A REMARK ON QUIVER VARIETIES AND WEYL GROUPS

ANDREA MAFFEI

Abstract. In this paper we define an action of the Weyl group on the quiver varieties M_{m,λ}(d, v) with generic (m, λ). To do it we describe a set of generators of the projective ring of a quiver variety. We also prove connectedness for the smooth quiver variety M(d, v) and normality for M_{0}(d, v) in the case of a quiver of finite type and d − v a regular weight.

In [13, 14] Nakajima defined quiver varieties and show how to use them to give a geometric construction of integrable representation of Kac-Moody algebras. Luckily these varieties can be used also to give a geometric construction of representations of Weyl groups. In [9], Lusztig constructed a representation of the Weyl group on the homology of quiver varieties. His construction is similar to the construction of Springer representations. In [14], Nakajima gave an construction of isomorphism Φ_{σ,ζ}(d, v) : 𝔐_{ζ}(d, v) ⟶ 𝔐_{σζ}(d, σ(v − d) + d) in the case of a quiver of finite type. His construction was analytic and relies on a description of quiver varieties as moduli spaces of instantons on ALE spaces. The main result of this paper is a direct and algebraic construction of these isomorphism which works for a general quiver without simple loops. To do it we also describe a set of generators of the algebra of covariant functions.

The paper is organized as follows. In the first section we fix the notation and we give the definition of a quiver variety: M_{m,λ}(d, v) where m, λ are two parameter, d is a weight of the algebra associated to the quiver and v an element of the root lattice. We are interested to quiver varieties as algebraic varieties but to explain one of the applications we need to give also the hyperKähler construction of a quiver variety. We use a result of Migliorini [11] to explain the connection between the two constructions.

Algebraic quiver varieties are defined as the Proj scheme of a ring of covariants. In the second section we describe a set of generators of this ring. In a special case which is not directly to Nakajima’s quiver varieties we are also able to give a more precise results and to describe a basis of the vector space of χ-covariants functions.

In the third section we use this description to generalize a construction of Lusztig [9]. Namely for any element of the Weyl group we construct an isomorphism Φ_{σ} between M_{m,λ}(d, v) and M_{σm,σλ}(d, σ(v − d) + d) if m, λ are generic.

In the fourth section, following Nakajima [13], we show how to use the action constructed in section 3 (and the connection between the hyperKähler construction and the algebraic construction) to describe an action of the Weyl group on the homology of a class of quiver varieties. This action is different from the one constructed by Lusztig in [9].

The research was supported by the MSRI, Berkeley.

In the fifth section we give a result which reduce the study of geometric and algebraic properties of quiver varieties M_{0,0}(d, v) to the case d − v dominant.

In the sixth section we prove the normality of the quiver variety M_{0}(d, v) and the connectedness of M(d, v) in the case of a quiver of finite type and d − v a regular weight.

1. Notations and definitions

In this section we give the definition of quiver varieties. Except some minor change all definition are due to Nakajima [13, 14].

1.1. The graph. Let (I, H) be a finite oriented graph: I is the set of vertices that we suppose of cardinality n, H the set of arrows and the orientation is given by the two maps

h ⟼ h_{0} and h ⟼ h_{1}

from H to I. We suppose also that:

1. ∀ h ∈ H h_{0} ≠ h_{1},

2. an involution h ↦ h̄ of H without fixed points and satisfying h̄_{0} = h_{1} is fixed,

3. a map ε : H ⟶ {−1, 1} is given such that ε(h̄) = −ε(h). We define Ω = {h ∈

H : ε(h) = 1\} and \overline{Ω} = \{h ∈ H : ε(h) = −1\}. Observe that given a symmetric graph without loops is always possible to define ε and an involution ¯ as above.

1.2. The Cartan matrix and the Weyl group. Let A be the matrix whose entries are the numbers

a_{ij} = card{h ∈ H : h_{0} = i and h_{1} = j}.

We define a generalized symmetric Cartan matrix by C = 2I − A. Following [7] an X, Y -regular root datum (I, X, X^{∨}, < , >) with Cartan matrix equal to C is defined in the following way:

1. X^{∨} and X are finitely generated free abelian groups,

2. < , >: X × X^{∨} ⟶ ℤ is a perfect bilinear pairing,

3. two linearly independent sets Π = {α_{i} : i ∈ I} ⊂ X and Π^{∨} = {α_{i}^{∨} : i ∈ I} ⊂ X^{∨} are fixed and we set Q = ⟨Π⟩ and Q^{∨} = ⟨Π^{∨}⟩,

4. <α_{i} , α_{j}^{∨}>= c_{ij},

5. (nonstandard) rank X = rank X^{∨} = 2n − rank C,

6. (nonstandard) a linearly independent set {ω_{i} : i ∈ I} of X such that <ω_{i}, α_{j}^{∨}>=

δ_{ij} is fixed. Once C is given it is easy to construct a data as above. We call 𝔥 the complexification of X^{∨} and we observe that through the bilinear pairing < , > we can identify 𝔥^{∗} with the complexification of X. We observe also that the triple (𝔥, Π, Π^{∨}) is a realization of the Cartan matrix C ([5] pg.1).

The Weyl group W attached to C is defined as the subgroup of Aut(X) ⊂ GL(𝔥^{∗}) generated by the reflections

s_{i} : x ⟼ x−<x, α_{i}^{∨}> α_{i}.  (1)

Observe that the dual action is given by s_{i}(y) = y−<α_{i}, y> α_{i}^{∨} and that the lattices

Q and Q^{∨} are stable for these actions. So the annihilator Q^{∨}^{∘} = {x ∈ X : <x, y> = 0 ∀y ∈ Q^{∨}} is also stable by W and we can consider the action of W on the lattice

P = X / Q^{∨}^{∘} ≃ Hom_{ℤ}(Q, ℤ) and we call x ↦ \overline{x} the projection from X to P. We observe also that this projection is an isomorphism from the lattice P̃, that is not W -stable, spanned by {ω_{i} : i ∈ I} and P. Finally we observe that

\overline{α}_{i} = ∑_{j∈I} c_{ij}\overline{ω}_{j}.

1.3. d, v and the space of all matrices. For the exposition it will be useful to identify the set I with the set of integers {1, …, n}.

Let d = (d_{1}, …, d_{n}) and v = (v_{1}, …, v_{n}) be two n-tuples of integers. We also think of d, v as elements of X in the following way:

d = ∑_{i∈I} d_{i}ω_{i} and v = ∑_{i∈I} v_{i}α_{i} ;  (2)

and through this identification we define an action of W on v. We define also v^{∨} = ∑_{i∈I} v_{i}α_{i}^{∨} ∈ Q^{∨}. Once d, v are fixed we fix complex vector spaces D_{i} and V_{i} of dimensions d_{i} and v_{i} and we define the following spaces of maps:

S_{Ω}(d, v) = ⨁_{i∈I} Hom(D_{i}, V_{i}) ⊕ ⨁_{h∈Ω} Hom(V_{h_{0}}, V_{h_{1}}),  (3a)
S\overline{Ω}(d, v) = ⨁_{i∈I} Hom(V_{i}, D_{i}) ⊕ ⨁_{h∈\overline{Ω}} Hom(V_{h_{0}}, V_{h_{1}}),  (3b)
S(d, v) = S_{Ω}(d, v) ⊕ S\overline{Ω}(d, v).  (3c)

More often, when it will not be ambiguous we will write S_{Ω}, S\overline{Ω} and S instead of S_{Ω}(d, v), S\overline{Ω}(d, v) and S(d, v).

For each h ∈ H (resp. i ∈ I) we define the projection B_{h} (resp. γ_{i} and δ_{i}) from S to Hom(V_{h_{0}}, V_{h_{1}}) (resp. Hom(D_{i}, V_{i}) and Hom(V_{i}, D_{i})) with respect to the decomposition described in (3).

When an element s of S is fixed we will often write B_{h} (resp. γ_{i}, δ_{i}) instead of B_{h}(s) (resp. γ_{i}(s) and δ_{i}(s)). We will also use γ for (γ_{1}, …, γ_{n}), δ for (δ_{1}, …, δ_{n}) and B for (B_{h})_{h∈H} and often we will write an element of S as a triple (B, γ, δ).

Once D_{i}, V_{i} and an element s of S are fixed we define also:

T_{i} = D_{i} ⊕ ⨁_{h: h_{1}=i} V_{h_{0}} ,  (4a)
a_{i} = a_{i}(s) = (δ_{i}(s), (Bh̄(s))_{h: h_{1}=i}) : V_{i} ⟶ T_{i} ,  (4b)
b_{i} = b_{i}(s) = (γ_{i}(s), (ε(h)B_{h}(s))_{h: h_{1}=i}) : T_{i} ⟶ V_{i} .  (4c)

We will identify the dual of space of the ℂ-linear maps Hom(E, F) between two finite dimensional vector spaces with Hom(F, E) through the pairing <φ, ψ>= Tr(φ∘ ψ). So we can describe S also as S_{Ω} ⊕ S_{Ω}^{∗} = T^{∗}S_{Ω} and we observe that a natural symplectic structure ω is defined over S by

ω((s_{Ω}, s\overline{Ω}), (t_{Ω}, t\overline{Ω})) =<s_{Ω}, t\overline{Ω}> − <t_{Ω}, s\overline{Ω}> .

1.4. Hermitian structure on S. We suppose now that the spaces D_{i}, V_{i} are endowed with hermitian metrics. So we can speak of the adjoint φ^{∗} of a linear map between these spaces, and we have a positive definite hermitian structure h on S with explicit formula:

h((B, γ, δ), (B̃, γ̃, δ̃)) = ∑_{h∈H} Tr(B_{h}B̃_{h}^{∗}) + ∑_{i∈I} Tr(γ_{i}γ̃_{i}^{∗} + δ̃_{i}^{∗}δ_{i})  (5)

= ∑_{i∈I} Tr(a_{i}ã_{i}^{∗} + b̃_{i}^{∗}b_{i})

and an associated real and closed symplectic form ω_{I}(s, t) = Re h(is, t) − Im h(s, t).

1.5. Group actions and moment maps. We can define an action of the groups G = GL(V) = ∏ GL(V_{i}) and GL(D) = ∏ GL(D_{i}) on the set S in the following way:

g(B_{h}, γ_{i}, δ_{i}) = (g_{h_{1}}B_{h}g_{h_{0}}^{−1}, g_{i}γ_{i}, δ_{i}g_{i}^{−1})  for g = (g_{i}) ∈ GL(V),  (6)
g(B_{h}, γ_{i}, δ_{i}) = (B_{h}, γ_{i}g_{i}^{−1}, g_{i}δ_{i})  for g = (g_{i}) ∈ GL(D).  (7)

Observe that these actions commute and that ω is GL(V) invariant. Moreover if U = U(V) = ∏ U(V_{i}) is the group of unitary transformations in GL(V) the real symplectic form ω_{I} is U(V) invariant.

Define μ, μ_{I}: S ⟶ 𝔤 = ⊕gl(V_{i}) by the following explicit formulas:

μ_{i}(B, γ, δ) = ∑_{h∈H : h_{1}=i} ε(h)B_{h}Bh̄ + γ_{i}δ_{i} = b_{i}a_{i} ,

μ_{I,i}(B, γ, δ) = \frac{i}{2} ( ∑_{h∈H : h_{1}=i} B_{h}B_{h}^{∗} − B∗ ; h̄Bh̄ + γ_{i}γ_{i}^{∗} − δ_{i}^{∗}δ_{i}) = \frac{i}{2}(b_{i}b_{i}^{∗} − a_{i}^{∗}a_{i}),

If we identify 𝔤^{∗} = Hom_{ℂ}(𝔤, ℂ) (resp. 𝔲^{∗} = Hom_{ℝ}(𝔲, ℝ)) with 𝔤 = ⊕gl(V_{i}) (resp. 𝔲) through the pairing <(x_{i}) , (y_{i})>= ∑_{i} Tr(x_{i}y_{i}), we can observe that μ is a moment map for the action of G on the symplectic manifold (S, ω) and that μ_{I} is a moment map for the action of U on the symplectic manifold (S, ω_{I}). It is common to group all these moment maps together and to define an hyperKähler moment map

μ̃ = (μ_{I}, μ) : S ⟶ 𝔲 ⊕ 𝔤 = (ℝ ⊕ ℂ) ⊗_{ℝ} 𝔲.

1.6. Quiver varieties as hyperKähler quotients. Let ζ_{i} = (ξ_{i}, λ_{i}) ∈ ℝ ⊕ ℂ and ζ = (ζ_{1}, …, ζ_{N}). We define:

𝔏_{ζ}(d, v) = {s ∈ S : μ(s) − λ_{i}Id_{V_{i}} = 0 and μ_{I,i}(s) − iξ_{i}Id_{V_{i}} = 0}.

We observe that 𝔏_{ζ}(d, v) is stable for the action of U(V), so, at least as a topological Hausdorff space we can define the quiver variety of type ζ as

𝔐_{ζ}(d, v) = 𝔏_{ζ}(d, v)/U(V).

It will be convenient to define also 𝔐_{ζ}(d, v) = ∅ if d, v ∈ ℤ^{n} and there exists i such that v_{i} < 0 or d_{i} < 0 for some i. We call ℨ = ℝ^{n} ⊕ ℂ^{n} and we observe that we can identify it to (ℝ ⊕ ℂ) ⊗_{ℤ} P through:

(ξ_{1}, …, ξ_{n}, λ_{1}, …, λ_{n}) ⟷ ∑_{i∈I}(ξ_{i}, λ_{i})\overline{ω}_{i}.  (8)

In particular we consider an action of the Weyl group W on ℨ through this identification.

Remark 1. There is a surjective map from ℨ to Z_{U}(𝔲) ⊕ Z_{G}(𝔤):

(ξ_{1}, …, ξ_{n}, λ_{1}, …, λ_{n}) ⟶ ∑_{i∈I}(iξ_{i}, λ_{i})Id_{V_{i}}

Observe that 𝔏_{ζ} is the fiber of μ̃ over the image of ζ in Z_{U}(𝔲) ⊕ Z_{G}(𝔤).

Remark 2. If v, d ≥ 0 define: I^{∗} = {i ∈ I : v_{i} ≠ 0}, H^{∗} = {h ∈ H : h_{0}, h_{1} ∈ I^{∗}}, ε^{∗} = ε|_{H^{∗}}, v^{∗} = (v_{i})_{i∈I^{∗}}, d^{∗} = (d_{i})_{i∈I^{∗}}, and ζ^{∗} = (ζ_{i})_{i∈I^{∗}} then it is clear that

𝔏_{ζ}^{∗}(d^{∗}, v^{∗}) ≃ 𝔏_{ζ}(d, v) and 𝔐_{ζ}^{∗}(d^{∗}, v^{∗}) ≃ 𝔐_{ζ}(d, v).

Except for the last equivalence which is trivial in our case the following is a general well known fact ([12] ch. 8).

Lemma 3. Let s ∈ 𝔏_{ζ} then

dμ̃_{s} is surjective ⟺ dμ_{s} is surjective ⟺ dμ_{I} is surjective ⟺
⟺ dim Stab_{G}{s} = 0 ⟺ Stab_{G}{s} = {1_{G}}

Definition 4. If u ∈ ℤ^{n} = Q^{∨} A ⊂ Q^{∨} we define

H_{u} = {ζ = (ξ, λ) ∈ ℨ : <ξ , u^{∨}>=<λ , u^{∨}>= 0} and H_{A} = ⋃_{u∈A} H_{u}.

Let now U_{v} = {u ∈ ℕ^{n} − {0} such that 0 ≤ u_{i} ≤ v_{i}} and H = H_{U_{v}}. H is a union of a finite number of real subspace of ℨ of codimension 3.

Lemma 5 (Nakajima, [13]). If ζ ∈ ℨ − H and μ̃(s) = ζ then Stab_{G}{s} = 1_{G}.

As a consequence of the above lemma and general results on on hyperKähler manifolds (for example [3] or [4]) we obtain the following corollary.

Corollary 6. If ζ ∈ 𝔷 − H then if it is not empty 𝔐_{ζ}(d, v) is a smooth hyperKähler manifold of real dimension 2 <2d − v, v^{∨}>.

1.7. Geometric invariant theory and moment map. In this section we explain the relation between the moment map and the GIT quotient proved by Kempf, Ness [6], Kirwan [12] and others. To be more precise we need a generalization of their results in the case of an action on an affine variety proved by Migliorini [11].

Let X be an affine variety over ℂ and G a reductive group acting on X. We can assume that X is a closed sub-variety of a vector space V where G acts linearly. Let h be an hermitian form on V invariant by the action of a maximal compact group U of G and define a real U-invariant symplectic form on V by

η(x, y) = Re h(ix, y).

