Populations of Solutions to Cyclotomic Bethe Equations

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.


Introduction
Let g be a complex Kac-Moody Lie algebra and σ : g → g an automorphism of order M ∈ Z ≥1 .
Let ω ∈ C × be a primitive M th root of unity. We may choose a Cartan subalgebra h ⊂ g such that σ(h) = h. We have the canonical pairing ·, · : h * ⊗ h → C, and the simple roots α i ∈ h * and coroots α ∨ i ∈ h, where i runs over the set I of nodes of the Dynkin diagram. Consider the following system of equations in m ∈ Z ≥0 variables t = (t 1 , . . . , t m ) ∈ C m and labels c = (c(1), . . . , c(m)) ∈ I m : where Λ 0 , Λ 1 , . . . , Λ N ∈ h * are weights (with σΛ 0 = Λ 0 ) and z 1 , . . . , z N are non-zero points arXiv:1505.07582v2 [math.QA] 14 Nov 2015 in the complex plane whose orbits, under the action of the cyclic group ω Z , are pairwise disjoint. When σ = id, ω = 1 and Λ 0 = 0, these equations reduce to the following well-known set of equations in mathematical physics: 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 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 (Z/M 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 C of affine hyperplanes in a complex affine space of finite dimension, together with an assignment of a number a(H) ∈ C to each hyperplane H ∈ C. Indeed, for each H ∈ C, let H = 0 be an affine equation for H; then the master function is Φ = H∈C a(H) log H .
The cyclotomic master function corresponds to a hyperplane arrangement in C m whose hyperplanes include t i = ω k t j , 1 ≤ i < j ≤ m, for each k ∈ Z/M Z. By contrast, the extended master function corresponds to a hyperplane arrangement in C mM , but has only those hyperplanes corresponding to the type A root system, i.e., t i = t j , 1 ≤ i < j ≤ 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 g is simple then every population is isomorphic to the flag variety of the Langlands dual Lie algebra L 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 g a general Kac-Moody Lie algebra and σ any diagram automorphism of g satisfying the linking condition. (We also place certain restrictions on the weight Λ 0 ; see Section 4.1.) The linking condition [7] states that, for every node i ∈ I, the restriction of the Dynkin diagram to the orbit σ 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 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 σ. 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 σ Z (i). There are two cases: L i = 1 and L i = 2.
First, suppose i ∈ I is a node with linking number L i = 1. A critical point (t, c) is represented by a tuple of polynomials, y = (y i (x)) i∈I , where the roots of the polynomial y i (x), i ∈ I, are the Bethe variables t s of "colour" i, i.e., those such that c(s) = i. Following [15], one defines a function of x, depending on a parameter c ∈ C . Here T i (x), i ∈ I, are certain functions encoding the "frame" data, i.e., the points z 1 , . . . , z N and the weights Λ 1 , . . . , Λ 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 y (i) (c) in which y i (x) is replaced by y Next, suppose i ∈ I is a node with linking number L i = 2. Then for every copy of the A 2 diagram, with nodes say j and, one must perform the sequence of generation steps j,, j. Doing this for each copy of 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, Λ 0 . Throughout Section 4, motivated by [26], we assume that Λ 0 , α ∨ i 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 s i · Λ 0 , before eventually being shifted back to Λ 0 . See Proposition 4.10 and compare [17].
In either case, L i = 1 or L i = 2, we write y (i,σ) (c) for the tuple of polynomials representing the new cyclotomic critical point. It depends on a single parameter c. The replacement y → y (i,σ) (c) is the elementary cyclotomic generation, in the direction of the orbit σ Z (i).
To a critical point (t, c) represented by a tuple of polynomials y one can associate a weight Λ ∞ . See (2.4) and (4.10). For fixed Λ 0 , Λ 1 , . . . , Λ N , we may regard Λ ∞ as encoding the number of roots t s of each "colour" i ∈ I, i.e., the degrees of the polynomials y i (x). It is known that Λ ∞ (y (i) (c)) is equal either to Λ ∞ (y) or to s i · Λ ∞ (y), where s i · denotes the shifted action of the Weyl reflection in root α i . See [15]. We have an analogous statement in the cyclotomic case. Namely, there is a "folded" Weyl group W σ with generators s σ i . See Section 2.3. And we show that Λ ∞ (y (i,σ) (c)) is equal either to Λ ∞ (y) or to s σ i · Λ ∞ (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 ∈ Z ≥1 , for the master functions associated to marked points z 1 , . . . , z N and integral dominant weights Λ 1 , . . . , Λ 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 K of polynomials. The ramification points of K are z 1 , . . . , z N and ∞, and the ramification data at these points are specified by the weights Λ 1 , . . . , Λ N and an integral dominant weightΛ ∞ . Given a full flag i=1 of polynomials adjusted to this flag, i.e., such that F k = span C (u 1 (x), . . . , u k (x)). Then define a tuple of functions y F = (y F k (x)) R k=1 by where -as in (1.3) above -the (T i (x)) R i=1 are functions encoding the "frame" data z 1 , . . . , z N and Λ 1 , . . . , Λ N , and where Wr(u 1 (x), . . . , u k (x)) denotes the Wronskian determinant. The ramification properties of K ensure that the y F k (x) are in fact polynomials. Moreover the map F → y F is an isomorphism of varieties from the variety of full flags in K to the population associated with K. The space K is the kernel of a certain linear differential operator D of order R + 1 (essentially a type A oper). This operator D can be defined in terms of the (T i (x)) R i=1 together with the polynomials (y i (x)) R i=1 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 Λ 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 K of quasi-polynomials: that is, polynomials in x 1 2 . The local behaviour of these quasi-polynomials near the origin is encoded in Λ 0 . The remaining ramification points are z 1 , . . . , z N , −z 1 , . . . , −z N , and ∞. See Definition 5.2.
The space of quasi-polynomials K admits a natural Z 2 gradation K = K O ⊕ K 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 F ∈ FL S (K) of a certain preferred type S, (5.9), are sent to polynomials under the map F → y F . This map of varieties FL S (K) → P(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 y F , F ∈ FL S (K), such that y i (x) y R+1−i (−x), i = 1, . . . , R. The question is: which flags in FL S (K) map to cyclotomic tuples?
To answer this question we introduce the notion of a cyclotomically self-dual space of quasipolynomials. The space K has a natural dual space K † of quasi-polynomials -see Section 5.5and we say K is cyclotomically self-dual if for all v(x) ∈ K, v(−x) ∈ K † . (Compare the very similar notion of a self-dual space of polynomials in [15].) We show that a sufficient condition for K to be cyclotomically self-dual is that there exists at least one full flag F in K such that y F is cyclotomic (Theorem 5.14). If K is cyclotomically self-dual then it admits a canonical nondegenerate bilinear form B. We show that, for all full flags F in K, the tuple y F is cyclotomic if and only if F is isotropic with respect to B (Theorem 5.17).
Therefore the cyclotomic population is isomorphic to the variety FL ⊥ S (K) of isotropic flags of type S in K. The bilinear form B is symmetric on K O and skew-symmetric on K Sp , and these subspaces are mutually orthogonal with respect to B (Theorem 5.23). Hence this variety FL ⊥ S (K) is isomorphic to the direct product of spaces of isotropic flags FL ⊥ (K Sp ) × FL ⊥ (K O ).
2 Master functions and cyclotomic symmetry

