Orbits and invariants of super Weyl groupoid

We study the orbits and polynomial invariants of certain affine action of the super Weyl groupoid of Lie superalgebra $\mathfrak {gl}(n,m)$, depending on a parameter. We show that for generic values of the parameter all the orbits are finite and separated by certain explicitly given invariants. We also describe explicitly the special set of parameters, for which the algebra of invariants is not finitely generated and does not separate the orbits, some of which are infinite.


Introduction
Let G be a finite group acting linearly on a finite dimensional vector space V over field of characteristic zero and Λ G = S(V * ) G be the algebra of the polynomial invariants. It is well-known that this algebra is finitely generated and separates the orbits, so that for any two orbits there exists an invariant f ∈ Λ G , which takes different values on these orbits, see e.g. [1]. Classical example is a finite Coxeter group generated by the hyperplane reflections in a real Euclidean space V , when by Chevalley theorem the corresponding algebra of invariants is freely generated by basic invariants [5].
In this paper instead of a finite group we consider the super Weyl groupoid W n,m , which is a finite groupoid introduced in [10] in relation with the Grothendieck ring of the Lie superalgebra gl(n, m). We will consider a special action of this groupoid depending on a parameter θ, which arised from the theory of the deformed quantum Calogero-Moser systems [8]. The algebra of invariants in that case is isomorphic to the algebra of the corresponding quantum integrals.
The case θ = 1 corresponds to the Lie superalgebra gl(n, m), when the corresponding invariants are known as supersymmetric polynomials and generated by the characters of the polynomial representations [7]. The corresponding algebra is not finitely generated and does not separate the orbits, some of which turned out to be infinite, which is quite different from the case of finite groups.
It turns out that for generic θ the situation is pretty similar to the case of finite groups. More precisely, we prove the following result.
Let τ θ be the action of the super Weyl groupoid W n,m described in the next section. We say that parameter θ is special if θ = p q for some 1 ≤ p ≤ m, 1 ≤ q ≤ n. Theorem 1.1. All the orbits of W n,m are finite if and only if θ is nonspecial. The algebra of polynomial invariants in that case is finitely generated and for non-rational θ separates the orbits.
For the special values of θ the algebra of invariants is not finitely generated and does not separate the orbits. We demonstrate this in the last section in the special cases θ = 1 and θ = 1/2.

Super Weyl groupoid and its actions
A groupoid G is a small category with all morphisms being invertible (see [3,13] for the details). The set of objects is denoted as B and called the base, the set of morphisms is usually denoted G as groupoid itself.
If the base B consists of one element G has a group structure. More generally, for any x ∈ B one can associate an isotropy group G x consisting of all morphisms g ∈ G from x into itself. For any groupoid we have a natural equivalence relation on the base B, when x ∼ y if there exists a morphism g ∈ G from x to y.
For any set X one can define the following groupoid S(X), whose base consists of all possible subsets Y ⊂ X and the morphisms are all possible bijections between them. By the action of a groupoid G on a set X we will mean the homomorphism of G into S(X) (which is a functor between the corresponding categories). If X is affine space, Y ⊂ X are the affine subspaces and morphisms are affine bijections, the we will call it affine action.
Following [10] we define now the super Weyl groupoid related to any basic classical Lie superalgebra g. The corresponding roots form a generalised root system R ⊂ V in Serganova's sense [6], which is certain generalisation of the root system in the presence of the isotropic roots. For isotropic roots one can not define the reflections, which leads to the absence of the Weyl groups in this case. The reflections with respect to the non-isotropic roots generate the small Weyl group W 0 , which describes a partial symmetry of the system.
Consider the following groupoid T iso with the base R iso , which is the set of all the isotropic roots in R. The set of morphisms from α → β is nonempty if and only if β = ±α in which case it consists of just one element.
The super Weyl groupoid is defined [10] as the disjoint union of the group W 0 considered as a groupoid with a single point base [W 0 ] and the semi-direct product groupoid W 0 ⋉T iso with the base R iso . Note that the disjoint union is a well defined operation on the groupoids [3].