Then we can define a moment map ν : V ⟶ 𝔲^{∗} = Hom_{ℝ}(𝔲, ℝ):

<ν(x), u>= \frac{1}{2}η(u · x, x).

We observe that the real symplectic form η restricted to a complex sub-manifold is always non degenerate and that μ restricted to the non singular locus of X is a moment map for the action of U on X.

Now let χ be a multiplicative character of G. We observe that for all g ∈ U we have |χ(g)| = 1 so i dχ : u ⟶ ℝ. In particular we can think to i dχ as an element of 𝔲^{∗}. Moreover we observe that it is invariant by the dual adjoint action, hence it makes sense to consider the quotient:

𝔐 = ν^{−1}(i dχ)/U.

As we saw our variety are a particular case of this construction.

On the other side we can consider the GIT quotient. Let us remind the definition. If φ is a character of G we consider the line bundle L_{φ} = V × ℂ on V with the following G-linearization:

g(x, z) = (g · x, φ(g)z).

An invariant section of L_{φ} is determined by an algebraic function f : V ⟶ ℂ such that f(gx) = φ(g)f(x) for all g ∈ G and x ∈ V. We use the same symbol L_{φ} also for the restriction of L_{φ} to X.

Given a rational action of G on ℂ-vector space A we define

A_{φ,n} = {a ∈ A : g · a = φ^{−n}(g)a for all g ∈ G},
A_{φ} = ⨁_{n=0}^{∞} A_{φ,n} as a graded vector space.

Hence we have that H^{0}(X, L_{φ})^{G} = ℂ[X]_{φ,1}. We observe that if I is the ideal of algebraic function on V vanishing on X then

H^{0}(X, L_{φ})^{G} = \frac{H^{0}(V, L_{φ})^{G}}{I_{φ,1}} .

This last fact can be proved easily for example averaging a φ equivariant function f on X in the following way:

f̃(v) = ∫_{U} φ^{−1}(u)f(u · v) du.

Definition 7. A point x of X is said to be χ-semi-stable if there exist n > 0 and f ∈ H^{0}(X, L_{χ}^{⊗n})^{G} such that f(x) ≠ 0. We observe that by the remark above a point of X is χ-semi-stable if and only if is χ-semi-stable as a point of V. We call X_{χ}^{ss} (resp. V_{χ}^{ss}) the open subset of χ-semi-stable points of X (resp. V).

Proposition 8 ( [12, 15]). There exists a good quotient of X_{χ}^{ss} by the action of G and we have that

X_{χ}^{ss}//G = Proj ℂ[X]_{χ}.

Moreover Proj ℂ[X]_{χ} is a finitely generated ℂ-algebra and a natural projective map

π : X_{χ}^{ss}//G ⟶ X//G = Spec ℂ[X]^{G}

is defined.

In the case of χ ≡ 1 the following fact is well known:

Proj ℂ[X]_{χ} = Spec ℂ[X]^{G} = ν^{−1}(0)/U.

The following result is less well known, and its proof requires some adjustment of the classical proof for the case χ ≡ 1 (see for example an appendix of [11] or [10] ).

Proposition 9 (Migliorini, [11]). Let x ∈ X then

∃g ∈ G : ν(gx) = idχ ⟺ Gx is a closed orbit in X_{χ}^{ss}.

Proposition 10 (Migliorini, [11]). The inclusion ν^{−1}(idχ) ⊂ X_{χ}^{ss} induces an homeomorphism

ν^{−1}(idχ)/U ≃ X_{χ}^{ss}//G.

1.8. Quiver varieties as algebraic varieties. If m = (m_{1}, …, m_{n}) ∈ ℤ^{N} we define a character χ_{m} of G_{v} by χ_{m} = ∏_{i∈I} det_{GL(V_{I})}^{m_{i}}.

If λ = (λ_{1}, …, λ_{n}) ∈ ℂ^{n} and m = (m_{1}, …, m_{n}) ∈ ℤ^{n} then we define the varieties:

Λ_{λ}(d, v) = {s ∈ S : μ_{i}(s) − λ_{i}Id_{V_{i}} = 0 for all i},
Λ_{m,λ}(d, v) = {s ∈ Λ_{λ}(d, v) : s is χ_{m} − semi-stable}.

and the associated quiver varieties

M_{m,λ}(d, v) = Λ_{m,λ}(d, v)//G_{v} and
M_{λ}^{0}(d, v) = M_{0,λ}(d, v) = Λ_{λ}(d, v)//G_{v}

We call p_{m,λ}^{d,v}: Λ_{m,λ}(d, v) ⟶ M_{m,λ}(d, v) the quotient map. Observe that the inclusion Λ_{m,λ}(d, v) ⊂ Λ_{λ}(d, v) induces a projective morphism

π_{m,λ}^{d,v} : M_{m,λ}(d, v) ⟶ M_{λ}^{0}(d, v).

Finally it will be convenient to define M_{m,λ}(d, v) = ∅ if d, v ∈ ℤ^{n} and v_{i} < 0 or d_{i} < 0 for some i.

We identify ℤ^{n} with P and Z = ℂ^{n} with ℂ ⊗_{ℤ} P through

(m_{1}, …, m_{n}) ⟶ ∑ m_{i}\overline{ω}_{i} and (λ_{1}, …, λ_{n}) ⟶ ∑ λ_{i}\overline{ω}_{i}.

Remark 11. As in 1 we have a surjective map from Z to Z(𝔤) and Λ_{λ}(d, v) is the fiber of μ over the image of λ in Z(𝔤).

Remark 12. Remark 2 holds without changes also in this case.

Remark 13. Observe that P ⊕ Z ⊂ ℨ. Observe also that the map m ⟶ χ_{m} define a surjective morphism from P to Hom(G_{v}, ℂ^{∗}) and that the following diagram commute:

[figure: P; Hom(G, ℂ^{∗}); ∋; χ; ℝ^{n}; Z(𝔲) ≃ (𝔲^{∗})^{U}; ∋; idχ]

In particular we can apply 10 to the action of G_{v} on Λ_{λ}(d, v) and we obtain:

𝔐_{(m,λ)}(d, v) ≃ M_{m,λ}(d, v).

Proposition 14. Let (m, λ) ∉ H and s ∈ Λ_{m,λ}(d, v) then Stab_{G_{v}}(s) = {1}.

Proof. As we have already claimed it is enough to prove dim Stab_{G}(s) = {1}. We know that there is a good quotient of Λ_{m,λ}(d, v) so it is enough to prove that any closed orbit has maximal dimension. By Proposition 9 if G_{v}s is closed in Λ_{m,λ}(d, v) then there exists g ∈ G_{v} such that μ_{I}(gs) = idχ. The thesis follows now form (m, λ) ∉ H and Lemma 5.

Corollary 15. If (m, λ) ∉ H and M_{m,λ}(d, v) ≠ ∅ then it is a smooth algebraic variety of dimension <v^{∨}, 2d − v>.

1.9. Path algebra and b-path algebra. To describe functions on quiver varieties we need some notation about the path algebra.

Definition 16. A path α in our graph is a sequence h^{(m)} … h^{(1)} such that h^{(i)} ∈ H and h_{1}^{(i)} = h_{0}^{(i+1)} for i = 1, …, m − 1. We define also α_{0} = h_{0}^{(1)}, α_{1} = h_{1}^{(m)} and we say that the length of α is m. If α_{0} = α_{1} we say that α is a closed path. We consider also the empty paths ∅_{i} for i ∈ I and we define (∅_{i})_{0} = (∅_{i})_{1} = i. The product of path is defined in the obvious way.

A b-path [β] in our graph is a sequence [i_{m+1}^{r_{m+1}}α^{(m)}i_{m}^{r_{m}} … α^{(1)}i_{1}^{r_{1}}], that we write between square brackets such that i_{j} ∈ I, α^{(j)} are B-path, r_{j} ∈ ℕ and α_{0}^{(j)} = i_{j} and α_{1}^{(j)} = i_{j+1} for j = 1, …, m. We consider also the “empty” b-paths indexed by elements of I: [∅_{i}]. We define [β]_{0} = i_{1}, [β]_{1} = i_{m+1} and [∅_{i}]_{0} = [∅_{i}]_{1} = i. The length of [β] is ∑_{j=1}^{m+1} r_{j} + ∑_{j=1}^{m} length(α^{j}) and the product of b-paths is defined in the obvious way:

[β] · [β′] = \begin{cases}0 & if [β′]_{1} ≠ [β]_{0} \\ [ββ′] & if [β′]_{1} = [β]_{0} = i \end{cases}

Given a path α = h^{(m)} … h^{(1)} and a b-path β = [i_{m+1}^{r_{m+1}}α^{(m)} . . . . . . i_{1}^{r_{1}}] we define an evaluation of α and β on S in the following way: if s = (B, γ, δ) ∈ S then

∅_{i}(s) = 0 ∈ Hom(V_{i}, V_{i}) and [∅_{i}](s) = 0 ∈ Hom(V_{i}, V_{i}),
α(s) = B_{h^{(m)}} ∘…∘B_{h^{(1)}} ∈ Hom(V_{α_{0}}, V_{α_{1}}),
β(s) = (γ_{i_{m+1}} ∘δ_{i_{m+1}})^{r_{m+1}} ∘α^{(m)}(s)∘(γ_{i_{m}} ∘δ_{i_{m}})^{r_{m}} ∘…∘
∘ ⋯ ∘ α^{(1)}(s)∘(γ_{i_{1}} ∘δ_{i_{1}})^{r_{1}} ∈ Hom(D_{β_{0}}, D_{β_{1}}).

The path algebra R is the vector space spanned by paths with the product induced by the product of paths. If i, j ∈ I we say that an element in R is of type (i, j) if it is in the linear span of the paths with source in i and target in j.

The b-path algebra Q is the vector space spanned by b-paths with the product induced by the product of b-paths described above. If i, j ∈ I we say that an element in R is of type (i, j) if it is in the linear span of the b-paths with source in i and target in j.

Remark 17. We observe that the evaluation on S is a morphism of algebra from R to the algebra defined by the morphisms of the category of vector spaces. Moreover if f is of type (i, j) we observe that f(s) ∈ Hom(V_{i}, V_{j}).

2. Generators of the projective ring of a quiver variety

In this section we want to describe a set of generators of the graded ring ℂ[S]_{χ} and by consequence of the projective ring of a quiver variety ℂ[Λ_{λ}]_{χ}. More precisely we will give a set of generators as ℂ[S]^{G}-module of its l-homogeneous part: ℂ[S]_{χ,l}. This result is a generalization of the one obtained by Lusztig in the case of invariants: χ ≡ 1. First of all recall his result.

Theorem 18 (Lusztig, [8] theorem 1.3). The ring ℂ[S]^{G} is generated by the polynomials:

s ⟼ Tr (α(s)) and s ⟼ φ (δ_{β_{1}}(s)β(s)γ_{β_{0}}(s))

for α a closed path, β a path and φ ∈ (Hom(D_{β_{0}}, D_{β_{1}}))^{∗}.

2.0.1. Determinants. To describe our result we do first some general remark. Forget for a moment our quiver, and suppose to have a finite set of finite dimensional vector spaces X_{1}, …, X_{k} of dimensions u_{1}, …, u_{k} and a pair of nonnegative integers (m_{i}^{+}, m_{i}^{−}) for each of them. Finally let m^{+}, m^{−} two nonnegative integers such that

N = ∑_{i=1}^{k} m_{i}^{+}u_{i} + m^{+} = ∑_{i=1}^{k} m_{i}^{−}u_{i} + m^{−},

and two vector spaces M^{+} and M^{−} of dimension m^{+}, m^{−}. Construct the vector spaces:

Y = ⨁_{i=1}^{k} ℂ^{m_{i}^{−}} ⊗ X_{i} ⊕ M^{−}, Z = ⨁_{i=1}^{k} ℂ^{m_{i}^{+}} ⊗ X_{i} ⊕ M^{+}

and observe that dim Y = dim Z = N. Define an action of the general linear group GL(X_{i}) of X_{i} on Y by

g_{i} · (∑_{j=1}^{k} v_{j} ⊗ x_{j} + m) = ∑_{j≠i} v_{j} ⊗ x_{j} + m + v_{i} ⊗ g_{i}x_{i},

and also a similar action on Z. Hence the vector space Hom(Y, Z) acquires a natural structure of G_{X} = ∏_{i=1}^{k} GL(X_{i}) module. If we choose an isomorphism σ between Hom(⋀^{N} Y, ⋀^{N} Z) and ℂ we can define a function det on Hom(Y, Z) by

det(A) = σ (⋀^{n} A) .

For simplicity we do not emphasize the role of σ on this definition, so strictly speaking, det is a function defined only up to a nontrivial constant factor. We observe also that

⋀^{n} Y ≃ (⋀^{u_{i}} X_{1})^{⊗m_{1}^{−}} ⊗ ⋯ ⊗ (⋀^{u_{k}} X_{k})^{⊗m_{k}^{−}} ⊗ ⋀^{m^{−}} M^{−} (and similarly for Z) so an isomorphism σ is determined if we choose orientations, or basis, of X_{j}, M^{+}, M^{−}. Finally observe that for any g = (g_{j}) ∈ G_{X} we have

det(g · A) = ∏_{i=1}^{k}(det_{GL(X_{i})}(g_{i}))^{m_{i}^{+}−m_{i}^{−}} det(A).

2.0.2. Description of generators. We go back now to our quiver and we describe a set of covariant polynomials on S. Any character χ of the group G_{v} = GL(V) is of the form χ = χ_{m} = ∏_{i∈I} det_{GL(V_{i})}^{m_{i}}. We fix such a character and we define

I^{+} = {i ∈ I : m_{i} > 0} and m̃_{i} = m_{i} if i ∈ I^{+},
I^{0} = {i ∈ I : m_{i} = 0} and m̃_{i} = 0 if i ∈ I^{0},
I^{−} = {i ∈ I : m_{i} < 0} and m̃_{i} = −m_{i} if i ∈ I^{−}.

We use now the construction explained in 2.0.1 in the case X_{i} = V_{i} and m_{i}^{+}−m_{i}^{−} = m_{i}. We choose ordered sets A = (a_{1}, …, a_{m^{−}}) ⊂ (⋃ D_{i})^{m^{−}} and B = (b_{1}, …, b_{m^{+}}) ⊂ (⋃ D_{i}^{∗})^{m^{+}} and we define a function I : A, B ⟶ I by a ∈ D_{I(a)}, b ∈ D_{I(b)}^{∗}. In the

framework described above it is then possible to set M^{−} = ⨁_{i=1}^{m^{−}} ℂ_{a_{i}} and M^{+} = ⨁_{i=1}^{m^{+}} ℂ_{b_{i}}. In particular we have

Y = ⨁_{i∈I} ⨁_{h=1}^{m_{i}^{−}} V_{i}^{(h)} ⊕ ⨁_{i=1}^{m^{−}} ℂ_{a_{i}}, Z = ⨁_{i∈I} ⨁_{k=1}^{m_{i}^{+}} V_{i}^{[k]} ⊕ ⨁_{i=1}^{m^{+}} ℂ_{b_{i}}

where V_{i}^{(l)}, V_{i}^{[l]} are isomorphic copies of V_{i}. We choose now elements of the b-path algebra as follows:

1. for any i, j ∈ I and for any 1 ≤ h ≤ m_{i}^{−}, 1 ≤ k ≤ m_{j}^{+} we choose an element α_{j,k}^{i,h} of the b-path algebra of type (i, j),

2. for any i ∈ I, 1 ≤ h ≤ m_{i}^{−} and for any 1 ≤ l ≤ m^{+} we choose an element α_{l}^{i,h} of the b-path algebra of type (i, I(b_{l})), 3. for any 1 ≤ l ≤ m^{−} and for any j ∈ I, 1 ≤ k ≤ m_{j}^{+} we choose an element α_{j,k}^{l}

of the b-path algebra of type (I(a_{l}), j),

4. for any 1 ≤ l ≤ m^{−} and for any 1 ≤ l′ ≤ m^{−} we choose an element α_{l′}^{l} of the

b-path algebra of type (I(a_{l}), I(b_{l′})). We call such a data Δ = ({(m_{i}^{+}, m_{i}^{−})}_{i∈I}, (m^{+}, m^{−}), A, B, α_{j,k}^{i,h}, α_{l}^{i,h}, α_{j,k}^{l}, α_{l′}^{l}) a χ-data and we attach to it a χ-covariant function on S:

f_{Δ}(s) = det (Ψ_{Δ}(s))

where Ψ_{Δ} is a linear map from Y to Z defined by

[Ψ_{Δ}]_{V_{j}^{[k]}}^{V_{i}^{(h)}}(s) = α_{j,k}^{i,h}(s),
[Ψ_{Δ}]_{ℂ_{b_{l}}}^{V_{i}^{(h)}}(s) = b_{l}∘δ_{I(b_{l})}∘α_{l}^{i,h}(s),
[Ψ_{Δ}]_{V_{j}^{[k]}}^{ℂ_{a_{l}}}(s) = α_{j,k}^{l}(s)∘γ_{I(a_{l})}|_{ℂa_{l}},
[Ψ_{Δ}]_{ℂ_{b_{l′}}}^{ℂ_{a_{l}}}(s) = b_{l′} ∘δ_{I(b_{l′})}∘α_{l′}^{l}(s)∘γ_{I(a_{l})}|_{ℂa_{l}}.