Kac-Moody algebras
Let I be a finite set of indices and A = (a i,j ) i,j∈I a generalized Cartan matrix, i.e., a i,i = 2 and a i,j ∈ Z ≤0 whenever i = j, with a i,j = 0 if and only if a j,i = 0. Let g := g(A) be the corresponding complex Kac-Moody Lie algebra [11,Section 1], h ⊂ g a Cartan subalgebra, and g = n − ⊕ h ⊕ n + a triangular decomposition. Let α i ∈ h * , α ∨ i ∈ h, i ∈ I be collections of simple roots and coroots respectively. We have dim h = |I| + dim ker A = 2|I| − rank A. By definition, where ·, · : h * ⊗ h → C is the canonical pairing. We assume that A is symmetrizable, i.e., there exists a diagonal matrix D = diag(d i ) i∈I , whose entries are coprime positive integers, such that the matrix B = DA is symmetric. Let (·, ·) be the associated symmetric bilinear form on h * . We have (α i , α j ) = d i a i,j and The form (·, ·) is non-degenerate. Therefore it gives an identification h ∼ = C h * and hence a non-degenerate symmetric bilinear form on h which we also write as (·, ·). Let P := {λ ∈ h * : λ, α ∨ i ∈ Z} be the integral weight lattice and P + := {λ ∈ h * : λ, α ∨ i ∈ Z ≥0 } the set of dominant integral weights.
Let W ⊂ End(h * ) be the Weyl group. It is generated by the reflections s i , i ∈ I, given by Let ρ ∈ h * be a vector such that ρ, α ∨ i = 1 for i ∈ I. We use · to denote the shifted action of the Weyl group, i.e.,

Diagram automorphism
Suppose σ is an automorphism of the Dynkin diagram [11,Section 4.7] of A. That is, σ is a permutation of the index set I such that a σi,σj = a i,j .
Let M be the order of σ and let ω ∈ C × be a primitive M th root of unity.
To such a permutation is associated a diagram automorphism g → g of the Kac-Moody Lie algebra [7], which we shall also write as σ. We have where E i ∈ n, F i ∈ n − , i ∈ I, are a set of Chevalley generators of [g, g]. This defines σ on the derived subalgebra [g, g] of g. For the action of σ on the derivations, i.e., on a complement of [g, g] in g, see [7,Section 3.2]. This action may be chosen to ensure that σ : g → g has order M and respects the bilinear form (·, ·) on h: The action of σ on h * is defined by σλ := λ • σ −1 so that σλ, σX = λ, X for all λ ∈ h * , X ∈ h. Note that then σα i = α σi for all i ∈ I. Let g σ ⊂ g be the Lie subalgebra of elements invariant under σ. We have