In [8] we defined the admissible deformations of generalised root systems. The roots remain the same, but the bilinear form B on V is deformed and depends for the classical series on one parameter, which following [9] we denote θ (in the notations of [8] θ = −k).The case θ = 1 corresponds to the original generalised root system.
Let X = V with the deformed bilinear form ( , ) and define the following affine action π of the Weyl groupoid W(R) on it. The base point The elements of the group W 0 are acting in a natural way and the element τ α acts as a shift As it was shown in [8] the algebra of invariants of this action Λ R,B is isomorphic to the algebra of quantum integrals of the corresponding deformed Calogero-Moser systems. In fact, the definition of the super Weyl groupoid was mainly motivated by this isomorphism.
In the case of Lie superalgebra gl(n, m) the root system in the basis The small Weyl group W 0 = S n × S m acts by separately permuting ε i , i = 1, . . . , n and δ p , p = 1, . . . , m. The corresponding super Weyl groupoid we denote as W n,m .
The deformed bilinear form is determined by For α = ε i − δ p ∈ R iso the corresponding hyperplanes Π ±α have the equations The corresponding element τ α ∈ W n,m acts on the hyperplane Π −α by the formula where z = (u 1 , . . . , u n , v 1 , . . . , v n ).
In the simplest example n = m = 1 we have two lines L ± on the plane defined by and the shifts: . In this case we have a very simple groupoid T 2 with the base consisting of two points connected by two morphisms, which are inverse to each other. For θ = 1 the orbits are single points outside of these lines and the pairs of points (u, v), (u + 1, v − 1) on these lines. The case θ = 1 is special: in this case two lines are the same line L given by u + v = 0 and preserved by the shifts. The orbits are still single points outside L, but on the line they are infinite and consist of the points (u + k, −u − k), k ∈ Z.
For general m and n the situation is more complicated, but as we will show now is still similar.

Orbits and invariants
The invariants of the action form the subalgebra consisting of polynomials f , which are symmetric in u 1 , . . . , u n and v 1 , . . . , v m separately and satisfy the invariance (or, in the terminology of [8], quasiinvariance) conditions on each hyperplane u i + θv j = 0 for i = 1, . . . , n and j = 1, . . . , m. This algebra first appeared in [8] as the image Λ n,m,θ of the algebra D(n, m, θ) of quantum integrals of the deformed Calogero-Moser system under the Harish-Chandra homomorphism.
Theorem 3.1. [8] For non-rational θ the algebra Λ n,m,θ is finitely generated and is generated by the deformed Bernoulli sums where B k (x) are classical Bernoulli polynomials.
Recall (see e.g. [14]) that Bernoulli polynomials and satisfy the following property: which implies that b k (u, v, θ) satisfy the quasi-invariance conditions (5).
Recall that θ is non-special if θ = p q for all 1 ≤ p ≤ m, 1 ≤ q ≤ n. Theorem 3.2. For non-special θ all the orbits are finite. The converse statement is also true.
Proof. Since b k (u, v, θ) are constants on the orbits, it is enough to prove that the system b k (u, v, θ) = c k , k = 1, . . . , n + m (7) for non-special θ has only finite number of solutions for all c 1 , . . . , c n+m .
To prove this we use the following result.
has only zero solution in C N , then the system (8) has only finitely many solutions in C N with the sum of the multiplicities equal to m 1 . . . m N .
Proof. 1 Assume that the system (8) has infinitely many solutions, then the corresponding algebraic solution set must contain an algebraic curve. The closure of the curve in the projective space CP N must intersect the infinite hyperplane. But the intersection is described the system (9), which has no solution. Contradiction means that the system (8) has only finitely many solutions, whose number is given by the classical projective Bèzout's theorem (see e.g. [12]).