The function f_{Δ} are a set of generators as ℂ[S]^{G}-module of ℂ[S]_{χ,1}, but we will need to define a smaller set of generators. To define this set we give a notion of good Δ.

Definition 19. A data Δ as above is said to be χ-good if it satisfies the following conditions:

1. m_{i}^{+} + m_{i}^{−} = m̃_{i} for all i ∈ I,

2. α_{l′}^{l} = 0 for all l, l′,

3. α_{∗}^{∗} is an element of the path algebra (and not just an element of the b-path algebra which is obviously bigger),

4. card{(j, k) : α_{j,k}^{i,h} ≠ 0} + card{l : α_{l}^{i,h} ≠ 0} ≤ v_{i} for all i, h,

5. card{(i, h) : α_{j,k}^{i,h} ≠ 0} + card{l : α_{j,k}^{l} ≠ 0} ≤ v_{j} for all j, k,

6. for all l there exists at most one pair (i, h) such that α_{l}^{i,h} ≠ 0,

7. for all l there exists at most one pair (j, k) such that α_{j,k}^{l} ≠ 0.

For the applications the only important point will be the first one.

Proposition 20. The set of polynomials f_{Δ} with Δ χ-good generates ℂ[S]_{χ,1} as a ℂ[S]^{G_{v}}-module.

Remark 21. Prof. Weyman said me that in the case D = 0 a similar proposition has been proved by him and Derksen for arbitrary characteristic in the case of a quiver without loops: [2].

2.1. Some remark on the invariant theory of GL(n). If V is a finite dimensional representation of a linearly reductive Lie group G and S is a simple representation of S we write V [S] for the S-isotipic component of type S of V.

We now fix n and we make some remark on the representations of GL(n). To any partition of height less or equal to n we associate an irreducible representation of GL(n) in the usual way. If we multiply these representations by a power of the inverse of determinant representation we obtain a complete list of irreducible representations of GL(n). If λ is a partition we call λ the transpose partition as usual and we define λ^{op} = (λ_{1} − λ_{n}, λ_{1} − λ_{n−1}, …, λ_{1} − λ_{1}). We call δ the determinant representation of GL(n) and ε = 1^{n} the associated partition. Finally we call V the natural representation.

Lemma 22. 1. L_{λ}^{∗} = δ^{−λ_{1}} ⊗ L_{λ}^{op},

2. Hom_{GL(n)}(δ^{m}, L_{λ} ⊗ L_{μ}) = \begin{cases}ℂ & if λ = μ^{op} + (m − μ_{1})ε, \\ 0 & otherwise, \end{cases}

3. Hom_{GL(n)}(δ^{m}, L_{λ} ⊗ L_{μ}^{∗}) = \begin{cases}ℂ & if λ = μ + mε, \\ 0 & otherwise. \end{cases}

Proof. We prove only 2).

Hom_{GL(n)}(δ^{m}, L_{λ} ⊗ L_{μ}) = Hom_{GL(n)}(δ^{m} ⊗ L_{μ}^{∗}, L_{λ})
= Hom_{GL(n)}(δ^{m−μ_{1}} ⊗ L_{μ^{op}}, L_{λ})

If m ≥ μ_{1} the last group is isomorphic to Hom_{GL(n)}(L_{μ^{op}+(m−μ_{1})ε}, L_{λ}) and if m < μ_{1} is isomorphic to Hom_{GL(n)}(L_{μ^{op}}; L_{λ+(μ_{1}−m)ε}). In any case the thesis follows.

We want now to describe Hom_{GL(n)} (δ^{m}, V^{⊗i} ⊗ (V^{∗})^{⊗j}). To do it we will use Schur-duality. Remind that the irreducible representations of the groups S_{m} are parametrized by partitions of m and call S_{λ} the irreducible representation associated with λ. Consider now the action of S_{m} on V^{⊗m} given by permuting the factors. This action commute with the GL(n) action. Schur duality asserts that the action of the group S_{m} × GL(n) on V^{⊗m} decomposes in the following way:

V^{⊗m} = ⨁_{\begin{matrix}λ⊢m \\ ht(λ)≤n \end{matrix}} S_{λ} ⊗ L_{λ}.

We describe a set of elements of Hom_{GL(n)} (δ^{m}, V^{⊗i} ⊗ (V^{∗})^{⊗j}). Let m be a nonnegative integers a choose a permutation σ of {1, …, i + mn}. To σ we associate maps:

Φ_{σ} : (V^{⊗i} ⊗ (V^{∗})^{⊗i})^{GL(n)} ⟶ V^{⊗i+mn} ⊗ (V^{∗})^{⊗i}[δ^{m}]
Ψ_{σ} : (V^{⊗i} ⊗ (V^{∗})^{⊗i})^{GL(n)} ⟶ V^{⊗i} ⊗ (V^{∗})^{⊗i+mn}[δ^{−m}]

by

Φ_{σ}(t ⊗ s) = σ(o ⊗ ⋯ ⊗ o ⊗ t) ⊗ s
Ψ_{σ}(t ⊗ s) = t ⊗ σ(o^{∗} ⊗ ⋯ ⊗ o^{∗} ⊗ s)

where o is a nonzero vector in ⋀^{n} V and o^{∗} is a non zero vector in ⋀^{n} V^{∗} and t ∈ V^{⊗i}, s ∈ (V^{∗})^{⊗i}.

Lemma 23. 1) If i ≠ j + mn then

Hom_{GL(N)} (δ^{m}, V^{⊗i} ⊗ (V^{∗})^{⊗j}) = 0.

2) If m > 0 then

V^{⊗i+mn} ⊗ (V^{∗})^{⊗i}[δ^{m}] = ∑_{σ} Im Φ_{σ}.

3) If m > 0 then

V^{⊗i} ⊗ (V^{∗})^{⊗i+mn}[δ^{−m}] = ∑_{σ} Im Ψ_{σ}.

Proof. 1) follows directly from lemma 22.

2) Let M = V^{⊗i+mn} ⊗(V^{∗})^{⊗i}[δ^{m}] and N = (V^{⊗i} ⊗(V^{∗})^{⊗i})^{G}. N is a S_{i} ×S_{i} module, M is a S_{i+mn} × S_{i}- module and the maps Φ_{I,σ} are equivariant with respect the S_{i} action on (V^{∗})^{⊗i}. In particular it is enough to prove that if λ is a partition of i, M_{λ} is the S_{λ}-isotipic component of M w.r.t. the S_{i} action and N_{λ} the S_{λ}-isotipic component of N w.r.t. the S_{i} action on (V^{∗})^{⊗i} then

M_{λ} = ∑_{σ} Φ_{σ}(N_{λ}).

By point 3 of lemma22 we have that

M = ⨁_{\begin{matrix}λ⊢i \\ ht(λ)≤n \end{matrix}} (S_{λ+mε} ⊗ L_{λ+mε}) ⊗ (S_{λ} ⊗ L_{λ})^{∗} [δ^{m}]

= ⨁_{\begin{matrix}λ⊢i \\ ht(λ)≤n \end{matrix}} S_{λ+mε} ⊗ S_{λ} ⊗ (δ^{m} ⊗ (L_{λ} ⊗ L_{λ}^{∗})^{G})

In particular M_{λ} = S_{λ+mε} ⊗S_{λ} is an irreducible representation of S_{i+mn} ×S_{i}. Observe ∑_{σ} Φ(N_{λ}) is a S_{i+mn} × S_{i}-submodule of M_{λ} and that it is clearly nonzero. So M_{λ} = ∑_{σ} Φ_{σ}(N_{λ}) as claimed.

The proof of 3) is equal to the previous one.

We want now to give a slightly different formulation of the lemma above. Let M = V^{⊗i} ⊗ (V^{∗})^{⊗j} we want to describe M_{δ^{m}}^{∗} = {φ ∈ M^{∗} : g · φ = δ^{−m}(g)φ}. Of course this problem is completely equivalent to the previous one. What we want to do is to reformulate in a more convenient way for our purposes the description of a set of generators of M_{δ^{m}}^{∗}. Let m ≥ 0 and choose I = {I_{1}, …, I_{m}} a collection of m disjoint subsets of {1, …, i + mn} of cardinality n. Let I_{j} = {i_{j1} < ⋯ < i_{jn}} and {1, …, i + mn} − ⋃ I = {j_{1} < ⋯ < j_{i}}. To I and to a permutation σ ∈ S_{i} we associate elements

ϕ_{I,σ} ∈ (V^{⊗i+mn} ⊗ (V^{∗})^{⊗i})_{δ^{m}}^{∗} and ψ_{I,σ} ∈ (V^{⊗i} ⊗ (V^{∗})^{⊗i+mn})_{δ^{−m}}^{∗}

defined by

ϕ_{I,σ}(v_{1} ⊗ … v_{i+mn} ⊗ φ_{1} … φ_{i}) = ∏_{j=1}^{m} <o^{∗}, v_{j1} ∧ ⋯ ∧ v_{jn}> · ∏_{h=1}^{i} <v_{j_{h}} , φ_{σ_{h}}>

ψ_{I,σ}(v_{1} ⊗ … v_{i} ⊗ φ_{1} … φ_{i+mn}) = ∏_{j=1}^{m} <o, φ_{j1} ∧ ⋯ ∧ φ_{jn}> · ∏_{h=1}^{i} <v_{σ_{h}} , φ_{j_{h}}>

where o is a nonzero vector in ⋀^{n} V and o^{∗} is a non zero vector in ⋀^{n} V^{∗}.

Lemma 24. 1) If i ≠ j + mn then (V^{⊗i} ⊗ (V^{∗})^{⊗j})_{δ^{m}}^{∗} = 0.

2) If m ≥ 0 then (V^{⊗i+mn} ⊗ (V^{∗})^{⊗i})_{δ^{m}}^{∗} is generated by the functions ϕ_{I,σ}.

3) If m ≥ 0 then (V^{⊗i} ⊗ (V^{∗})^{⊗i+mn})_{δ^{−m}}^{∗} is generated by the functions ψ_{I,σ}.

Proof. The proof is clear by the previous lemma.

2.2. A special case. In this section we prove a special case of Proposition 20 in which we are able to give a more precise result. To simplify the exposition of the proof of Proposition 20 we will also prove another lemma.

Here and in the following we will use polarization. If V is finite dimensional vector space then we can define a map

℘ : (V^{⊗n})^{∗} ⟶ S^{n}(V^{∗}) ⊂ ℂ[V] through ℘(φ)(v) = φ(v ⊗ ⋯ ⊗ v).

Lemma 25. ℘ is surjective, moreover if V is a finite dimensional representation of a reductive group Γ, and χ is a character of Γ then

℘((V^{⊗n})_{χ}^{∗}) = S^{n}(V^{∗})_{χ}

where E_{χ} is the isotipic component of type χ^{−1} of a G module E.

Lemma 26. For i = 1, …, n let Γ_{i} be a reductive group, χ_{i} be a character of Γ_{i} and E_{i} be a f.d.representation of Γ_{i}. Let Γ = ∏ Γ_{i}, then E = ⊗_{i}E_{i} is a representation of Γ and χ = ∏ χ_{i} is a character of Γ. Then

E_{χ}^{∗} = (E_{1})_{χ_{1}}^{∗} ⊗ ⋯ ⊗ (E_{n})_{χ_{n}}^{∗}.

Let J^{+}, J^{−} be two sets of indexes and define J̃^{+} = {0} ∐ J^{+}, J̃^{−} = {0} ∐ J^{−} and J = J^{+} × J^{−}, J̃ = J̃^{+} × J̃^{−} − {(0, 0)}. For each i ∈ J̃^{+} (resp. j ∈ J̃^{+}) choose a vector space Y_{i} (resp. X_{j}) and define X = ⨁_{j∈J^{−}} X_{j} and Y = ⨁_{i∈J^{+}} Y_{i}. Consider the group

G_{XY} = ∏_{i∈J^{+}} GL(Y_{i}) × ∏_{j∈J^{−}} GL(X_{j})

and its character c = ∏_{i∈J^{+}} det_{GL(Y_{i})} × (∏_{j∈J^{−}} det_{GL(X_{j})})^{−1}.

We fix a matrix r = (r_{ij})_{i∈J̃^{+},j∈J̃^{−}} of integers such that r_{i0} = 1 = r_{0j} for all i, j and r_{00} = −1 and we consider the vector spaces:

H^{XY} = H = ⨁_{(i,j)∈J̃}Hom(X_{j}, Y_{i})^{⊕r_{ij}} and H_{0}^{XY} = H_{0} = ⨁_{(i,j)∈J}Hom(X_{j}, Y_{i})

where we adopt the convention E^{⊕n} = ℂ^{n} = 0 if n < 0. When the spaces X, Y will be clear from the context we will write H and H_{0} instead of H^{XY} and H_{0}^{XY}. We fix a basis e_{m}^{ij} of ℂ^{r_{ij}} so we have a canonical identification

H = ⨁_{(i,j)∈J̃}Hom(X_{j}, Y_{i}) ⊗ ℂ^{r_{ij}}.  (9)

We want to study c-equivariant polynomials on H. If we choose two finite dimensional vector spaces Ã, B̃, linear maps α : Ã ⟶ X_{0} , β : Y_{0} ⟶ B̃, and elements φ_{ij} ∈ (ℂ^{r_{ij}})^{∗} for all i, j then we can define a map Φ_{φ,α,β} : H ⟶ H_{0} ⊕ Hom(Ã, Y) ⊕ Hom(X, B̃) ⊂ Hom(X ⊕ Ã, Y ⊕ B̃) by

Φ_{φ,α,β}( ∑_{(i,j)∈J̃}A_{ij} ⊗ v_{ij}) = ∑_{(i,j)∈J} φ_{ij}(v_{ij})A^{ij} + ∑_{i∈J^{+}} A^{i0}∘α + ∑_{j∈J^{−}} β∘A^{0j} (10)

where A_{ij} ⊗ v_{ij} ∈ Hom(X_{j}, Y_{i}) ⊗ ℂ^{r_{ij}}.

The following is a special version of 20.

Lemma 27. ℂ[H]_{c} is generated as a vector space by the following functions:

s ⟼ det(Φ_{φ,α,β}(s))

where Φ_{φ,α,β} : H ⟶ H_{0} is as above.

2.2.1. The special case. We will study an even more special case in which we are able to prove a better result that I find nice. In the above setting suppose that X_{0} = Y_{0} = 0 and that r_{ij} = 1 for all i, j.

Define the following set of matrices:

S_{n} = {S = (s_{ij}) ∈ ℕ^{J^{+}×J^{−}} : ∑_{i,j} s_{ij} = n}
S^{XY} = S = {S = (s_{ij}) ∈ ℕ^{J^{+}×J^{−}} : ∑_{j} s_{ij} = dim Y_{i} ∀i ∈ J^{+}

and ∑_{i} s_{ij} = dim X_{j} ∀j ∈ J^{−} }

As for H we will write S when the spaces X_{j}, Y_{i} will be clear from the context. Observe that S = ∅ if ∑_{j} dim X_{j} ≠ ∑_{i} dim Y_{i} and that if N = ∑ dim X_{j} = ∑ dim Y_{i} then S ⊂ S_{N}. For each card(J^{+}) × card(J^{−}) matrix S = (s_{ij}) we consider φ_{ij} ∈ ℂ^{∗} given by φ_{ij}(λ) = s_{ij}λ and we define

Φ_{S} = Φ_{φ,0,0} and f_{S} = f_{S}^{XY} = det(Φ_{S}).

Proposition 28. {f_{S}}_{S∈S^{XY}} is a basis of ℂ[H]_{c}.

Proof. We have to compute S^{n}(H^{∗})_{c} = (S^{n}(H))_{c}^{∗} for all n. For all S ∈ S_{n} define

E_{S} = ⨂_{(i,j)∈J^{+}×J^{−}} S^{s_{ij}} (Hom(X_{j}, Y_{i})) .