The linking condition and the folded diagram
For any i ∈ I let be the length of the orbit of the node i under the automorphism σ of the Dynkin diagram A. Define Note that L i ≥ 1. Following [7], we say that σ obeys the linking condition if and only if 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 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 A 2n , n ∈ Z ≥1 , we in fact have L i = 1 for every node i: that is, no two distinct nodes in the same σ-orbit are ever linked by an edge of the Dynkin diagram. In type A 2n the non-trivial diagram automorphism gives L i = 2 for i ∈ {n, n + 1} and L i = 1 otherwise: n − 1 n 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 A (1) n , n ∈ Z ≥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 I σ ⊆ I consisting of exactly one representative of each σ-orbit. Then the Cartan matrix A σ = (a σ i,j ) i,j∈Iσ of the folded diagram is given by Remark 2.3. Compare Section 3.3 of [7], noting that our convention a j,i = α i , α ∨ j differs from that of [7]. Lemma 2.4 ([7]). If σ obeys the linking condition then A σ (and its transpose) is a symmetrizable Cartan matrix whose type (finite, affine, or indefinite) is the same as that of A.
For each i ∈ I σ let us define also Then we have Thus α ∨,σ i , E σ i , F σ i , i ∈ I σ generate a copy of (the derived subalgebra of) the Kac-Moody Lie algebra g(A σ ) inside g σ := {X ∈ g : σX = X}. Next, for all i ∈ I σ , if we let Define W σ to be the group generated by the elements s σ i ∈ End(h * ) given by Lemma 2.5. W σ is a subgroup of W . Indeed, we have

The cyclotomic master function
be a collection of nonzero points z i ∈ C × such that ω Z z i ∩ ω Z z j = ∅ whenever i = j. We shall call Λ i the weight at z i . In addition, we pick a weight Λ 0 ∈ h σ, * . We call Λ 0 the weight at the origin. Let c = (c(j)) m j=1 be an m-tuple of elements of I, and introduce variables t = (t j ) m j=1 . We shall say that t j is a variable of colour c(j).
We define the cyclotomic master function Φ = Φ g,σ (t; c; z; Λ, Λ 0 ) associated to these data to be A point t with complex coordinates is called a critical point of the cyclotomic master function if or equivalently (in view of Lemma 2.6 below) if the following equations are satisfied: for j = 1, . . . , m. Call this system of equations the cyclotomic Bethe equations.
Define Λ ∞ , the weight at infinity, to be The group S m acts on pairs of m-tuples (t, c) by permuting indices:

The extended master function
The equations (2.3) admit another, closely related, interpretation. Recall the definition of the (usual) master function [19]. Namely, letΛ = (Λ i )Ñ i=0 be a collection ofÑ + 1 ∈ Z ≥0 weights Λ i ∈ h * , and letz = (z i )Ñ i=0 be a collection of nonzero pointsz i ∈ C × . Pickm ∈ Z ≥0 , let c = (c(j))m j=1 be anm-tuple of elements of I and introduce variables t = (t j )m j=1 . The master function associated to these data is It is a function of the variables t, depending on the parameters c,z andΛ. The critical points of the master function are those points t with complex coordinates such that ∂ Φ/∂t j = 0 for j = 1, . . . ,m, i.e., those points such that the following equations are satisfied: In this paper we are concerned with the following special case. LetÑ = N M , choose (z i ) N M i=0 to bez and choose the weights at these points to bẽ where z i , Λ i , i = 1, . . . , N , and Λ 0 are as in Section 2.4. We call the master function in this case the extended master function, Φ = Φ g,σ (t; c; z; Λ; Λ 0 ). It is given by (2.8) and the critical point equations (2.6) take the form The group Sm acts on pairs ofm-tuples (t, c) by permuting indices: Let us call a point (t, c) ∈ Cm × Im a cyclotomic point if we havem = M m for some m ∈ Z ≥0 and, by acting with some permutation in Sm, we can arrange that

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 g is semisimple, and that σ is an automorphism of g of order M > 1. Recall [8,9] that the quadratic Gaudin Hamiltonians are the followingÑ + 1 elements of U (g) ⊗(Ñ +1) : where I a , a = 1, . . . , dim g, is a basis of g, I a is the dual basis with respect to the non-degenerate invariant bilinear form (·, ·) : g × g → C, and we write X (i) for X acting in the ith tensor factor. (For convenience we number these factors starting from 0.) For Λ ∈ h * , let M Λ denote the Verma module over g with highest weight Λ, M Λ := Ind g h⊕n + Cv Λ . Let us represent theH (i) as linear maps in End Ñ i=0 MΛ i . Then the following can be shown using the techniques of the Bethe Ansatz.
(The fact that this simultaneous eigenvector is nonzero is proved for g = sl n nondegenerate critical points in [14], for g = sl n isolated critical points in [13], and for semisimple g and isolated critical points in [25]. See also [24].) In [26] 1 , B. Vicedo and one of the present authors defined cyclotomic Gaudin Hamiltonians. The quadratic cyclotomic Gaudin Hamiltonians are the elements of U (g) ⊗N given by 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 = 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, , 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 Λ i , Λ i ∈ h * . In other words, let us represent the Hamiltonians (3.3) as linear maps Let (in this section, Section 3) Λ 0 ∈ h σ, * be the weight given by The explicit form of the eigenvector ψ t is

(It is an interesting open problem to determine under what circumstances the vector ψ 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 writeH (i) as H The following is then a corollary of Theorem 3.1.
Proof . Let t be the corresponding (by Lemma 2.9) cyclotomic critical point of the extended master function Φ. Then the result is a special case of Theorem 3.1, by substituting (2.7) and (2.11) into (3.1). (To see that H (0) ext has eigenvalue zero note that In summary, we have the following observation.

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 ∈ 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 Λ 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".

