neq R+2$. This implies in particular that
${\mathcal{K}}_{\textup{Sp}}$ and ${\mathcal{K}}_{\textup{O}}$ are mutually
orthogonal. By Lemma 5.12 it also gives

$\displaystyle
B(r_{k},r_{R+2-k})B(r_{R+2-k},r_{k})=(-1)^{\langle\Lambda,\alpha^{\vee}_{1}+\dots+\alpha^{\vee}_{R}\rangle},\qquad
k=1,\dots,R+1.$

Recall the definition of $\Lambda$, (5.2). Now
$\langle\Lambda_{s}+\sigma\Lambda_{s},\alpha^{\vee}_{1}+\dots+\alpha^{\vee}_{R}\rangle\in
2{\mathbb{Z}}$ for each $s=1,\dots,N$, since $\Lambda_{s}$ is integral.
Therefore it follows from (5.1) that

$\displaystyle\langle\Lambda,\alpha^{\vee}_{1}+\dots+\alpha^{\vee}_{R}\rangle\in\begin{cases}2{\mathbb{Z}}+1,&p>0,\\\
2{\mathbb{Z}},&p=0.\end{cases}$

Consider the case $p>0$. Then we have

$\displaystyle B(r_{k},r_{R+2-k})B(r_{R+2-k},r_{k})=-1,\qquad k=1,\dots,R+1.$
(5.22)

Recall from (5.7) that $\deg r_{k}$ and $\deg r_{R+2-k}$ are both half odd
integers if $k=p+1,\dots,R+1-p$, and are integers otherwise. Hence, by (5.21),
$B(r_{k},r_{R+2-k})$ and $B(r_{R+2-k},r_{k})$ lie in
$\\{(-1)^{\frac{1}{2}},(-1)^{-\frac{1}{2}}\\}$ if $k=p+1,\dots,R+1-p$ and in
$\\{1,-1\\}$ otherwise. Combining this statement with (5.22) we find

$\displaystyle
B(r_{k},r_{R+2-k})=\begin{cases}-B(r_{R+2-k},r_{k}),&k=1,\dots,p,R+2-p,\dots,R+1,\\\
+B(r_{R+2-k},r_{k}),&k=p+1,\dots,R+1-p,\end{cases}$

which is the required result.

Finally, consider the case $p=0$. Then

$\displaystyle B(r_{k},r_{R+2-k})B(r_{R+2-k},r_{k})=1,\qquad k=1,\dots,R+1,$

and since in this case $\deg r_{k}$ is integral for all $k$, this implies

$\displaystyle B(r_{k},r_{R+2-k})=B(r_{R+2-k},r_{k}),\qquad k=1,\dots,R+1$

as required. ∎

The following are corollaries of Theorem 5.23 together with Lemma 5.20.

######  Corollary 5.24.

Every Witt basis $(r_{k})_{k=1}^{R+1}$ of ${\mathcal{K}}$ is decomposable.

A basis $(r_{k})_{k=1}^{R+1}$ of $K$ such that

$\displaystyle B_{ij}:=B(r_{i},r_{j})=\delta_{R+2-i,j}b_{i}$

with

$\displaystyle b_{k}:=\begin{cases}(-1)^{k},&k=1,\dots,p,\\\
\,\,+1,&k=p+1,\dots,R+1-p,\\\ (-1)^{R+1-k},&k=R+2-p,\dots,R+1\end{cases}$

is called a _reduced Witt basis_. By Lemma 5.20, reduced Witt bases are Witt
bases.

######  Corollary 5.25.

Any Witt basis can be transformed to a reduced Witt basis by a suitable
diagonal transformation followed by a suitable permutation of the basis
vectors.

######  Corollary 5.26.

For any Witt basis $(r_{k})_{k=1}^{R+1}$ of ${\mathcal{K}}$, the full flag
${\mathcal{F}}=\\{F_{1}\\!\subset F_{2}\\!\subset\\!\cdots\\!\subset
F_{R+1}={\mathcal{K}}\\}$ given by
$F_{k}=\operatorname{span}_{\mathbb{C}}(r_{1},\dots,r_{k})$, $k=1,\dots,R+1$,
is isotropic $($and hence the corresponding tuple $\bm{y}^{\mathcal{F}}$ is
cyclotomic by Theorem 5.17).

Conversely, given any isotropic full flag ${\mathcal{F}}=\\{F_{1}\subset