Observe that S^{n}(H) = ⨁_{S∈S_{n}} E_{S} as a G-module. So S^{n}(H)_{c}^{∗} = ⨁_{S∈S_{n}}(E_{S})_{c}^{∗}. Observe now that E_{S} is a quotient of

Ẽ_{S} = ⨂_{(i,j)∈J^{+}×J^{−}}(X_{j}^{∗})^{⊗s_{ij}} ⊗ Y_{i}^{⊗s_{ij}}.  (11)

By the lemmas in the previous section we have that

(Ẽ_{S})_{c}^{∗} = \begin{cases}0 & if S ∉ S^{XY} , \\ ℂ & if S ∈ S^{XY} . \end{cases}

So in particular (E_{S})_{c}^{∗} = 0 if S ∉ S^{XY}. Hence dim S^{n}(H)_{c}^{∗} ≤ card(S^{XY}).

The function f_{S} are clearly c-equivariant so the only thing that we have to prove is that they are linearly independent. To prove it we will prove a generalization of it.

If i ∈ J^{+} and j ∈ J^{−} let E_{ij} be the card(J^{+}) × card(J^{−}) matrix with a 1 in the (i, j) position and 0 elsewhere.

For each i ∈ J^{+}, j ∈ J^{−}, m ∈ ℕ and N ∈ ℕ we consider the following sentence P_{i,j,m,N}:

If ∑_{j} X_{j} = N = ∑_{i} Y_{i} then {f_{S+mE_{ij}}}_{S∈S^{XY}} is linearly independent.

In the case m = 0 we call this proposition P_{0,N} since it does not depend on i, j and observe that ∀N P_{0,N} is equivalent to our thesis.

For each N ∈ ℕ we consider also the following sentence Q_{N}:

If ∑_{j} X_{j} = N = ∑_{i} Y_{i} then P_{i,j,m,N} is true for all i ∈ J^{+}, j ∈ J^{−} and m ∈ ℕ.

First remark: N = 1 is true. Second remark: let S_{0}^{XY} = {S ∈ S^{XY} : s_{ij} = 0 for all i ∈ J^{+} and j ∈ J^{−} such that dim Y_{i}, dim X_{j} ≥ 2. Observe that {f_{S}}_{S∈S̃^{XY}} is linearly independent.

Now we prove Q_{N} by induction on N. First step: Q_{N−1} ⇒ P_{0,N}. Suppose that there exists c_{S} ∈ ℂ such that

∑_{S∈S^{XY}} c_{S}f_{S} = 0.

If dim X_{j_{0}}, dim Y_{i_{0}} ≥ 2 choose a nonzero element x_{j_{0}} ∈ X_{j_{0}} (resp. y_{i_{0}} ∈ Y_{i_{0}}) and an hyper-plane X′_{j_{0}} ⊂ X_{j_{0}} (resp. Y′_{i_{0}} ⊂ Y_{i_{0}}) such that X_{j_{0}} = ℂx_{j_{0}} ⊕ X′_{j_{0}} (resp. Y_{j_{0}} = ℂy_{i_{0}} ⊕ Y′_{i_{0}}) and define:

X̃_{j} = \begin{cases}X_{j} & if j ≠ j_{0} \\ X′_{j_{0}} & if j = j_{0} \end{cases} and Ỹ_{i} = \begin{cases}Y_{i} & if i ≠ i_{0} \\ Y′_{i_{0}} & if i = i_{0} \end{cases}  (12)

and define Ψ : H^{X̃Ỹ} ⟶ H^{XY} by

Ψ(T)|X̃_{j} = T and Ψ(T)(x_{j_{0}}) = y_{j_{0}}.  (13)

Then we see that

0 = ∑_{S∈S^{XY}} c_{S}f_{S}^{XY} (Ψ(T)) = ∑_{S∈S^{XY} : s_{i_{0}j_{0}}≠0} s_{i_{0}j_{0}}c_{S}f_{S}^{X̃Ỹ} (T)

= ∑_{S∈S^{X̃Ỹ}} (s_{i_{0}j_{0}} + 1)c_{S+E_{i_{0}j_{0}}}f_{S+E_{i_{0}j_{0}}}^{X̃Ỹ}(T)

By induction P_{i,j,1,N−1} is true for all i, j so we see that c_{S} = 0 for all S ∈ S such that there exists i_{0}, j_{0} such that s_{i_{0},j_{0}} ≥ 1 and dim X_{j_{0}}, dim Y_{i_{0}} ≥ 2. Now we conclude by the second remark. Second step: Q_{N−1} ⇒ P_{i_{0},j_{0},m,N} if dim X_{j_{0}}, dim Y_{i_{0}} ≥ 2 and m ≥ 1. Suppose that ∑_{S∈S^{XY}} c_{S}f_{S+mE_{i_{0}j_{0}}}^{XY} = 0. We can construct X̃_{j}, Ỹ_{i}, Ψ as in the first step and we see that

0 = ∑_{S∈S^{XY}} c_{S}f_{S+mE_{i_{0}j_{0}}}^{XY}(Ψ(T)) = ∑_{S∈S^{XY}} (s_{i_{0}j_{0}} + m)c_{S}f_{S+mE_{i_{0}j_{0}}}^{X̃Ỹ}(T)

= ∑_{S∈S^{X̃Ỹ}} (s_{i_{0}j_{0}} + m + 1)c_{S+E_{i_{0}j_{0}}}f_{S+(m+1)E_{i_{0}j_{0}}}^{X̃Ỹ}(T)

and by P_{i_{0},j_{0},m+1,N−1} we deduce c_{S} = 0 for all S. Third step: Q_{N−1} ⇒ Q_{N}. By the previous two step we have only to prove P_{i_{0},j_{0},m,N} for m ≥ 1 and dim X_{j_{0}} = 1 or dim Y_{i_{0}} = 1. We will suppose dim X_{j_{0}} = 1, the other case is completely similar. Suppose that ∑_{S∈S^{XY}} c_{S}f_{S+mE_{i_{0}j_{0}}} = 0. Set

S̃_{i} = {S ∈ S^{XY} : s_{ij_{0}} = 1}

and observe that since dim X_{j_{0}} = 1 then S^{XY} = ∐ S̃_{i}. Now choose a non zero vector x_{j_{0}} ∈ X_{j_{0}} and for all i ∈ J^{+} choose a non zero vector y_{i} ∈ Y_{i} and an hyper-plane Y′_{i} of Y_{i} such that Y_{i} = ℂy_{i} ⊕ Y′_{i}.

Now fix i_{1} ≠ i_{0} such that dim Y_{i_{1}} ≥ 2 and consider J̃^{+} = J^{+} and J̃^{−} = J^{−} − {j_{0}}. For all i ∈ J̃^{+} and for all j ∈ J̃^{−} define:

X̃_{j} = X_{j}, and Ỹ_{i} = \begin{cases}Y_{i} & if i ≠ i_{1}, \\ Y′_{i_{1}} & if i = i_{1}. \end{cases}

For any S ∈ S̃_{i_{1}} we define also t(S) ∈ S^{X̃Ỹ} by t(S)_{ij} = s_{ij} for all i ∈ J̃^{+}, j ∈ J̃^{−}. S ⟼ t(S) is a bijection between S̃_{i_{1}} and S^{X̃Ỹ}: we call t^{−1} the inverse map. Finally we define Ψ : H^{X̃Ỹ} ⟶ H^{XY} as in the previous step and we observe that if S ∈ S then f_{S+mE_{i_{0}j_{0}}} ∘Ψ = 0 if S ∉ S̃_{i_{1}}. Hence

0 = ∑_{S∈S^{XY}} c_{S}f_{S+mE_{i_{0}j_{0}}}(Ψ(T)) = ∑_{S∈S̃_{i}} c_{S}f_{t(S)}(T)

= ∑_{S∈S^{X̃Ỹ}} c_{t^{−1}(S)}f_{S}(T)

and applying P_{0,N−1} we obtain c_{S} = 0 for all S ∈ S̃_{i_{1}} if dim Y_{i_{1}} ≥ 2 and i_{1} ≠ i_{0}. In a similar way we prove c_{S} if S ∈ S̃_{i_{1}} and dim Y_{i_{1}} = 1 and i_{1} ≠ i_{0}.

Finally we observe that if S ∈ S̃_{i_{0}} then f_{S+mE_{i_{0}j_{0}}} = (m+1)f_{S}, hence c_{S} = 0 follows now from P_{0,N} that we already know to be true.

2.2.2. Proof of Lemma 27. We study first (H^{⊗n})_{c}^{∗} and then we apply polarization. As in the previous section we can decompose H^{⊗n} in summands of the following form:

E = ⨂_{(i,j)∈J̃}(Hom(X_{j}, Y_{i}) ⊗ ℂ^{r_{ij}})^{⊗s_{ij}} = ⨂_{(i,j)∈J̃}(X_{j}^{∗})^{⊗s_{ij}} ⊗ Y_{i}^{⊗s_{ij}} ⊗ (ℂ^{r_{ij}})^{⊗s_{ij}}  (14)

where s_{ij} are nonnegative integers such that ∑_{i,j} s_{ij} = n. Observe that the order of the factors is not important for us since we will apply polarization.

We can describe easily E_{c}^{∗} using the lemma in the previous section. In particular a necessary and sufficient condition for the existence of c-covariants is ∑_{i∈J̃^{+}} s_{ij} = dim X_{j} for all j ∈ J^{−} and ∑_{j∈J̃^{+}} s_{ij} = dim Y_{i} for all i ∈ J^{+}. Moreover

E_{c}^{∗} ≃ ⨂_{(i,j)∈J} ((ℂ^{r_{ij}})^{∗})^{⊗s_{ij}} ⊗ ⨂_{j∈J^{−}}(Y_{0}^{∗})^{⊗s_{0j}} ⊗ ⨂_{i∈J^{+}} X_{0}^{⊗s_{i0}}

To write explicit formulas we choose an order on the factors of E for example choosing a lexicographic order in i ∈ J̃^{+}, j ∈ J̃^{−} and 1 ≤ q ≤ s_{ij}:

E = \underbrace{X_{1}^{∗} ⊗ Y_{1} ⊗ ℂ^{r_{11}}}_{q=1} ⊗ ⋯ ⊗ \underbrace{X_{1}^{∗} ⊗ Y_{1} ⊗ ℂ^{r_{11}}}_{q=s_{11}} ⊗ \underbrace{X_{1}^{∗} ⊗ Y_{2} ⊗ ℂ^{r_{12}}}_{q=1} ⊗ ⋯

Once we have chosen such an order we can write an element of E as linear combination elements of the form ⊗_{(i,j,q)∈K̃}x^{i,j,q} ⊗y^{i,j,q} ⊗v^{i,j,q} with x^{i,j,q} ∈ X_{j}^{∗}, y^{i,j,q} ∈ Y_{i} v^{i,j,q} ∈ ℂ^{r_{ij}} and we settled K̃ = {(i, j, q) ∈ J̃× ℕ : 1 ≤ q ≤ s_{ij}}. We define also K = {(i, j, q) ∈ K̃ : (i, j) ∈ J}. Using this convention if

ϕ = ⨂_{(i,j,q)∈K̃} ϕ^{i,j,q} ∈ ⨂_{(i,j,q)∈K}(ℂ^{r_{ij}})^{∗} ⊗ ⨂_{(0,j,q)∈K̃} Y_{0}^{∗} ⊗ ⨂_{(i,0,q)∈K̃} X_{0}  (15)

the corresponding c equivariant linear function on E is defined on an element s = ⊗_{(i,j,q)∈K̃}x^{i,j,q} ⊗ y^{i,j,q} ⊗ v^{i,j,q} by

ϕ(s) = ∏_{i∈J^{+}} < ⋀_{\overrightarrow{(i,j,q)∈K̃}}y^{i,j,q}, o_{i}^{∗}> ∏_{j∈J^{−}} < ⋀_{\overrightarrow{(i,j,q)∈K̃}}x^{i,j,q}, o_{j}>

∏_{(i,j,q)∈K} ϕ^{i,j,q}(v^{i,j,q}) ∏_{(i,0,q)∈K̃} ϕ^{i,0,q}(x^{i,0,q}) ∏_{(0,j,q)∈K̃} ϕ^{0,j,q}(y^{0,j,q})

Now consider the group

𝔖 = 𝔖_{1} × 𝔖_{2} × 𝔖_{3} = ∏_{(i,j)∈J} S_{s_{ij}} × ∏_{j∈J^{−}} S_{s_{0j}} × ∏_{i∈J^{+}} S_{s_{i0}}.

This group acts naturally on ⨂_{(i,j)∈J} ((ℂ^{r_{i_{1}j_{1}}})^{∗})^{⊗s_{ij}} ⊗⨂_{j∈J^{−}}(Y_{0}^{∗})^{⊗s_{0j}} ⊗⨂_{i∈J^{+}} X_{0}^{⊗s_{i0}} = E_{c}^{∗} by permuting the factors and we observe that

℘((σ_{1}, σ_{2}, σ_{3})ϕ) = ε(σ_{2})ε(σ_{3})℘(ϕ)

for all ϕ ∈ E_{c}^{∗} and for all (σ_{1}, σ_{2}, σ_{3}) ∈ 𝔖. So we have that

℘(E_{c}^{∗}) = ℘( ⨂_{(i,j)∈J} S^{s_{ij}}((ℂ^{r_{ij}})^{∗}) ⊗ ⨂_{j∈J−} ⋀^{s_{0j}} Y_{0}^{∗} ⊗ ⨂_{i∈J^{+}} ⋀^{s_{i0}} X_{0}).

In particular since S^{m}(V) is spanned by vectors of the form v ⊗ ⋯ ⊗ v, ℘(E_{c}^{∗}) is spanned by the functions ℘(ϕ) with ϕ of the following special form:

ϕ = ⨂_{(i,j)∈J}(ϕ^{i,j})^{⊗s_{ij}} ⊗ ⨂_{j∈J^{−}} ϕ^{0,j,1} ∧ ⋯ ∧ ϕ^{0,j,s_{0j}} ⊗ ⨂_{i∈J^{+}} ϕ^{i,0,1} ∧ ⋯ ∧ ϕ^{i,0,s_{i0}}.  (16)

The lemma now follows from the following claim: Claim: For each ϕ as in (16) ℘(ϕ) is a linear combination of the functions det(ϕ_{φ,α,β}).

We prove the claim as follows: we construct vector spaces A_{i}, B_{j} and A = ⨁_{i∈J^{+}} A_{i}, B = ⨁_{j∈J^{−}} B_{j} and

H̃_{0} = H_{0} ⊕ ⨁_{i∈J^{+}} Hom(A_{i}, Y_{i}) ⊕ ⨁_{j∈J^{−}} Hom(X_{j}, B_{j}) ⊂ Hom(X ⊕ A, Y ⊕ B) = H̃.

Observe that on H̃, H̃_{0} there is an action of G̃ = G_{XY} ×G_{AB} = G_{XY} ×∏_{i∈J^{+}} Gl(A_{i})× ∏_{j∈J^{−}} Gl(B_{j}) and we call c̃the character of G̃ given by (∏_{j} det_{GL(X_{j})} × ∏_{i} det_{GL(A_{i})})^{−1}× ∏_{i} det_{GL(Y_{i})} × ∏_{j} det_{GL(B_{j})}. We have an embedding of G_{XY} in G̃ such that σ^{∗}c̃ = c. Observe also that by Proposition 28 we know that the c̃-covariants functions on H are generated by the functions det(ΦS̃) with S ∈ S̃: we put a tilde to emphasize that we have to consider also the components {A_{i}} and {B_{j}}. Then we construct a G_{XY} -equivariant map ρ : H ⟶ H̃_{0} such that

1. there exists a c̃-covariant function f on H̃ such that ℘(ϕ) = f ∘ρ.

2. for all S̃ ∈ S̃ there exists φ, α, β as in equation (10) such that det(ΦS̃)∘ρ =

det(Φ_{φ,α,β}) . The claim now follows by Proposition 28.

For i ∈ J^{+} and j ∈ J^{−} define

A_{i} = ℂ^{s_{i0}},  A = ⨁_{i∈J^{+}} A_{i},  B_{j} = ℂ^{s_{0j}},  B = ⨁_{j∈J^{−}} B_{j}.