Proof. To prove this suppose that the system has non-zero solution. We can assume that all x i are non-zero. Let us change the variables x i → −θx i for i = 1, . . . , n, so that the system becomes Let us identify equal x i as {z 1 , . . . , z p }, p ≤ n + m, where all z j are different. Multiplicity of z j is a pair (k j , l j ), where k j shows how many times z j enters the set {x 1 , x 2 , . . . , x n } and l j is the same for the rest of the set x i . Consider the first p of these equations where a j = k j − θ −1 l j as a linear system on a j . Its determinant is of Vandermonde type and is not zero since all z j are different and non-zero. Hence all a j must be zero which may happen only if θ is special.
Combining this Lemma with the affine Bèzout's theorem we conclude that for non-special θ the system (7) has only finite number of solutions for all c k , and thus all the orbits are finite.
To prove the converse statement one should produce an infinite orbit for every special θ = p/q, which can be done. For example, in the case n = 2, m = 1, θ = 1/2 one can check that all the points belong to the same orbit (see also the last section).
Corollary 3.5. For non-special θ all the orbits consist of not more than (m + n)! elements.
The rest of the section is the proof of the remaining part of our main theorem 1.1.
We will assume in the rest of this section that θ is non-rational.
We are going to show that the deformed Bernoulli sums b k (u, v, θ), k ∈ Z separate the orbits. To prove this we need to study the orbits in more details.
(10) Define also where Proof. According to the properties Bernoulli polynomials we have te −θy j t e −θt − 1 , 6 where x i = u i + 1/2, y j = v j − 1/2 and similarly forx andỹ. So we see that the conditions (10) are equivalent to the relation which can be rewritten as This implies that Lemma now follows.
Let us define now the equivalence relation on the pairs of monic polynomials (f (t), g(t)) by writing (f, g) ∼ (f ,g) whenever they satisfy the relation Restricting this to the monic polynomials f (t), g(t) with the degrees n and m respectively we recover the previous equivalence relation (10). Let us call a pair of monic polynomials (f (t), g(t)) minimal if f (x) = 0 implies that g(x + θ − 1) = 0.
Define also Theorem 3.7. If θ is not rational that any equivalence class contains a unique minimal pair (f (t), g(t)). If such f (t) = n i=1 (t − x i ) then any equivalent pair (f (t),g(t)) has the form for some k 1 , . . . , k n ∈ Z ≥0 .
Proof. Let us prove first the existence of the minimal pair. Let x be any root of f (t) and assume that g(x + θ − 1) = 0. Consider It is easy to see that (f, g) ∼ (f 1 , g 1 ). If g 1 (x + θ − 1) = 0 we can apply this procedure to (f 1 , g 1 ) and continue in the same way to get a sequence of equivalent pairs (f, g), (f 1 , g 1 ), . . . . All pairs are different as the sum of roots of f i is equal to the sum of roots of f minus i. According to Theorem 3.5 the sequence contains only finite number of elements. The last element of the sequence is minimal. The rest of the proof follows from the following lemmas. We start with the following technical lemma.
Lemma 3.8. Let θ be a complex number and Z be a finite set of complex numbers. Suppose that there exists a map F : Z → Z such that F (z) = z + l(z) − θ, where l(z) ∈ Z for any z ∈ Z. Then θ is a rational number.
Proof. For any F : Z → Z we can find a subsetZ in Z such that the restriction of F onZ is a one to one correspondence onZ. LetZ = {z 1 , . . . , z N }. According to our assumptions we haveZ = F (Z) = {z 1 + l 1 − θ, . . . , z N + l N − θ}. Therefore so θ is rational. Proof. Assume that for any x,x such that f (x) = 0 andf (x) = 0 we have x −x / ∈ Z or x −x ∈ Z and |x −x| > deg g + degg. Let us rewrite the relation (12) in the following form where ψ(t) = f (t) f (t) , χ(t) =g (t) g(t) . For any function h(t) denote by h k (t) the following function h k (t) = h(t)h(t − 1) . . . h(t − k).
Multiply the relations (14) with t = t, t − 1, t − 2, . . . t − k to have ψ k (t)χ(t) = ψ k (t − θ)χ(t − k − 1), or equivalentlỹ Take k = deg g + degg. Let us note that according to our assumption polynomials f k andf k have no common roots. The same is true for f k (t − θ)