Conditions on Λ 0
In the remainder of the paper we assume that σ is a diagram automorphism obeying the linking condition (2.1). That means for each i ∈ I, either L i = 1 or L i = 2.
In addition, in this section, Section 4, we place the following conditions on the weight Λ 0 ∈ h σ, * .
For each i ∈ I such that L i = 1, we suppose that and For each i ∈ I such that L i = 2, we suppose that Remark 4.1. One can verify that these conditions are satisfied by the weight Λ 0 of (3.5) in the case of diagram automorphisms of finite-type Dynkin diagrams. Our assumptions on Λ 0 in the treatment of type A R in Section 5 below are weaker.

Tuples of polynomials
To any pair (t; c) with t ∈ Cm and c ∈ Im, we may associate a tuple of polynomials y = (y 1 (x), . . . , y r (x)), given by We say that this tuple y represents the pair (t; 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 y defines a point in the direct product P( Conversely, given any y ∈ P(C[x]) |I| we may extract the pair (t; c) ∈ Cm × Im such that (4.4) holds. This pair is unique up to permutation by an element of Sm; see (2.10). Define We say that a tuple of polynomials y = (y i (x)) i∈I ∈ P(C[x]) |I| is generic (with respect to (T i (x)) i∈I ) if for each i ∈ I, y i (x) has no root in common with T i (x), or with any y j (x), j ∈ I\{i}, such that α j , α ∨ i = 0. Note that if y represents a critical point of the extended master function Φ(t; c; z; Λ), (2.8), i.e., its roots obey (2.9), then the tuple 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.)

Elementary generation: the L i = 1 case
Throughout this subsection, we suppose i ∈ I is such that L i = 1. That means that the simple roots α σ k i , i = 1, . . . , M i , are mutually orthogonal. Equivalently it means that the reflections is a solution to the equation where Proposition 4.2. If y represents a critical point then y Proof . We have Λ 0 , α ∨ i ∈ Z ≥0 as in (4.1), and for each s ∈ {1, . . . , N }, Λ s is integral dominant so Λ s , α ∨ i ∈ Z ≥0 . So the integrand is a rational function with poles at most at the points t p , p ∈ {1, . . . , m}, for which c(p) = i. Consider such a point t p . Note that 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 which vanishes if (4.8) vanishes. This shows that y Thus, given any tuple y representing a critical point we have, for each value of a parameter c ∈ C, a new tuple of polynomials y (i) , obtained from the tuple y by replacing y i (x) with y . We say y (i) is obtained from y by generation in the ith direction, and we call y (i) the immediate descendant of y in the ith direction. The tuples y (i) describe a projective line in P(C[x]) |I| . It will be useful to have the following specific parameterization of this line. There exists a unique solution y (4.9) and define y (i) (c) ∈ P(C[x]) |I| to be the tuple obtained from the tuple y by replacing y i (x) with y Recall that there is a weight at infinity, Λ ∞ , associated to any critical point. For the critical point represented by y this weight is, cf. (2.4), (4.10) For fixed Λ 0 , Λ 1 , . . . , Λ N we can think of Λ ∞ as encoding the degrees of the polynomials y j . Note that deg y This establishes the following lemma.
. If generation in the ith direction is degree-increasing, then the weight at infinity associated with the critical point represented by y

Cyclotomic generation: the L i = 1 case
We continue to suppose that i is such that L i = 1.
If y represents a cyclotomic point then its immediate descendant y (i) in the ith direction generically does not. However if, starting from a cyclotomic critical point, we successively generate in each of the directions σ k i, k = 1, . . . , 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 denote equality up to a constant (independent of x) nonzero factor. Recall the definition (2.11) of a cyclotomic point.
for all j ∈ I. If y j (x) and y σj (x) share the same leading coefficient for all j ∈ I, then the tuple y represents a cyclotomic point if and only if For the rest of this subsection, we suppose y represents a cyclotomic critical point. Hence in is a parameterization of the space of solutions to (4.7).) Define y (i,σ) (c) to be the tuple of polynomials given by Proof . First let us show that y (i,σ) represents a cyclotomic point for all c ∈ C. Comparing our definition of y (i,σ) with the criterion in Lemma 4.5, one sees that it is enough to check that Inspecting (4.6), we see that this equality holds for all c ∈ C if and only if But now, given (4.10) and the assumption that Λ s , s = 1, . . . , n are integral, the following lemma implies that (4.11) holds if and only if we impose the condition (4.2) on Λ 0 .
Lemma 4.7. Suppose Λ ∈ h * is an integral weight. Then, for any j ∈ I, Now we show that y (i,σ) represents a critical point for all but finitely many c ∈ C. Note that from definition (4.5) we have Hence, in view of (4.10), 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

Elementary generation: the L i = 2 case
For this subsection we suppose that i ∈ I is such that L i = 2. That implies M i is even and the restriction of the Dynkin diagram to the nodes σ Z i consists of M i 2 ∈ Z ≥1 disconnected copies of the Dynkin diagram of type A 2 , as sketched below: Here, for brevity, we writeī := σ M i /2 i. We define y Here the limits x 0 mean that y (i) i (x) is holomorphic at x = 0. This condition defines the integral uniquely, since Λ 0 , α ∨ i / ∈ Z by our assumption (4.3).
Proposition 4.9. If y represents a critical point then y Proof . The proof is as for Proposition 4.2.
Let y (i) = (y (i) j (x)) j∈I be the tuple of polynomials whose ith component y i (x) is as above, and whose remaining components are the same as those of y, i.e., for all j ∈ I\{i}.
Let (t (i) ; c (i) ) denote the pair represented by this tuple in the sense of Section 4.2. It turns out that t (i) is not in general a critical point of the extended master function Φ(t (i) ; c (i) ; z; Λ), 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 y (i) is generic.
for each p.
Proof . By (2.9) for each root t p in the tuple t we have For all roots of colours j ∈ I such that α j , α ∨ i = 0 this is immediately equivalent to the required equation. So we must consider roots of colour i, and roots of colours j ∈ I such that α j , α ∨ i < 0. By definition of y or equivalently .