Define also α_{i} : A_{i} ⟶ X_{0} and (β_{j})^{t} : B_{j}^{∗} ⟶ Y_{0}^{∗} by

α_{i}(e_{l}) = ϕ^{i,0,l},  and α = ∐_{i∈J^{+}} α_{i} : A ⟶ X_{0}
(β_{i})^{t}(e^{l}) = ϕ^{0,j,l}  and β^{t} = ∏_{j∈J^{−}} B_{j} : B^{∗} ⟶ Y_{0},

where e_{l} (resp. e^{l}) is the canonical basis of ℂ^{m} (resp. (ℂ^{m})^{∗}). We define β_{i} (resp. β) as the transpose of (β_{i})^{t} (resp. β^{t}). Now define ρ^{ij} : Hom(X_{j}, Y_{i})⊗ℂ^{r_{ij}} ⟶ Hom(X_{j}, Y_{i}), ρ^{i0} : Hom(X_{0}, Y_{i}) ⟶ Hom(A_{i}, Y_{i}), ρ^{0j} : Hom(X_{j}, Y_{0}) ⟶ Hom(X_{j}, B_{j}) by

ρ^{ij}(T ⊗ v) = ϕ^{i,j}(v)T,  ρ^{i0}(T) = T ∘α_{i},  ρ^{0j}(T) = β_{j} ∘T,

and finally define ρ = ⨁_{i,j∈J̃} ρ^{ij} : H ⟶ H̃_{0}. Observe that ρ is G_{XY} -equivariant.

Observe now that H̃_{0}^{⊗n} = ⨁ ẼS̃ where S̃ ∈ S̃ and ẼS̃ is defined as in (11). In particular we choose the following summand of H̃_{0}^{⊗n}:

Ẽ = ⨂_{(i,j)∈J} Hom(X_{j}, Y_{i})^{⊗s_{ij}} ⊗ ⨂_{j∈J^{−}} Hom(X_{j}, B_{j})^{⊗s_{0j}} ⊗ ⨂_{i∈J^{+}} Hom(A_{i}, Y_{i})^{⊗s_{i0}}

and we observe that (Ẽ)∗ ; c̃ = ℂ. Choose a non zero element ϕ̃ ∈ (Ẽ)∗ ; c̃ and observe that up to a scalar we have

℘H̃(ϕ̃)∘ρ = ℘(ϕ).  (17)

To see this choose ϕ as in (16), and bases y_{h}^{i} of Y_{i}, x_{k}^{j} of X_{j}^{∗} (and its dual basis z_{k}^{j} of X_{j}) . Choose also a bases ε_{m}^{ij} of ℂ^{r_{ij}} such that ϕ^{i,j}(ε_{m}^{ij}) = δ_{m,1} and set A^{ij} = ρ^{ij}(s) = ∑_{h,k} a_{hk}^{ij}y_{h}^{i} ⊗ x_{k}^{j} for s ∈ H. Then

℘(ϕ)(t) = ∑_{h∈K_{Y} ,k∈K_{X}} ∏_{i∈J^{+}} < ⋀_{\overrightarrow{(i,j,q)∈K̃}} a_{h(i,j,q)k(i,j,q)}^{ij}y_{h(i,j,q)}^{i}, o_{i}^{∗}> ∏_{j∈J^{−}} < ⋀_{\overrightarrow{(i,j,q)∈K̃}} x_{k(i,j,q)}^{j}, o_{j}>

∏_{i∈J^{+}} < ⨂_{\overrightarrow{(i,0,q)∈K̃}} x_{k(i,0,q)}^{0}, ϕ^{i,0,1} ∧ ⋯ ∧ ϕ^{i,0,s_{i0}}>

∏_{j∈J^{−}} < ⨂_{\overrightarrow{(0,j,q)∈K̃}} a_{h(0,j,q)k(0,j,q)}^{0j}y_{h(0,j,q)}^{0}, ϕ^{0,j,1} ∧ ⋯ ∧ ϕ^{0,j,s_{i0}}>

= ∑_{k∈K_{X}} ∏_{i∈J^{+}} < ⋀_{\overrightarrow{(i,j,q)∈K̃}} A^{ij}z_{k(i,j,q)}^{j}, o_{i}^{∗}> ∏_{j∈J^{−}} < ⋀_{\overrightarrow{(i,j,q)∈K̃}} x_{k(i,j,q)}^{j}, o_{j}>

∏_{i∈J^{+}} < ⋀_{\overrightarrow{(i,0,q)∈K̃}} x_{k(i,0,q)}^{0}, ϕ^{i,0,1} ∧ ⋯ ∧ ϕ^{i,0,s_{i0}}>

∏_{j∈J^{−}} < ⋀_{\overrightarrow{(0,j,q)∈K̃}} A^{0j}z_{k(0,j,q)}^{j}, ϕ^{0,j,1} ∧ ⋯ ∧ ϕ^{0,j,s_{i0}}>

where the indexes are as follows:

K_{X} = {k : K̃ ⟶ ℕ : 1 ≤ k(i, j, q) ≤ dim X_{j}}
K_{Y} = {h : K̃ ⟶ ℕ : 1 ≤ h(i, j, q) ≤ dim Y_{i}}.

The lefthandside in (17) clearly furnishes the same expression.

Finally if we fix S̃ = (s_{MN})_{N∈{X_{j}}∪{A_{i}} and M∈{Y_{i}}∪{B_{j}}} ∈ S̃ and we choose φ_{ij} = s_{Y_{i}X_{j}}ϕ^{i,j} and α = ∐_{i∈J^{+}} s_{Y_{i}A_{i}}α_{i} : A ⟶ X_{0} and β = ∐_{j∈J^{−}} s_{B_{j}X_{j}}β_{j} : Y_{0} ⟶ B we have

det(ΦS̃)∘ρ = det(Φ_{φ,α,β}).

Remark 29. The basis of ℂ[H]_{c} we have described are different from the polarization of the natural basis of E_{c}^{∗}. The relation between the two basis is given by formulas of the following types

1. If A = (\begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}) and B = (\begin{matrix}b_{11} & b_{12} \\ b_{21} & b_{22} \end{matrix}) then

det (\begin{matrix}a_{11} & b_{12} \\ a_{21} & b_{22} \end{matrix}) + det (\begin{matrix}b_{11} & a_{12} \\ b_{21} & a_{22} \end{matrix}) = det(A + B) − det A − det B.

2. If A = (\begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}), B = (\begin{matrix}b_{11} & b_{12} \\ b_{21} & b_{22} \end{matrix}), C = (\begin{matrix}c_{11} & c_{12} \\ c_{21} & c_{22} \end{matrix}) and D = (\begin{matrix}d_{11} & d_{12} \\ d_{21} & d_{22} \end{matrix}) then

det (\begin{matrix}a_{11} & b_{11} \\ a_{21} & b_{21} \end{matrix}) det (\begin{matrix}c_{12} & d_{12} \\ c_{22} & d_{22} \end{matrix}) − det (\begin{matrix}a_{11} & b_{12} \\ a_{21} & b_{22} \end{matrix}) det (\begin{matrix}c_{12} & d_{11} \\ c_{22} & d_{21} \end{matrix}) +

− det (\begin{matrix}a_{12} & b_{11} \\ a_{22} & b_{21} \end{matrix}) det (\begin{matrix}c_{11} & d_{12} \\ c_{21} & d_{22} \end{matrix}) + det (\begin{matrix}a_{12} & b_{12} \\ a_{22} & b_{22} \end{matrix}) det (\begin{matrix}c_{11} & d_{11} \\ c_{21} & d_{21} \end{matrix}) =

= − det (\begin{matrix}A & B \\ C & D \end{matrix}) + det (\begin{matrix}A & 0 \\ 0 & D \end{matrix}) + det (\begin{matrix}0 & B \\ C & 0 \end{matrix})

The first type of formula correspond to the reduction of Lemma 27 to the case r_{ij} = 1 and X_{0} = Y_{0} = 1. The second type of formula correspond to the case of Proposition 28.

2.3. Proof of Proposition 20. Choose basis B_{i} (resp. B_{i}^{∗}) of the vector spaces D_{i} and D_{i}^{∗} and we write our vector space S(d, v) in the following way:

S = ⨁_{h∈H} V_{h_{0}}^{∗} ⊗ V_{h_{1}} ⊕ ⨁_{\begin{matrix}i∈I \\ b∈B_{i} \end{matrix}} V_{i,b} ⊕ ⨁_{i∈I,b^{∗}∈B_{i}^{∗}} V_{i,b^{∗}}^{∗}

where V_{i,b} (resp. V_{i,b^{∗}}^{∗}) is an isomorphic copy of V_{i} (resp. V_{i}^{∗}). We fix also a character χ_{m} and m_{i}, m̃_{i}, m_{i}^{+}, m_{i}^{−}, I^{+}, I^{−}, I^{0} as in 2.0.2 and we describe first the χ_{m}-covariants of S^{⊗n}. To do it we observe that we can decompose S^{⊗n} in the following way:

S^{⊗n} = ⨁_{ℓ} E_{1}^{(ℓ)} ⊗ ⋯ ⊗ E_{n}^{(ℓ)}

where each E_{i}^{(ℓ)} is a representation of G of one of the following types: V_{h_{0}}^{∗} ⊗ V_{h_{1}}, V_{i,b} or V_{i,b^{∗}}^{∗}. So it is enough to compute the χ-covariants of each piece E_{1}^{(ℓ)} ⊗ ⋯ ⊗ E_{n}^{(ℓ)}. We fix one of them: E = E_{1} ⊗ ⋯ ⊗ E_{n} and we compute E_{χ}^{∗}. Let I^{∗} be a copy of I and fix an isomorphism i ⟷ i^{∗} between the two sets. For each j = 1, …, n we define a subset S_{j} of I ∐ I^{∗} according to the following rule:

S_{j} = \begin{cases}\{h_{0}^{∗}, h_{1}\} & if E_{j} = V_{h_{0}}^{∗} ⊗ V_{h_{1}}, \\ \{i\} & if E_{j} = V_{i,b} for some b ∈ B_{i}, \\ \{i^{∗}\} & if E_{j} = V_{i,b^{∗}}^{∗} for some b^{∗} ∈ B_{i}^{∗}. \end{cases}

Let now be S = ∐_{j=1}^{n} S_{j}. An element of S can be thought as a couple (i, j) (or (i^{∗}, j)) where i (or i^{∗}) is in S_{j}. We consider now a special class of partitions of S: a collection 𝔉 = {C, M_{i}^{(l)} for i ∈ I and 1 ≤ l ≤ m_{i}} of disjoint subsets of 2^{S} is called m-special if:

1. ⋃ 𝔉 is a partition of S, 2. ∀ C ∈ C cardC = 2 and ∃i ∈ I, S_{j_{1}}, S_{j_{2}} such that i ∈ S_{j_{1}}, i^{∗} ∈ S_{j_{2}} and

C = {(i, j_{1}), (i^{∗}, j_{2})},

3. ∀M ∈ M_{i}^{(l)} we have M = {(i, j)} if i ∈ I^{+} and M = {(i^{∗}, j)} if i ∈ I^{−},

4. cardM_{i}^{(l)} = v_{i} = dim V_{i}. We can represents a special collection with an enriched graph whose vertices are

the sets S_{j} and completed according with the following rules:

1. we put an arrow from S_{j_{1}} to S_{j_{2}} if there exists C = {(i, j_{2}), (i^{∗}, j_{1})} ∈ C,

2. we put an indexed circle box ∘_{i}^{l} on S_{j} if there exists M = {(i, j)} ∈ M_{i}^{(l)}

3. we put an indexed square box □_{i}^{l} on S_{j} if there exists M = {(i^{∗}, j)} ∈ M_{i}^{(l)} 4. if E_{j} is of type V_{i,b} or V_{i,b^{∗}}^{∗} then we add the element b or b^{∗} at the left of the

corresponding vertex.

5. if E_{j} is of type V_{h_{0}}^{∗} ⊗ V_{h_{1}} then we write h at the left of the corresponding vertex. Observe that a vertex can be marked with a circle and a square but that it cannot be marked with two circles or two squares.

There is a perfect bijection between m-special collection 𝔉 and graphs as above such that:

1. the cardinality of vertexes marked with ∘_{i}^{l} is v_{i} for each i ∈ I^{+} and 1 ≤ l ≤ m_{i},

2. the cardinality of vertexes marked with □_{i}^{l} is v_{i} for each i ∈ I^{−} and 1 ≤ l ≤ −m_{i}. We will use the same letter 𝔉 to indicate the collection or the graph.

To a special collection 𝔉 as above we attach a function ϕ_{𝔉} on E. We define it by the formula

ϕ_{𝔉}(e_{1} ⊗ ⋯ ⊗ e_{n}) = ∏_{C∈C} ϕ_{C} · ∏_{i∈I^{+}} ∏_{l=1}^{m_{i}} <o_{i}^{∗}, ⋀ M_{i}^{(l)}> · ∏_{i∈I^{−}} ∏_{l=1}^{m_{i}} <o_{i}, ⋀ M_{i}^{(l)}>

where o_{i} is a non zero element in ⋀^{v_{i}} V_{i}, o_{i}^{∗} is a non zero element in ⋀^{v_{i}} V_{i}^{∗} and

e_{j} = \begin{cases}x_{j}^{∗} ∈ V_{i}^{∗} & if E_{j} = V_{i,b}^{∗}, \\ y_{j} ∈ V_{i} & if E_{j} = V_{i,b}, \\ x_{j}^{∗} ⊗ y_{j} ∈ V_{h_{0}}^{∗} ⊗ V_{h_{1}} & if E_{j} = V_{h_{0}}^{∗} ⊗ V_{h_{1}}, \end{cases}

ϕ_{C} =<x_{j_{1}}^{∗} , v_{j_{2}}>  if C = {(i^{∗}, j_{1}), (i, j_{2})}
⋀ M_{i}^{(l)} = y_{j_{1}} ∧ ⋯ ∧ y_{j_{v_{i}}} if M_{i}^{(l)} = {{(i, j_{1})}, …, {(i, j_{v_{i}})}} and i ∈ I^{+}
⋀ M_{i}^{(l)} = x_{j_{1}}^{∗} ∧ ⋯ ∧ x_{j_{v_{i}}}^{∗} if M_{i}^{(l)} = {{(i, j_{1})}, …, {(i, j_{v_{i}})}} and i ∈ I^{−}

Finally we extend ϕ_{𝔉} to all E by linearity. By the lemma above and the discussion in 2.1 we deduce easily the following lemma:

Lemma 30. E_{χ}^{∗} is generated by the functions ϕ_{𝔉}.

Proposition 20 now follows from lemma 30 and the following claim: claim: for any special collection 𝔉 the function ℘(ϕ_{𝔉}) is a ℂ[S]^{G_{v}}-linear combination of the functions f_{Δ} described in 2.0.2.

We consider the connected components of the graph. There are only five possible types of paths:

1. closed paths,

2. straight paths leaving from a non boxed vertex and arriving in a non boxed vertex,

3. straight paths leaving from a non boxed vertex and arriving in a circle boxed vertex,

4. straight paths leaving from a square boxed vertex and arriving in a non boxed vertex,

5. straight paths leaving from a square boxed vertex and arriving in a circle boxed vertex.

Let now 𝔉_{0} be the union of the connected components of the first two types and 𝔉_{1} be the union of the remaining components. Observe that

℘(ϕ_{𝔉}) = ℘(ϕ_{𝔉_{0}})℘(ϕ_{𝔉_{1}}).

Observe also that ϕ_{𝔉_{0}} is an invariant function (indeed this part of the graph corresponds to the situation studied by Lusztig in [9]). Since we are interested in generators of ℂ[S]_{χ,1}as a ℂ[S]^{G}-module, we can suppose for simplicity 𝔉 = 𝔉_{1}.

Observe now that each connected component Γ of the graph of the third type and with a circle ∘_{i_{1}}^{l} at the end, has an initial vertex which is an S_{j} = {i_{0}^{∗}} and that is marked with b ∈ B_{i_{0}}^{∗} on the left. All the other vertexes of the connected component are of type S_{j} = {h_{0}^{∗}, h_{1}} and they define a path α^{Γ} such that α_{0}^{Γ} = i_{0} and α_{1}^{Γ} = i_{1}. We call b = b(Γ) and l = L_{1}(Γ).

In the same way we see that:

1. each connected component Γ of the fourth type determines a path α^{Γ}, b^{∗} = b^{∗}(Γ) ∈ B_{α_{1}^{Γ}}^{∗} and l = L_{0}(Γ) such that 1 ≤ l ≤ −m_{α}_{0}^{Γ},

2. each connected component Γ of the fifth type determines a path α^{Γ}, l_{0} = L_{0}(Γ) and l_{1} = L_{0}(Γ) such that 1 ≤ l_{0} ≤ −m_{α}_{0}^{Γ} and 1 ≤ l_{1} ≤ m_{α}_{1}^{Γ}.

Now we prove the claim in the following way, we construct X_{j} and Y_{i} as in 2.2, a groups homomorphism σ : G_{v} ⟶ G_{XY} such that σ^{∗}c = χ_{m}, a G_{v} equivariant map ρ : S ⟶ H, and a G_{XY} c-covariant function f on H such that:

1. for all φ, α, β there exists a χ_{m}-good data such that det(Φ_{ϕ,α,β})∘ρ = f_{Δ},

2. ℘(ϕ_{𝔉}) = f ∘ρ

The claim will clearly follow.

Set

J^{−} = {(i, l) : i ∈ I^{−} and 1 ≤ l ≤ −m_{i}},
J^{+} = {(i, l) : i ∈ I^{+} and 1 ≤ l ≤ m_{i}}.

For all (i, l) ∈ J^{−} choose X_{(i,l)} = V_{i} and for each (i, l) ∈ J^{+} choose Y_{(i,l)} = V_{i}. For each (i_{0}, l_{0}) ∈ J^{−} and for each (i_{1}, l_{1}) ∈ J^{+} define:

r_{(i_{0},l_{0})(i_{1},l_{1})} = card{ connected component Γ of the fifth type such
that α_{0}^{Γ} = i_{0}, α_{1}^{Γ} = i_{1}, L_{0}(Γ) = l_{0} and L_{1}(Γ) = l_{1}}

We the connected component Γ of the set in the left handside as a basis e_{Γ} of the vector space ℂ^{r_{(i_{0},l_{0})(i_{1},l_{1})}}. This basis plays the role of the basis e_{m}^{ij} we used to give the identification in (9).

For each Γ of the third type choose a one dimensional vector space ℂ_{b(Γ)} and fix a generator b_{Γ}. For each Γ of the fourth type choose a one dimensional vector space ℂ_{b^{∗}(Γ)} and fix a generator b_{Γ}^{∗}.

X_{0} = ⨁_{Γ of the third type} ℂ_{b(Γ)} = ⨁_{Γ of the third type} ℂb_{Γ}
Y_{0} = ⨁_{Γ of the fourth type} ℂ_{b^{∗}(Γ)} = ⨁_{Γ of the fourth type} ℂb_{Γ}^{∗}.

Now for each connected component Γ of the third type define ρ^{Γ} : S ⟶ Hom(ℂ_{b(Γ)}, Y_{(α_{1}^{Γ},L_{1}(Γ))}) by

s ⟼ {λ ↦ α^{Γ}(s)γ_{α}_{0}^{Γ}(b(Γ))λ}

In a similar way define ρ^{Γ} if Γ is the fourth or of the fifth type. Finally define

ρ : S ⟶ H by ρ = ⨁_{Γ} ρ^{Γ}.

Define also a group homomorphism σ : G_{v} ⟶ G_{XY} by (σ(g_{i}))_{X_{(i_{0},l_{0})}} = g_{i_{0}} and (σ(g_{i}))_{Y_{(i_{1},l_{1})}} = g_{i_{1}}, and observe that ρ is G_{v} equivariant.

Now we describe ϕ ∈ (H^{⊗ñ})_{c}^{∗} (in general ñ is less or equal to n) such that

℘(ϕ_{𝔉})(s) = ℘(ϕ)(ρ(s)).  (18)

We describe ϕ by giving a summands Ẽ of H^{⊗ñ} as in (14) and ϕ ∈ Ẽ_{c}^{∗} as in (15).To define Ẽ we have to define s_{(i_{1},l_{1})(i_{0},l_{0})}, s_{(i_{1},l_{1})0} and s_{0(i_{0},l_{0})} for all (i_{1}, l_{1}) ∈ J^{+} and for all (i_{0}, l_{0}) ∈ J^{−}. We set

s_{(i_{1},l_{1})(i_{0},l_{0})} = r_{(i_{1},l_{1})(i_{0},l_{0})}
s_{(i_{1},l_{1})0} = card{ connected component Γ of the third type
such that α_{1}^{Γ} = i_{1} and L_{1}(Γ) = l_{1}}
s_{0(i_{0},l_{0})} = card{ connected component Γ of the fourth type
such that α_{0}^{Γ} = i_{0} and L_{0}(Γ) = l_{0}}

Observe that we can choose a bijection q ⟷ Γ_{q} between {1, …, s_{(i_{1},l_{1})(i_{0},l_{0})}} and the set of connected component Γ of the fifth type such that α_{1}^{Γ} = i_{1},L_{1}(Γ) = l_{1}, α_{0}^{Γ} = i_{0} and L_{0}(Γ) = l_{0}. So we can define ϕ^{(i_{1},l_{1}),(i_{0},l_{0}),q} by

ϕ^{(i_{1},l_{1}),(i_{0},l_{0}),q}(e_{Γ}) = δ_{Γ,Γ_{q}}.

Observe also that we can choose a bijection q ⟷ Γ_{q} between {1, …, s_{(i_{1},l_{1})0}} and the set of connected component Γ of the third type such that α_{1}^{Γ} = i_{1},L_{1}(Γ) = l_{1}. So we can define ϕ^{(i_{1},l_{1}),0,q} by

ϕ^{(i_{1},l_{1}),0,q}(b_{Γ}) = δ_{Γ,Γ_{q}}.

In a similar way define ϕ^{0,(i_{0},l_{0}),q}.

Up to a sign which depends on our choices and ordering equation (18) is a tautologically satisfied.

Observe now that by lemma 27 and linearity ℂ[H]_{c} is generated by functions s ⟼ det(Φ_{φ,α,β}(s)) where Ã, B̃, φ, α, β are as in (10) and moreover there exists a basis e_{1}, …, e_{r_{A}} of Ã and a basis ẽ_{1}, …, ẽ_{r_{B}} of B̃^{∗} such that for all i there exist a connected component of the third type Γ_{i}^{A} such that α(e_{i}) = b_{Γ}_{i}^{A} and for all i there exists a connected component of the fourth type Γ_{i}^{B} such that ẽ_{i}(β(b_{Γ}^{∗})) = δ_{Γ}_{Γ}_{i}^{B}. So it is enough to prove that if Ã, B̃, φ, α, β are as above then there exists a χ_{m}-good Δ such that

det(Φ_{φ,α,β})∘ρ = f_{Δ}.

We define

A = (b_{Γ}_{i}^{A})_{i=1,...,r_{A}} ∈ (⋃ D_{i})^{dim Ã}
B = (b_{Γ_{i}^{B}}^{∗})_{i=1,...,r_{B}} ∈ (⋃ D_{i}^{∗})^{dim B̃}
α_{jh}^{ik} = ∑_{1≤q≤s_{(j,h)(i,k)}} ϕ^{(j,h),(i,k),q}(e_{Γ})α^{Γ}
α_{l}^{ik} = δ_{i,(α^{^{Γ}^{B} l})_{0}}δ_{k,L_{0}(Γ_{l}^{B})}α^{Γ^{B}^{l}}
α_{jh}^{l} = δ_{j,(α^{^{Γ}^{A} l})_{1}}δ_{h,L_{1}(Γ_{l}^{A})}α^{Γ^{A}^{l}}

The equation (18) follows now by the very definition.

3. The action of the Weyl group

For any m ∈ P and for any λ ∈ Z we defined a variety M_{m,λ}(d, v). Observe that on both m, λ there is a natural action of the Weyl group W. We define an action of the Weyl group also on (d, v). We have already described d as an element of X and v as an element of Q. We can now define

σ(d, v) = (d, σ(v − d) + d).

Observe that σ(v − d) + d ∈ Q so the definition is well given. So it make sense to consider the variety M_{σm,σλ}(σ(d, v)) or the variety 𝔐_{σζ}(σ(d, v)) for ζ ∈ ℨ.

In [13] Nakajima used analytic methods to prove, in the case of a finite Dynkin diagram, that if ζ is generic then there exists a diffeomorphism of differentiable manifolds

Φ_{σ,ζ}: 𝔐_{ζ}(d, v) ⟶ 𝔐_{σζ}(σ(d, v))

and moreover that Φ_{σ′,σζ} ∘Φ_{σ,ζ} = Φ_{σ′σ,ζ}. In the same paper he also asserted that a similar construction could be obtained in the general case using reflection functors as indeed we are going to do.

In [9] Lusztig gave a purely algebraic construction of an isomorphism

M_{0,λ}(d, v) ≃ M_{0,s_{i}λ}(s_{i}(d, v))

whenever λ_{i} ≠ 0. In this paper we will give a generalization of Lusztig construction.

Definition 31. If u ∈ ℤ^{n} = Q^{∨} and A ⊂ Q^{∨} we define

H_{u} = {(m, λ) ∈ P ⊕ Z : <u,^{∨} , λ>=<u^{∨}, m>= 0} and H_{A} = ⋃_{a∈A} H_{a}

Let K = max{1, a_{ij}^{2} : i, j ∈ I}. If v ∈ ℤ^{n} we define

Ũ_{v} = {u ∈ ℕ^{I} : 0 ≤ u_{i} ≤ Kv_{i}} and H̃^{v} = HŨ_{v}.

We define also

U_{∞} = ⋃_{i∈I} W α_{i}^{∨} and H^{∞} = H_{U_{∞}}.

Finally we set G_{v} = {(m, λ) ∈ P × Z_{G} : σ(m, λ) ∉ H^{σ·v} for all σ ∈ W }. Both of the following definition of the set G will be fine for us:

G = {(v, m, λ) ∈ Q × P × Z_{G} : (m, λ) ∈ G_{v}} or
G = {(v, m, λ) ∈ Q × P × Z_{G} : (m, λ) ∉ H^{∞}}.

We observe that in any case G is W -stable.

Proposition 32. For all d, v, for all σ ∈ W and for all (m, λ) such that (m, λ, v) ∈ G_{v} there exists an algebraic isomorphism:

Φ_{σ,m,λ}^{d,v}: M_{m,λ}(d, v) ⟶ M_{σm,σλ}(σ(d, v)).

Moreover this isomorphism satisfies

Φ_{τ,σm,σλ}^{σ(d,v)}∘Φ_{σ,m,λ}^{d,v} = Φ_{τσ,m,λ}^{d,v}.  (19)

3.1. Generators. In this section we define the actions of the generators s_{i} of W following [9]. We fix i ∈ I and (d, v), λ ∈ Z and m ∈ P. We call (d, v′) = s_{i}(d, v), λ′ = s_{i}λ and m′ = s_{i}m. Through all this section we assume v, v′ ≥ 0. For the convenience of the reader we write explicit formula in this case:

λ′_{j} = λ_{j} − c_{ij}λ_{i}  m′_{j} = m_{j} − c_{ij}m_{i} for all j
v′_{i} = d_{i} − v_{i} + ∑_{j≠i} a_{ij}v_{j}  v′_{j} = v_{j} for all j ≠ i

Observe that we can choose

D′_{j} = D_{j} for all j and V′_{j} = V_{j} for all j ≠ i.

In particular we have

T_{i} = D_{i} ⊕ ⨁_{h_{1}=i} V_{h_{0}} = T′_{i}

since we suppose that our quiver has not simple loops.

Definition 33 (Lusztig [9]). Fix λ ∈ Z_{G} and define Z_{i}^{λ}(d, v) to be the sub-variety of S_{i}(d, v) × S_{i}(d, v′) of pairs (s, s′) = ((B, γ, δ), (B′, γ′, δ′)) such that the following conditions hold:

1. B_{h}(s) = B′_{h}(s′) for all h such that h_{0}, h_{1} ≠ i,

2. γ_{j}(s) = γ′_{j}(s′) for all j ≠ i ,

3. δ_{j}(s) = δ′_{j}(s′) for all j ≠ i, 4. the following sequence is exact:

0 ⟶ V′_{i} ⟶^{a′_{i}} T_{i} ⟶^{b_{i}} V_{i} ⟶ 0,  (20)

5. a′_{i}(s′)b′_{i}(s′) = a_{i}(s)b_{i}(s) − λ_{i}Id_{T_{i}},

6. s ∈ Λ_{λ}(d, v) and s′ ∈ Λ_{λ}′(d, v′).

Lemma 34. Let (s, s′) ∈ S_{i}(d, v) × S_{i}(d, v′) and suppose that it satisfies conditions 1), 2), 3), 4) , 5) above then:

1. s ∈ Λ_{λ}(d, v) ⟺ s′ ∈ Λ_{λ}′(d, v′),

2. if μ_{j}(s) − λ_{j}Id_{V_{j}} = 0 for all j ≠ i then s ∈ Λ_{λ}(d, v) ,

3. if μ_{j}(s′) = λ_{j}Id_{V′_{j}} for all j ≠ i then s′ ∈ Λ_{λ}′(d, v′).

Proof. 2) We have to prove b_{i}a_{i} −λ_{i}Id_{V_{i}} = 0 and by condition 4) it is enough to prove b_{i}a_{i}b_{i} = λ_{i}b_{i}. So b_{i}a_{i}b_{i} = b_{i}(a′_{i}b′_{i} − λ_{i}) = λ_{i}b_{i} by conditions 4) and 5).

The proof of 3) is equal to the proof of 2). We prove the implication ⇒ in 1). By

2) and 3) it is enough to prove that b′_{j}a′_{j} = λ′_{j} for j ≠ i.

b′_{j}a′_{j} = ∑_{h_{1}=j} ε(h)B′_{h}B′ ; h̄ + γ′_{j}δ′_{j} =

= ∑_{h_{1}=j,h_{0}≠i} ε(h)B_{h}Bh̄ + γ_{j}δ_{j} + ∑_{h_{1}=j,h_{0}=i} ε(h)B′_{h}B′ ; h̄ =

= b_{j}a_{j} + ∑_{h_{1}=j,h_{0}=i} ε(h) (B′_{h}B′ ; h̄ − B_{h}Bh̄)

= b_{j}a_{j} + ∑_{h_{0}=j,h_{1}=i} (Bh̄ε(h)B_{h} − B′ ; h̄ε(h)B′_{h})

= b_{j}a_{j} + ∑_{h_{0}=j,h_{1}=i} ([a_{i}b_{i}]_{V_{h_{0}}}^{V_{h_{0}}} − [a′_{i}b′_{i}]_{V_{h_{0}}}^{V_{h_{0}}})

= λ_{j} + ∑_{h_{0}=j,h_{1}=i} λ_{i} = λ′_{j}

The proof of the converse is completely analogous.

Lemma 35. Let λ ∈ Z_{G}, (s, s′) ∈ Z_{i}^{λ}(d, v) and α be an element of the path algebra algebra of type (α_{0}, α_{1}) then

1. if α_{0}, α_{1} ≠ i there exists an element α′ of the b-path algebra of type (α_{0}, α_{1}) such that α′(s′) = α(s),

2. if α_{1} ≠ i there exists an element α′ of the b-path algebra of type (α_{0}, α_{1}) such that α′(s′)γ′_{α_{0}} = α(s)γ_{α_{0}},

3. if α_{0} ≠ i there exists an element α′ of the b-path algebra of type (α_{0}, α_{1}) such that δ′_{α_{1}}α′(s′) = δ_{α_{1}}α(s),

4. there exists an element α′ of the b-path algebra of type (α_{0}, α_{1}) such that δ′_{α_{1}}α′(s′)γ′_{α_{0}} = δ_{α_{1}}α(s)γ_{α_{0}}, Proof. By induction on the length of α we can reduce the proof of this lemma to the following identities that are a consequence of condition 5) in definition 33:

B′_{h}B′_{k} = \begin{cases}B_{h}B_{k} & if h ≠ k̄ \\ B_{h}B_{k} − λ_{i} & if k = h̄ \end{cases}
δ′_{i}B′_{k} = δ_{i}B_{k}
B′_{h}γ′_{i} = B_{h}γ_{i}
δ′_{i}γ′_{i} = δ_{i}γ_{i} − λ_{i}

for h, k such that h_{0} = i = k_{1}.

Lemma 36. Let (s, s′) ∈ Z_{i}^{λ}(d, v) and suppose m_{i} ≥ 0 or λ_{i} ≠ 0 then

s is χ_{m} semistable ⟺ s′ is χ_{m′} semistable

Proof. We prove only ⇒. Let’s do first the case m_{i} ≥ 0. If s is χ_{m} semi-stable, then there exists Δ = {A, B, α_{∗}^{∗}} m-good such that f_{Δ}(s) ≠ 0. Using the notation in 2.0.2 we have φ_{Δ} = det Ψ_{Δ} where Ψ_{Δ} : Y ⟶ Z is a linear map. In our case we can write Z as ℂ^{m_{i}} ⊗ V_{i} ⊕ Z̃ and we observe that no V_{i} summands appear in Y or Z̃.