(4.15)
By definition of (t (i) , c (i) ), the left-hand side of (4.15) is Now suppose j ∈ I is such that α j , α ∨ i ∈ Z <0 . By definition y (i) j (x) = y j (x). Suppose t p is a root of y j (x), i.e., suppose c(p) = j. Since y represents a critical point, y must be generic, and hence t p is not a root of y i (x). By our assumption that 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 On adding this equation multiplied by α i , α ∨ j to the equation (4.13), we arrive at . It remains to consider roots of colour i. First note that y i (x) and y 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 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 y (i) is generic, it follows from (4.14) and Lemmas 4.12 and 4.13 below that which is the required equality.
where (s j ) J j=1 are all distinct and non-zero, then Proof . We have W Wr(f, g) = W Wr(f, g). Hence and hence the result.
To deal with the case in which y (i) fails to be generic, we shall also need the following observation, which follows from (4.14).
Lemma 4.14. For any j ∈ I such that α j , α ∨ i < 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 yī(x) and y Proof . By our assumption (4.3) that 2 Λ 0 , α ∨ i + 1 ∈ Z ≥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 yī(x) that are not also roots of y which must vanish, because according to Proposition 4.10 the following vanishes: The polynomial y We say generation in the ith direction from y is degree-increasing if deg y Let y (ī,i) ı (x; 0) be the unique solution to (4.17) whose coefficient of x deg yī is zero. The degree of y (ī,i) ı (x; 0) is always given by deg y (ī,i) ı (x; 0) = deg yī(x) + Λ ∞ + ρ, α ∨ i + α ∨ ı , whether or not generation is degree-increasing. (Note that Λ ∞ + ρ, α ∨ i + α ∨ ı is odd, by our assumption (4.3), and in particular not zero.)

Generation in the ith direction is degree-increasing if and only if
Let then y (ī,i) (c) = (y (ī,i) j (x; c)) j∈I be the tuple of polynomials whoseīth component is and whose remaining components are the same as those of y (i) , i.e., Let (t (ī.i) ; c (ī,i) ) denote the pair represented by this tuple in the sense of Section 4.2.
The following result says that whenever y (ī,i) (c) is generic, this new pair (t (ī,i) (c), c (ī,i) ) obeys the same form of equations as did (t (i) , c (i) ).
Proposition 4.16. If y represents a critical point then, for all c ∈ C such that y (ī,i) (c) is generic, we have Proof . The proof is analogous to that of Proposition 4.10.
Finally, we define y Here the limits    Let y (i,ī,i) (c) = (y (i,ī,i) j (x; c)) j∈I be the tuple of polynomials whose ith component is y (i,ī,i) (x; c) as above and whose remaining components are those of y (ī,i) (c), i.e., and y (i,ī,i) j (x) = y j (x) for all j ∈ I\{i,ī}.
Let (t (i,ī.i) ; c (i,ī,i) ) denote the pair represented by this tuple in the sense of Section 4.2.
Proposition 4.18. If y represents a critical point and y (i,ī,i) (c) is generic, then y (i,ī,i) (c) represents a critical point. That is, the pair (t i,ī,i (c), c i,ī,i ) obeys the equations Proof . The proof is analogous to that of Proposition 4.10.
We say y (i,ī,i) (c) is obtained from y by generation in the ith direction, and we call y (i,ī,i) (c) the immediate descendant of y in the ith direction. We have the following; cf. Lemma 4.4.

Lemma 4.19. Generation in the ith direction (with L i = 2) is degree-increasing if and only if
. If generation in the ith direction is degree-increasing, then the weight at inf inity associated with the critical point represented by y Proof . Recall that (4.19) holds if and only if (4.18) holds. Note also that deg y By direct calculation, one verifies that so we have the result.

Cyclotomic generation: the L i = 2 case
We continue to suppose that i ∈ I is such that L i = 2. Suppose for the rest of this subsection that y represents a cyclotomic critical point. Define y (i,σ) (c) to be the tuple of polynomials given by and y (i,σ) j (x; c) = y j (x) for j ∈ I\σ Z i.
Proof . First let us show that y (i,σ) (x; c) represents a critical point for all but finitely many c ∈ C. As in the proof of Theorem 4.6, we first observe that y (i,σ) is indeed the result of generating in each of the directions i, σi, . . . , σ M i /2−1 i (in any order). By Proposition 4.18 it is enough to check that y (i,ī,i) (c) is generic for all but finitely many c ∈ 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 y (i,σ) (x; c) represents a cyclotomic point. Given Lemma 4.5 and the definition (4.20), it is enough to check that 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 for all j ∈ I.
Proof . Note first that from (4.12) we have for all j ∈ I. It follows that Then, from the definition of y using σΛ 0 = Λ 0 and the property (4.1). Recall (4.19) and the fact that Λ ∞ + ρ, α ∨ i + α ∨ ı is odd, by the assumption (4.3). If Λ ∞ + ρ, α ∨ i + α ∨ ı < 0, then these are all the roots of y

Def inition of the cyclotomic population
Suppose y ∈ P(C[x]) |I| is a tuple of polynomials representing a cyclotomic critical point.
Recall the definition of y (i,σ) (c), from Section 4.5 when L i = 1 and from Section 4.6 when L i = 2. We say y (i,σ) (c) is obtained from y by cyclotomic generation in the direction i.
Let us define the cyclotomic population originated at y to be the Zariski closure of the set of all tuples of polynomials obtained from y by repeated cyclotomic generation, in all directions i ∈ I. 5 The case of type A R : vector spaces of quasi-polynomials

Type A data
Throughout this section we specialise to g = sl R+1 . We shall treat in parallel the cases where R = 2n − 1 and R = 2n, n ∈ Z ≥0 . We have the usual identification of h ∼ = h * with a subspace of (R + 1)-dimensional Euclidean space, given by is the standard orthonormal basis. Let σ : g → g be the unique non-trivial diagram automorphism, whose order is 2. The nodes I of the Dynkin diagram, and the action of σ on these nodes, are as shown below: Let (z i ) N i=1 be nonzero points z i ∈ C × such that z i ± z j = 0 whenever i = j. Let Λ 1 , . . . , Λ N be dominant integral weights.
We suppose the weight at the origin, Λ 0 ∈ h * , obeys σΛ 0 = Λ 0 (as always). That is, In addition, we pick and fix an integer p ∈ {0, 1, . . . , n}, and suppose that and Note the following particular cases: • If R = 2n is even and p = 0 then (5.1) just says that Λ 0 is dominant integral.
In the case p = n (and any R) our choice of Λ 0 obeys the assumptions set out in Section 4.1.

Vector spaces of quasi-polynomials
LetT (4.5). In view of (5.1), . We define the degree, deg p, of a Laurent polynomial p(x) ∈ C[x ± 1 2 ] to be the leading power of x (for large x) that appears in p(x) with non-zero coefficient.
In the remainder of this section, K will denote a decomposable vector space of quasi-polynomials with frameT 1 , . . . ,T R ;Λ ∞ .
Conditions (ii) and (iii) specify the ramification conditions of K at every point z ∈ C. Condition (i) specifies the ramification conditions at ∞. See [15,Section 5.5]. The degrees 0 ≤ d 1 < d 2 < · · · < d R+1 will be called the exponents of K at infinity.
Note that conditions (ii) and (iii) together imply in particular that K has no base points. That is, there is no z ∈ C such that u(z) = 0 for all u ∈ K. They also imply the following important lemma.
Since K is decomposable it follows from condition (i) that K admits a decomposable basis (u k ) R+1 k=1 such that deg u k = d k for each k. We call any such basis a special basis. (n i − n j ).
Lemma 5.6. Let (u i (x)) R+1 i=1 be a special basis of K. Then Proof . By Lemma 5.3, Wr † (u 1 , . . . , u R+1 ) ∈ C x 1 2 . We must show that it has degree zero and compute the constant term. From the condition that Λ −Λ ∞ ∈ Z ≥0 [α i ] i∈I it follows that The result follows by Lemma 5.5.
Let (u k ) R+1 k=1 be a special basis of K. Introduce the subspaces K Sp := span C (u 1 , . . . , u p ) ⊕ span C (u R+2−p , . . . , u R+1 ), By Lemma 5.4, these definitions of do not depend on the choice of special basis (u k ) R+1 k=1 . By Lemma 5.1 we have that, whenever p > 0, then Exceptionally, when p = 0, we have Given a decomposable subspace V , we write sdim V for the pair of numbers

Flags in K
Let FL(K) denote the space of full (i.e., R + 1-step) flags in K.
We say an r-step flag The space of decomposable full flags in K has R+1 2p connected components. These connected components are labeled by 2p-element subsets Q ⊂ {1, . . . , R + 1}. Define FL Q (K) to be the subset consisting of the flags F = {0 = F 0 ⊂ F 1 ⊂ · · · ⊂ F R+1 = K} such that for each k, We call elements of FL Q (K) flags of type Q. For each Q the variety FL Q (K) is isomorphic to the direct product of full flag spaces FL(K Sp ) × FL(K O ). The isomorphism sends a pair of flags F 1,+ ⊂ · · · ⊂ F 2p, Call a 2p-element subset Q ⊂ {1, . . . , R + 1} symmetric if Q is invariant with respect to the involution k → R + 2 − k. In particular, the following subset S is symmetric S := {1, . . . , p, R + 2 − p, . . . , R + 1}. (5.9) If (u k ) R+1 k=1 is a special basis of K then the full flag F = {0 = F 0 ⊂ F 1 ⊂ F 2 ⊂ · · · ⊂ F R+1 = K} defined by F k = span C (u 1 , . . . , u k ), k = 1, . . . , R + 1, (5.10) belongs to FL S (K). By Lemma 5.4 this flag is independent of the choice of special basis.
(we say such a basis is adjusted to F) and then let 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 Λ ∞ (y F ), as in (4.10), belongs to the shifted Weyl orbit ofΛ ∞ [15, Section 3.6]. It is equal toΛ ∞ if and only if F is the flag given in (5.10).
The map F → y F defines a morphism of varieties, This morphism β defines an isomorphism of FL(K) onto its image, as in Lemmas 5.14-5.16 of [15]. Proof . In the exceptional case p = 0 no fractional powers are present at all and the result is clear. Suppose p > 0. Let F ∈ FL S (K) and let (u F k ) R+1 k=1 be a basis of K adjusted to F. By inspection one sees that because F ∈ FL S (K), ) for precisely those k such that the productT k−1 Lemma 5.9. The tuple β(F) = y F is decomposable if and only if F is a decomposable flag. If F is a decomposable flag of type Q then where S Q := (S\Q) ∪ (Q\S) denotes the symmetric difference of S and Q. In particular y F is a tuple of polynomials if and only if Q = S.