Now we construct a new data Δ′ = {A′, B′, α′_{∗}^{∗}} such that f_{Δ}′(s′) ≠ 0 and f_{Δ}′ a χ′-covariant polynomial. Our strategy will be the following: we substitute each V_{i} with the space T_{i} in the space Z and we add m_{i} copies of V′_{i} to Y. Let’s do it more precise: first of all the new data will not be m′ good so we have to define m′_{j}^{+} and m′_{j}^{−}:

1. m′_{i}^{+} = 0 and m′_{i}^{−} = m_{i} = m_{i}^{+},

2. m′_{j}^{−} = m_{j}^{−} and m′_{j}^{+} = m_{j}^{+} + a_{ij}m_{i}^{+} for all j ≠ i,

3. m′^{−} = m^{−} and m′^{+} = m^{+} + d_{i}m_{i}^{+}. Observe that m′_{j}^{+} − m′_{j}^{−} = m′_{j} for all j so our data will furnish a χ′ equivariant function. Moreover if we define

Z′ = ℂ^{m_{i}} ⊗ V_{i} ⊕ Z̃ and Y′ = ℂ^{m_{i}} ⊗ T_{i} ⊕ Y

we observe that they have the numbers of V′_{j}, ℂ_{a}, ℂ_{b} factors specified by m′. Now we construct the new data Δ′ in such a way that with respect to the decompositions above we have:

[Ψ_{Δ}(s)]_{ℂ^{m_{i}}⊗V_{i}⊕Z̃}^{Y} = (\begin{matrix}(Id ⊗ b_{i})∘π \\ Φ \end{matrix}) ,
[Ψ_{Δ}′(s′)]_{ℂ^{m_{i}}⊗T_{i}⊕Z̃}^{ℂ^{m_{i}}⊗V′_{i}⊕Y} = (\begin{matrix}Id ⊗ a′_{i} & π \\ 0 & Φ \end{matrix}) .

If we construct a data with this property we observe that Ψ_{Δ}(s) is an isomorphism if and only if Ψ_{Δ}′(s′) is an isomorphism. Hence f_{Δ}(s) ≠ 0 implies f_{Δ}′(s′) ≠ 0 and the lemma is proved.

To construct the new data we choose a basis e_{1}, …, e_{d_{i}} of D_{i} and we define the other elements of the data according to the following rules

1. A′ = A,

2. if B = (b_{1}, …, b_{m^{+}}) we set B′ = (b_{1}, …, b_{m^{+}}, \underbrace{e_{1}, …, e_{1}}_{m_{1} times}, …, \underbrace{e_{d_{i}}, … e_{d_{i}}}_{m_{1} times}),

3. α′_{j_{2},h_{2}}^{j_{1},h_{1}} for j_{1} ≠ i and h_{2} ≤ m_{j_{2}}^{+} is an element constructed according to case 1) in the previous lemma,

4. α′_{j_{2},h_{2}}^{a} for h_{2} ≤ m_{j_{2}}^{+} is an element constructed according to case 2) in the previous lemma,

5. α′_{b_{l}}^{j_{1},h_{1}} for j_{1} ≠ i and l ≤ m^{+} is an element constructed according to case 3) in the previous lemma.

6. α′_{b_{l}}^{i,h} = α′_{j,k}^{i,h} = 0 if l ≤ m^{+} and k ≤ m_{j}^{+}.

In this way we guarantee that the projection of Ψ_{Δ}′(s′) onto Z̃ is equal to (0 Φ). To define the remaining part of the new data we do not give details on the indexes, but we explain how to construct it. It is clear that we can choose α′_{∗}^{i,h} for the remaining indexes * in such a way that the projection of Ψ_{Δ}′(s′)|_{ℂ^{m_{i}}⊗V′_{i}} on ℂ^{m_{i}} ⊗ T_{i} is equal to Id ⊗ a′_{i}. Finally we observe that a path β from V_{j} to V_{i} with j ≠ i has to go through a summand of T_{i} so there exists a path α such that β(s) = b_{i}∘α(s). Now we use the previous lemma to change α with a α′ such that β(s) = b_{i}∘α′(s′). More generally if β is an element of the path algebra of type (j, i) with j ≠ i then there exists an element of the b-path algebra α′ such that β(s) = b_{i}∘α′(s). In this way we define the elements of the b-path algebra connecting summands of Y and summands of ℂ^{m_{i}} ⊗ T_{i}.

In the case m_{i} < 0 we proceed in a similar way: we choose Δ m-good and we have

Y = ℂ^{−m_{i}} ⊗ V_{i} ⊕ Ỹ, Y′ = ℂ^{−m_{i}} ⊗ T_{i} ⊕ Ỹ, Z′ = ℂ^{−m_{i}} ⊗ V′_{i} ⊕ Z.

As in the previous case we can find a new data Δ′ such that:

[Ψ_{Δ}(s)]_{Z}^{ℂ^{−m_{i}}⊗V_{i}⊕Ỹ} = (π∘(Id ⊗ a_{i}) Φ) ,
[Ψ_{Δ}′(s′)]_{ℂ^{−m_{i}}⊗V′_{i}⊕Z}^{ℂ^{−m_{i}}⊗T_{i}⊕Ỹ} = (\begin{matrix}Id ⊗ b′_{i} & 0 \\ π & Φ \end{matrix}) .

Now to conclude that Ψ_{Δ}′(s′) is an isomorphism if Ψ_{Δ}(s) is we need to know that b′_{i} is an epimorphism and this is not guarantee by (s, s′) ∈ Z_{i}^{λ}(d, v). But if λ_{i} ≠ 0 then, since b′_{i}a′_{i} = −λ_{i}, we have that b′_{i} is surjective.

Definition 37. Let p (resp. p′) be the projections of Z_{i}^{λ}(d, v) on Λ_{λ}(d, v) ⊂ S(d, v) (resp. Λ_{λ}′(d, v′) ⊂ S(d, v′)). Suppose that m_{i} > 0 or λ_{i} ≠ 0 then we define

Z_{i}^{m,λ} = p^{−1}(Λ_{m,λ}(d, v)) = p′^{−1}(Λ_{m′,λ′}(d, v′)).

We define also

G_{i,v} = ∏_{j≠i} GL(V_{j}) × GL(V_{i}) × GL(V′_{i}).

Observe that there are natural projections from G_{i,v} to G_{v} and G_{v′}, therefore there are natural actions of G_{i,v} on S_{i}(d, v), S_{i}(d, v′). Observe that here is a natural action of G_{i,v} on Z_{i}^{λ} and Z_{i}^{m,λ} such that the projections p, p′ are equivariant.

Lemma 38. Let s ∈ Λ_{λ,m}(d, v) then 1. if λ_{i} ≠ 0 then b_{i} is epic and a_{i} is mono, 2. if m_{i} > 0 then b_{i} is epic,

3. if m_{i} < 0 then a_{i} is mono.

Proof. If λ_{i} ≠ 0 then the result is clear by b_{i}a_{i} = λ_{i}. Suppose now that λ_{i} = 0 and m_{i} > 0. Let U_{i} = Im b_{i} and let V_{i} = U_{i} ⊕ W_{i}. Define now a one parameter subgroup g(t) of G_{V} in the following way:

[g_{i}(t)]_{U_{i}⊕W_{i}}^{U_{i}⊕W_{i}} = (\begin{matrix}1 & 0 \\ 0 & t^{−1} \end{matrix}) and g_{j} ≡ 1 for j ≠ i

Since Im b_{i} ⊂ U_{i} we have that there exists the limit lim_{t→0} g(t) · s = s_{0}. Let now n > 0 and f a χ^{n}-covariant function on S such that f(s) ≠ 0. Then

f(s_{0}) = lim_{t→0} f(g(t) · s) = lim_{t→0} det_{GL(V_{i})}^{nm_{i}}f(s) = lim_{t→0} t^{−nm_{i} dim W_{i}}f(s)

So we must have dim W_{i} = 0. The proof of the third case is completely similar to this one.

Lemma 39 (see also Lusztig [9]). If m_{i} > 0 or λ_{i} ≠ 0 then

1. p : Z_{i}^{m,λ}(d, v) ⟶ Λ_{m,λ}(d, v) is a principal GL(V′_{i}) bundle,

2. p′ : Z_{i}^{m,λ}(d, v) ⟶ Λ_{m′,λ′}(d, v′) is a principal GL(V_{i}) bundle.

Proof. Lusztig’s proof extend to this case without changes. Let’s prove for example 1. We have to prove: i. that the action on the fiber is free, ii. that it is transitive. First of all we observe that by the previous lemma if s ∈ Λ_{m,λ} then b_{i}(s) is epic. In particular there exists a′_{i} : V′_{i} ⟶ T_{i} such that sequence (20) is exact, and clearly a′_{i} is univoquely determined up to the action of GL(V′_{i}), moreover this action is free. So i. and ii. reduce to the following fact: if s ∈ Λ_{m,λ} and a′_{i} is such that sequence (20) is exact, then there exists a unique, b′_{i} such that a′_{i}b′_{i} = a_{i}b_{i} − λ_{i}. Since a′_{i} is mono the unicity is clear. To prove the existence we observe that it is equivalent to Im a′_{i} ⊃ Im(a_{i}b_{i} − λ_{i}). But the last statement is clear since we have: Im a′_{i} = ker b_{i} and b_{i}(a_{i}b_{i} − λ_{i}) = 0.

Proposition 40. If m_{i} > 0 or λ_{i} ≠ 0 then the projections p, p′ induces algebraic isomorphisms p̄ , p̄′:

Λ_{m,λ}(d, v)//G_{v} ⟵_{p̄}^{∼} Z_{i}^{m,λ}(d, v)//G_{i,v} ⟶_{p̄′}^{∼} Λ_{m′,λ′}(d, v′)//G_{v′}

Proof. This proposition is a straightforward consequence of the previous lemma and the following general fact (see for example [12] Proposition 0.2): let G be an algebraic groups over ℂ and X, Y two irreducible algebraic variety over ℂ; if G acts on X and φ : X ⟶ Y is such that for all y ∈ Y the fiber X_{y} contains exactly one G-orbit then φ is a categorical quotient. If we apply this lemma to the projection p, (resp. p′) and to the group GL(V′_{i}) (resp. GL(V_{i})) we obtain the required result.

We can use this proposition to define the action of the generators of the Weyl group.

Definition 41. Let i, λ, m, d, v, λ′, m′, v′ be as above, and suppose d_{j} ≥ 0, v_{j}, v′_{j} ≥ 0 for all j then we define an isomorphism of algebraic variety

Φ_{s_{i},λ,m}^{d,v}: M_{m,λ}(d, v) ⟶ M_{m′,λ′}(d, v′)

in the following way:

1. if m_{i} > 0 or λ_{i} ≠ 0 then we set Φ_{s_{i},λ,m}^{d,v} = p̄′p̄^{−1},

2. if m_{i} < 0 then we exchange the role of v, v′ in the previous construction: more precisely we observe that m′_{i} > 0 so we can define Φ_{s_{i},λ′,m′}^{d,v′}: M_{m′,λ′}(d, v′) ⟶ M_{m,λ}(d, v) and we define Φ_{s_{i},λ,m}^{d,v} = (Φ_{s_{i},λ′,m′}^{d,v′})^{−1}.

Remark 42. To see that Φ_{s_{i},λ,m}^{d,v} is univoquely defined we have to verify that if λ_{i} ≠ 0 and m_{i} < 0 the two definitions above coincide. This fact reduces easily to the following remark: if λ_{i} ≠ 0 then

(s, s′) ∈ Z_{i}^{λ}(d, v) ⟺ (s′, s) ∈ Z_{i}^{λ′}(d, v′).

Let us prove, for example, the ⇒ part. Since a_{i}b_{i} = a′_{i}b′_{i} + λ_{i} = a′_{i}b′_{i} − λ′_{i} the only thing we have to verify is that the sequence

0 ⟶ V_{i} ⟶^{a_{i}} T_{i} ⟶^{b′_{i}} V′_{i} ⟶ 0

is exact. The surjectivity of b′_{i} and the injectivity of a_{i} are a consequence of λ_{i} ≠ 0. Since dim T_{i} = dim V_{i} + dim V′_{i} we need only to prove that b′_{i}a_{i} = 0. Observe that b′_{i}a_{i} = 0 if and only if a′_{i}b′_{i}a_{i} = 0 since also a′_{i} is injective. Finally a′_{i}b′_{i}a_{i} = (a_{i}b_{i} − λ_{i})a_{i} = 0.

3.2. Preliminaries. We saw how to define

Φ_{s_{i},m,λ}^{d,v}: M_{m,λ}(d, v) ⟶ M_{s_{i}(m),s_{i}(λ)}(s_{i}(d, v))

in the case that (λ_{i}, m_{i}) ≠ 0 and d, v, s_{i}v ≥ 0. To define an action of the Weyl group we have now to guarantee that coxeter relations hold. We will prove these relations in the next paragraph. Before doing it we observe that we have to guarantee some conditions on m, λ such that we will be able to define Φ_{s_{i},σm,σλ}^{σ(d,v)} for any element σ ∈ W: this condition will be (m, λ) ∈ G_{v} (31). We have also to say something about the case d_{i} < 0 or v_{i} < 0 for some i ∈ I.

In the case that d_{i} < 0 for some i then M_{m,λ}(σ(d, v)) = ∅ for all σ, m, λ by the very definition, so there is nothing to define.

The second trivial case is d = v = 0. Indeed in this case we have M_{σm,σλ}(σ(d, v)) = {0} so the definition is trivial.

The other two cases are treated in the two lemmas below.

In the following we fix d such that d_{i} ≥ 0 for all i. It will be convenient to define an affine action of W on Q by σ · v = σ(v − d) + d.

Lemma 43. Let d ≥ 0 and (m, λ) ∈ G_{v} if there exists σ such that σ · v ≱ 0 then M_{m,λ}(d, v) = ∅.

Proof. Suppose that σ is an element of minimal length such that σ · v ≱ 0 and let l = ℓ(σ). We prove the lemma by induction on l. The case l = 0 is trivial. Initial step: l = 1. If s_{i} · v ≱ 0 then we have 0 ≤ d_{i} + ∑ a_{ij}v_{j} < v_{i}. Hence

dim T_{i} < dim V_{i}, u = (0, …, 1^{i}, 0, …) ∈ Ũ_{v} and (λ_{i}, m_{i}) ≠ 0. So M_{m,λ}(d, v) = ∅ by

lemma 38. Inductive step: if l ≥ 2 then l−1 ⇒ l. Let σ = τs_{i} with ℓ(τ) = l−1 and v′ = s_{i}·v, λ′ = s_{i}λ, m′ = s_{i}m. By induction M_{m′,λ′}(d, v′) = ∅ and, since l ≥ 2, v′ ≥ 0. If (m_{i}, λ_{i}) ≠ 0 then we can apply Proposition 40 and we obtain M_{m,λ}(d, v) ≃ M_{m′,λ′}(d, v′) = ∅. If

(m_{i}, λ_{i}) = 0 then u = (0, …, 1^{i}, 0, …) ∉ Ũ_{v}, hence v_{i} = 0. Moreover λ′ = λ and

m′ = m so (m′_{i}, λ′_{i}) = 0 and u = (0, …, 1^{i}, 0, …) ∉ U_{v′}. Hence v′_{i} = 0 so v′ = v and

τv ≱ 0 against the minimality of σ.

Lemma 44. Let (I, H) be connected, (m, λ) ∈ G_{v} and suppose d ≥ 0 and σ·v ≥ 0 for all σ ∈ W. If there exists i ∈ I, σ ∈ W such that σ(m, λ) = (m′, λ′) and (m′_{i}, λ′_{i}) = 0 then d = v = 0.

Proof. Without loss of generality we can assume σ = 1.

First step: v_{i} = 0. This is clear since otherwise u = (0, …, 1^{i}, 0, …) ∈ U_{v}. Second step: d_{i} = 0 and v_{j} = 0 for all j such that a_{ij}. Let v′ = s_{i} · v and observe that s_{i}λ = λ and s_{i}m = m. Then as in first step we have 0 = v′_{i} = d_{i} + ∑_{j} a_{ij}v_{j} from which the claim follows.