Fundamental dif ferential operator and the recovery theorem
To any given a tuple y = (y i (x)) R i=1 ∈ P C x 1 2 R of quasi-polynomials, we may associate a differential operator D(y), defined by with the understanding that y 0 = y R+1 = 1. Here ∂ := ∂/∂x and log f : Theorem 5.10 ([15, Lemma 5.6]). Let y ∈ β(FL(K)). Then K = ker D.

The dual space K †
Let K † be the complex vector space The space K † is a space of quasi-polynomials by Lemma 5.3. The spaces K † and K are dual with respect to the pairing (·, ·) : Given any basis ( where u i denotes omission. We have be the numbers given by by an argument as for Lemma 5.1. Lemma 5.11. Let (u i (x)) R+1 i=1 be a special basis of K. Then deg W k = d † k , k = 1, . . . , R + 1, and the basis (W k ) R+1 k=1 is decomposable.

Cyclotomic points and cyclotomic self-duality
Let us fix (−1) m := e mπi for m ∈ Z/2. Then given a monomial q(x) = x m , m ∈ Z/2, we define q(−x) := (−1) m x m . We extend the transformation q(x) → q(−x) to Laurent polynomials in x 1 2 by linearity. We say that K is cyclotomically self-dual if Lemma 5.12. If K is cyclotomically self-dual then Proof . If K is cyclotomically self-dual then we must have d k = d † R+2−k , k = 1, . . . , R + 1. Comparing (5.4) and (5.12) we see that this implies that and hence Therefore and so, because the right-hand side here does not depend on k, Recall (5.6) and the definition (5.2) of Λ. Using now the fact that Λ, Thus, given (5.14), we have the result.
If K is cyclotomically self-dual then there is a non-degenerate bilinear form B on K defined by i.e., Let us call a tuple of quasi-polynomials y ∈ P C x Proposition 5.13. Let F ∈ FL(K). If the tuple β(F) ∈ P C x 1 2 R is cyclotomic then F is a decomposable flag.
Proof . Let y F = β(F). To prove that F is decomposable it is enough to show that each entry y F k of this tuple lies in C[x] or in from which we conclude that for each k at least one of a k (x) and b k (x) must vanish.
Proof . We shall need the following identity among Wronskian determinants. where f denotes omission.
To prove Theorem 5.14 we argue as for Theorem 6.8 in [15]. Let F ∈ FL(K) be a full flag in K and (u i (x)) R+1 i=1 a basis of K adjusted to this flag. Let y = y F be the corresponding tuple of quasi-polynomials as in (5.11), and (W i (x)) R+1 i=1 the corresponding basis of K † as in (5.5). Then Theorem 5.14 follows from the case k = R + 1 of the following lemma.
R is a nonzero constant by Lemma 5.6 we therefore have Now we may use again the fact that y is cyclotomic, so y k (−x) y R+1−k (x). In view of (5.11), that implies . This completes the proof of Theorem 5.14.
Given a subspace U ⊂ K, let U ⊥ := {v ∈ K : B(u, v) = 0 for all u ∈ U } denote its orthogonal complement in K with respect to the bilinear form B. Recall that a full flag . . , R. Theorem 5.17. Suppose K is cyclotomically self-dual. A full flag F ∈ FL(K) is isotropic if and only if the associated tuple y F 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 F ∈ FL(K) is isotropic then F is decomposable.

Witt bases and the symmetries of the bilinear form B
We say that (r k ) R+1 k=1 is a Witt basis of the cyclotomically self-dual space K if The following lemma gives a useful alternative characterization of Witt bases. Proof . Suppose (u k ) R+1 k=1 is a basis of K and let (W k ) R+1 k=1 be as in (5.5). Then (W i (x)) R+1 i=1 and (u i (−x)) R+1 i=1 are two bases of K † and so u i (−x) = R+1 j=1 C ij W j (x), for some invertible matrix C ij .
Theorem 5.21. Every cyclotomically self-dual space K has a special basis (r k ) R+1 k=1 which is also Witt basis, and in which in fact Proof . Let (u k (x)) R+1 k=1 be a special basis of K. We may suppose that the u k (x) all have leading coefficient 1. Let (W k (x)) R+1 k=1 be the basis of K † as in (5.5). By Lemma 5.11, deg W k = d † k . By Lemma 5.5, we have where the ellipsis indicates terms of lower degree in x and where Since K is cyclotomically self-dual we must have Now from (5.4) we have using which one verifies that Given this equality, if we set In this way, we arrive at for some constants c j k . That is, we have , · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · .
The following are corollaries of Theorem 5.23 together with Lemma 5.20.
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.
Conversely, given any isotropic full flag F = {F 1 ⊂ F 2 ⊂ · · · ⊂ F R+1 = K} there is a Witt basis (r k ) R+1 k=1 such that F k = span C (r 1 , . . . , r k ), k = 1, . . . , R + 1. If in addition F is of type S then this basis can be chosen to be a reduced Witt basis.
Lemma 5.27. The full flag F given in (5.10) is isotropic and hence the corresponding tuple y F is cyclotomic.
Proof . We can choose the special basis (u k ) R+1 k=1 defining F to be the Witt basis of Theorem 5.21. Then the result follows from Corollary 5.26. Lemma 5.29. If Q is symmetric then the variety FL ⊥ Q (K) of isotropic flags is isomorphic to the direct product of spaces of isotropic flags FL ⊥ (K Sp ) × FL ⊥ (K O ) and the isomorphism of these varieties is given by the map η Q defined in (5.8).

Isotropic f lags
In view of these lemmas and Theorem 5.17, we have the following description of the subspace of all cyclotomic tuples within the image β(FL(K)) ⊂ P C x

Inf initesimal deformation of isotropic f lags of type S
The connected Lie group of endomorphisms of K preserving B acts transitively on the variety of isotropic full flags of type Q, FL ⊥ Q (K), for each symmetric subset Q ⊂ {1, . . . , R + 1}. In particular it acts transitively on FL ⊥ S (K), and hence on the cyclotomic tuples of polynomials in the image β(FL ⊥ S (K)) ⊂ P(C[x]) R . We shall describe the infinitesimal action of this group on β(FL ⊥ S (K)). The connected Lie group of endomorphisms of K preserving B preserves each of the subspaces K Sp and K O . Thus this group is the product Sp(K Sp ) × SO(K O ) of the group of special symplectic transformations in End(K Sp ) and the group of special orthogonal transformations in End(K O ). Its Lie algebra sp(K Sp ) ⊕ so(K O ) consists of all traceless endomorphisms X of K such that B(Xu, v) + B(u, Xv) = 0 for all u, v ∈ K.
This choice of basis gives identifications K Sp ∼ = C 2p and K O ∼ = C R+1−2p and hence sp(K Sp ) ∼ = sp 2p and so(K O ) ∼ = so R+1−2p . The Lie algebra sp 2p has root system of type C p . The Lie algebra so R+1−2p has root system of type D n−p if R = 2n − 1 is odd and of type B n−p if R = 2n is even.
Let (E i,j ) R+1 i,j=1 be the basis of End(K) defined by E i,j r k = δ ik r j .
These generators define linear transformations belonging to End(K Sp ) ⊕ End(K O ).
Remark 5.31. The Lie algebra so(K O ) ⊕ sp(K Sp ) is contained in the simple Lie superalgebra osp(K) of all orthosymplectic transformations of the space K. See [10] for the definition. It would be interesting to understand whether this superalgebra plays a role here.
For any k = 1, . . . , p and all c ∈ C, the basis e cX k r is again a Witt basis of K. Let e cX k F denote the corresponding isotropic flag and β(e cX k F) the corresponding tuple representing a cyclotomic point. Let us describe the dependence on c of this tuple.

Populations of cyclotomic critical points in type A
Recall the definition of the extended master function Φ, (2.8). In the setting of the present section (see Section 5.1) it has the explicit form (α c(j) , Λ R+1−i ) log(t j + z i ) + 1≤i<j≤m (α c(i) , α c(j) ) log(t i − t j ) (5.25) and the critical point equations (2.9) become Given a tuple of polynomials y ∈ P(C[x]) R , we have the pair (t, c) ∈ Cm × Im represented by y in the sense of Section 4.2. We say the tuple y represents a critical point of Φ if t is a critical point of Φ(t; c; z; Λ; Λ 0 ), i.e., if (t, c) satisfy the equations (5.26).
The following theorem says that we can go from cyclotomic critical points of the extended master function Φ, (5.25), to decomposable cyclotomically self-dual vector spaces of quasipolynomials.
The kernel ker D(y) of the fundamental differential operator D(y), Section 5.4, is a decomposable cyclotomically self-dual vector space of quasi-polynomials with frameT 1 , . . . ,T R ;Λ ∞ , whereΛ ∞ is the unique dominant weight in the orbit of Λ ∞ (y), (4.10), under the shifted action of the Weyl group of type A R .
Conversely, we have the following, arguing as in Lemmas 3.1, 3.2 and 5.15 in [15] and using Theorem 5.17.
Suppose there exists an isotropic flag F ∈ FL ⊥ S (K) such that the tuple y F is generic. Then y F represents a cyclotomic critical point of Φ, (5.25).
Since being generic is an open condition, the set of generic tuples in the image β(FL ⊥ S (K)) is either empty or it is open and dense in β(FL ⊥ S (K)). Starting from an initial tuple y that represents a cyclotomic critical point of Φ, (5.25), we may let K = ker D(y) as in Theorem 5.32. Then we have the variety β(FL ⊥ S (K)) ∼ = FL ⊥ (K Sp ) × FL ⊥ (K O ), (5.27) where the isomorphism is by Theorem 5.30. Almost all of the tuples in β(FL ⊥ S (K)) are generic and hence represent cyclotomic critical points of Φ. Call this variety β(FL ⊥ S (K)) ⊂ P(C[x]) R the cyclotomic population originated at y.
This establishes the lemma.