Let now W′ =<{s_{j} : a_{ij} ≠ 0 and j ≠ i}>. If (d, v) ≠ 0 then there exists j ∈ I and σ ∈ W′ such that a_{ij} ≠ 0 and

n = d_{j} + ∑_{h∈I} a_{jh}ṽ_{h} > 0.

where ṽ = σ · v. Since (σλ)_{i} = λ_{i} = 0 = m_{i} = (σm)_{i} we can assume σ = 1. Let now v′ = s_{i}s_{j} · v, λ′ = s_{i}s_{j}λ and m′ = s_{i}s_{j}m, we have:

v′_{i} = a_{ij}n  λ′_{i} = −a_{ij}λ_{j}  m_{i} = −a_{ij}m_{j}
v′_{j} = n  λ′_{j} = (a_{ij}^{2} − 1)λ_{j}  m_{j} = (a_{ij}^{2} − 1)m_{j}.

Hence u = (0, …, a_{ij}^{j}, 0, …, a_{ij}^{2} − 1^{i}, 0, …) ∈ U_{v′} and <u^{∨}, λ′>=<u^{∨}, m′>= 0 against

(m, λ) ∈ G_{v}.

Remark 45. the analogous lemma in the case of G = {(m, λ, v) : (m, grl) ∉ H^{∞}} are more simple.

3.3. Relations. In this section we define an isomorphism of algebraic variety

Φ_{σ,m,λ}^{d,v}: M_{m,λ}(d, v) ⟶ M_{σm,σλ}(σ(d, v)).

in the case (m, λ) ∈ G_{v} or (m, λ) ∉ H^{∞}. In the case d ≱ 0 or in the case in which there exists σ ∈ W such that σv ≱ 0 or in the case d = v = 0 we have seen in the previous section that there is nothing to define or that the definition is trivial. In the remaining cases we observe that for all τ, i we have (τ(m)_{i}, τ(λ)_{i}) ≠ 0 by lemma 44. Hence we can define Φ_{σ,m,λ}^{d,v} by induction on ℓ(σ) by the formula

Φ_{σ,m,λ}^{d,v} = Φ_{s_{i},τm,τλ}^{τ(d,v)}∘Φ_{τ,m,λ}^{d,v}.  (21)

Of course we have to prove that this definition is well given by checking Coxeter relations:

s_{i}^{2} = Id, s_{i}s_{j} = s_{j}s_{i} if a_{ij} = 0 and s_{i}s_{j}s_{i} = s_{j}s_{i}s_{j} if a_{ij} = 1

which in our situation take the following form:

Φ_{s_{i},s_{i}λ,s_{i}m}^{s_{i}(d,v)}∘Φ_{s_{i},λ,m}^{d,v} = Id  (22a)
Φ_{s_{i},s_{j}λ,s_{j}m}^{s_{j}(d,v)}∘Φ_{s_{j},λ,m}^{d,v} = Φ_{s_{j},s_{i}λ,s_{i}m}^{s_{i}(d,v)}∘Φ_{s_{i},λ,m}^{d,v}  (22b)
Φ_{s_{i},s_{j}s_{i}λ,s_{j}s_{i}m}^{s_{j}s_{i}(d,v)}∘Φ_{s_{j},s_{i}λ,s_{i}m}^{s_{i}(d,v)}∘Φ_{s_{i},λ,m}^{d,v} = Φ_{s_{j},s_{i}s_{j}λ,s_{i}s_{j}m}^{s_{i}s_{j}(d,v)}∘Φ_{s_{i},s_{j}λ,s_{j}m}^{s_{j}(d,v)}∘Φ_{s_{j},λ,m}^{d,v}.  (22c)

The first of the two equations is clear by the very definition and remark 42. The second equation is trivial. We need to prove the third equation. We will need the following two simple lemmas of linear algebra which proofs are trivial.

Lemma 46. Let V, W, X, Y, Z be finite dimensional vector spaces and α, β, γ, δ, ε, φ linear maps between them as in the diagrams below. The diagram

0 ⟶ V \underrightarrow{(\begin{matrix}α \\ β \\ γ \end{matrix})} W ⊕ X ⊕ Y \underrightarrow{(\begin{matrix}δ & 0 & −1 \\ 0 & ε & φ \end{matrix})} Y ⊕ Z ⟶ 0

is exact if and only if the diagram

0 ⟶ V \underrightarrow{(\begin{matrix}α \\ β \end{matrix})} W ⊕ X \underrightarrow{(φδ ε)} Z ⟶ 0

is exact and γ = δα.

Lemma 47. Let U, V, W, X, Y, Z be finite dimensional vector spaces and α, β, γ, δ, ε, φ, ψ, ρ, σ linear maps between them as in the diagrams below such that ψ ⊕ ρ : W ⊕ X ⟶ Z is an epimorphism. Then the diagram

0 ⟶ U \underrightarrow{(\begin{matrix}α \\ β \\ γ \end{matrix})} V ⊕ W ⊕ X \underrightarrow{(\begin{matrix}δ & 0 & 1 \\ ε & φ & 0 \\ 0 & ψ & ρ \end{matrix})} X ⊕ Y ⊕ Z \underrightarrow{(ρ σ −1)} Z ⟶ 0

is exact if and only if γ = −δα , ψ = σϕ , ρδ + σε = 0 and the diagram

0 ⟶ U \underrightarrow{(\begin{matrix}α \\ β \end{matrix})} V ⊕ W \underrightarrow{(ε φ)} Y ⟶ 0

is exact.

We fix now and i, j such that a_{ij} = 1 and we verifies (22c). Let

λ′ = s_{i}λ  m′ = s_{i}m  v′ = s_{i}v
λ′′ = s_{j}λ′  m′′ = s_{j}m′  v′′ = s_{j}v′
λ′′′ = s_{i}λ′′  m′′′ = s_{i}m′′  v′′′ = s_{i}v′′
λ̃ = s_{j}λ  m̃ = s_{j}m  ṽ = s_{j}v
˜λ̃ = s_{i}λ̃  ˜  m̃ = s_{i}m̃  ˜  ṽ = s_{i}ṽ

First of all we observe that since relation (22a) holds we can assume that:

1. λ_{i} ≠ 0 or m_{i} > 0 and λ_{j} ≠ 0 or m_{j} > 0,

2. λ′_{j} ≠ 0 or m′_{j} > 0 and λ̃_{i} ≠ 0 or m̃_{i} > 0,

3. λ′′_{i} ≠ 0 or m′′_{i} > 0 and ˜ λ̃_{j} ≠ 0 or ˜ m̃_{j} > 0. Define

Z_{iji} = {(s′′′, s) ∈ Λ_{m′′′,λ′′′}(d, v′′′) × Λ_{m,λ}(d, v) : ∃s′′ ∈ S(d, v′′),
and s′ ∈ S(d, v′) such that (s′′′, s′′) ∈ Z_{i}^{m′′,λ′′}(d, v′′),
(s′′, s′) ∈ Z_{j}^{m′,λ′}(d, v′) and (s′, s) ∈ Z_{i}^{m,λ}(d, v)}
Z_{jij} = {(s′′′, s) ∈ Λ_{m′′′,λ′′′}(d, v′′′) × Λ_{m,λ}(d, v) : ∃˜s̃ ∈ S(d, ˜ ṽ),
and s̃ ∈ S(d, ṽ) such that (s′′′, ˜ s̃) ∈ Z_{j}^{˜m̃,˜λ̃}(d, ˜ ṽ),
(˜s̃, s̃) ∈ Z_{i}^{m̃,λ̃}(d, ṽ) and (s̃, s) ∈ Z_{j}^{m,λ}(d, v)}

Observe that (s′′′, s) ∈ Z_{iji} ⟺ p_{m′′′.λ′′′}^{d,v′′′}(s′′′) = Φ_{s_{i}}Φ_{s_{j}}Φ_{s_{i}}(p_{m,λ}^{d,v}(s)) and that (s′′′, s) ∈ Z_{jij} ⟺ p_{m′′′.λ′′′}^{d,v′′′}(s′′′) = Φ_{s_{j}}Φ_{s_{i}}Φ_{s_{j}}(p_{m,λ}^{d,v}(s)). So relation (22c) is equivalent to Z_{iji} = Z_{jij}.

Let now R_{i} = D_{i} ⊕ ⨁_{h: h_{1}=i,h_{0}≠j} V_{h_{0}}, R_{j} = D_{i} ⊕ ⨁_{h: h_{1}=i,h_{0}≠j} V_{h_{0}} and observe that T_{i} = R_{i} ⊕ V_{j} and T_{j} = R_{j} ⊕ V_{i}. Let k be the only element of H such that k_{0} = j and k_{1} = i. Let ε = ε(k). Define also A = A(s) = B_{k}(s), B = B(s) = Bk̄(s) and for l = i, j and {l′, l} = {i, j} set c_{l} = c_{l}(s) = π_{R_{l}}^{R_{l}⊕V_{l′}}a_{l}(s) and d_{l} = d_{l}(s) = b_{l}(s)|_{R_{l}}.

Let now (s, s′′′) ∈ Λ_{λ}(d, v)×Λ_{λ}′′′(d, v′′′) and set A^{∗} = A(s^{∗}), B^{∗} = B(s^{∗}), c_{l}^{∗} = c_{l}(s^{∗}) and d_{l}^{∗} = d_{l}(s^{∗}) for l ∈ {i, j} and ∗ ∈ { ,′′′ }.

If we apply lemmas 46 and 47 to our situation we obtain the following result: (s, s′′′) ∈ Z_{iji} of and only there exists vector spaces V′_{i}, V′_{j}, V′′_{i}, V′′_{j} and linear maps A′, B′, c′_{i}, d′_{i}, c′_{j}, d′_{j}, A′′, B′′, c′′_{i}, d′′_{i}, c′′_{j}, d′′_{j} such that:

1. dim V_{l}^{∗} = v_{l}^{∗} for l ∈ {i, j} and ∗ ∈ {′,′′ },

2. for each ∗ ∈ {′, ′′} and l ∈ {i, j} A^{∗} ∈ Hom(V_{i}^{∗}, V_{j}^{∗}), B^{∗} ∈ Hom(V_{j}^{∗}, V_{i}^{∗}), c_{l} ∈ Hom(V_{l}^{∗}, R_{l}^{∗}) and d_{l} ∈ Hom(R_{l}^{∗}, V_{l}^{∗}),

3. V′′′_{j} = V′′_{j}, c′′′_{j} = c′′_{j}, d′′′_{j} = d′′_{j} and

c′′′_{i}d′′′_{i} = c_{i}d_{i} − λ_{i} − λ_{j}  c′′′_{i}B′′′ = c′_{i}B′′
A′′′d′′′_{i} = A′′d′_{i}  εA′′′B′′′ = εA′′B′′ − λ_{j}

4. V′′_{i} = V′_{i}, c′′_{i} = c′_{i}, d′′_{i} = d′_{i} and

c′′_{j}d′′_{j} = c_{j}d_{j} − λ_{i} − λ_{j}  c′′_{j}A′′ = c_{j}A′
B′′d′′_{j} = B′d_{j}  εA′′B′′ = εA′B′ + λ_{i} + λ_{j}

5. V′_{j} = V_{j}, c′_{j} = c_{j}, d′_{j} = d_{j} and

c′_{i}d′_{i} = c_{i}d_{i} − λ_{i}  c′_{i}B′ = c_{i}B
A′d′_{i} = Ad_{i}  εA′B′ = εAB − λ_{i}

6. εc′_{i}B′′A′′′ + c′_{i}d′_{i}c′′′_{i} = 0 and εA′B′′ = d_{j}a′′_{j},

7. the following diagrams are exact

\begin{matrix}0 & ⟶ & V′′′_{i} & \underrightarrow{(\begin{matrix}c′′′_{i} \\ c′′_{j}A′′′ \end{matrix})} & R_{i} ⊕ R_{j} & \underrightarrow{(Ad_{i} d_{j})} & V_{j} & ⟶ & 0 \\ 0 & ⟶ & V′′_{j} & \underrightarrow{(\begin{matrix}c′′_{j} \\ c′_{i}B′′ \end{matrix})} & R_{j} ⊕ R_{i} & \underrightarrow{(Bd_{j} d_{i})} & V_{i} & ⟶ & 0 \\ 0 & ⟶ & V′_{i} & \underrightarrow{(\begin{matrix}c′_{i} \\ A′ \end{matrix})} & R_{i} ⊕ V_{j} & \underrightarrow{(d_{i} εB)} & V_{i} & ⟶ & 0 \end{matrix}

Remark 48. The first condition in point 6) is equivalent to εB′′A′′′ = d′_{i}c′′′_{i}. Indeed this condition is certainly sufficient. To prove the necessity observe that by the injectivity of a′_{i} = (c′_{i} A′)^{t} it is enough to prove εc′_{i}B′′A′′′ +c′_{i}d′_{i}c′′′_{i} = 0 and εA′B′′A′′′ + A′d′_{i}c′′′_{i} = 0. The first equation is the first condition in point 6) and the second one is a consequence of εA′B′′ = d_{j}c′′′_{i}, A′d′_{i} = Ad_{i} and the exactness of the first sequence.

Remark 49. The condition (s, s′′′) ∈ Z_{jij} can be expressed in a similar way. In the previous conditions we have only to change i with j and ε with −ε.

We will prove now Z_{iji} ⊂ Z_{jij}. To do it we suppose that A′, …, d′′_{j} are given as

above and we construct Ã, B̃, c̃_{i}, d̃_{i}, c̃_{j}, d̃_{j}, ˜ Ã, ˜ B̃, ˜ c̃_{i}, ˜ d̃_{i}, ˜ c̃_{j}, ˜ d̃_{j} such that they satisfy the conditions. for (s, s′′′) ∈ Z_{jij}. First step: construction of Ã, B̃, c̃_{i}, c̃_{j}, d̃_{i}, d̃_{j}. Choose s̃ such that (s̃, s) ∈ Z_{j}^{χ,λ} and define Ã = A(s̃), B̃ = B(s̃), c̃_{l} = c_{l}(s̃) and d̃_{l} = d_{l}(s̃) for l ∈ {i, j} .

Now I claim that there exists unique ˜ Ã: V′′′_{i} ⟶ Ṽ_{j} and ˜ B̃ : Ṽ_{j} ⟶ V′′′_{i} such that:

\begin{cases}c̃_{j} ˜Ã = c′′_{j}A′′′ \\ B̃ ˜Ã = −εd_{i}c′′′_{i} \end{cases} and \begin{cases}˜Ã ˜B̃ = ÃB̃ − ελ_{i} − ελ_{j} \\ c′′′_{i} ˜B̃ = c_{i}B̃ \end{cases}

Unicity of ˜ Ã: since the map ã_{j} = (c̃_{j} − εB̃)^{t} is injective the unicity is clear. Existence of ˜ Ã: to prove the existence of ˜ Ã is enough to prove:

Im (\begin{matrix}c′′_{j}A′′′ \\ −εd_{i}c′′′_{i} \end{matrix}) ⊂ Im (c̃_{j} ; B̃) = ker (d_{j} −εA) .

So the thesis follows from d_{j}c′′_{j}A′′′ + Ad_{i}c′′′_{i} = 0.

Let now ˜ ã_{i} = (c′′′_{i} ˜ Ã)^{t}. I claim that ˜ ã_{i} is injective and that Im ˜ ã_{i} = ker(d_{i} εB̃) = ker b̃_{i}. First of all observe that since m̃_{i} > 0 or λ_{i} ≠ 0, b̃_{i} is surjective. Observe also that

(\begin{matrix}c̃_{j} & 0 \\ 0 & Id_{V′′′_{i}} \end{matrix})∘( ˜ Ã ; c′′′_{i}) = (\begin{matrix}c′′_{j}A′′′ \\ c′′′_{i} \end{matrix}) .

So ˜ ã_{i} is injective as claimed. Now since dim R_{i} + dim Ṽ_{j} = dim V′′′_{i} + dim V_{i} to prove the last part of the claim it is enough to check that b̃_{i}˜ã_{i} = 0. Indeed

b̃_{i}˜ã_{i} = d_{i}c′′′_{i} + εB̃ ˜Ã = 0.

Unicity of ˜ B̃: this is a consequence of ˜ ã_{i} injective. Existence of ˜ B̃: As for the existence of ˜ Ã this is equivalent to

Im (\begin{matrix}c_{i}B̃ \\ ÃB̃ − ελ_{i} − ελ_{j} \end{matrix}) ⊂ Im (c′′′_{i} ; ˜Ã) = ker (d_{i} εB̃) .

So the thesis follows from εB̃ÃB̃ − λ_{i}B̃ − λ_{j}B̃ + d_{i}c_{i}B̃ = 0.

Finally we set

˜Ṽ_{i} = V′′′_{i}  ˜  c̃_{i} = c′′′_{i}  ˜  d̃_{i} = d′′′_{i}
˜Ṽ_{i} = Ṽ_{j}  ˜  c̃_{j} = c̃_{j}  ˜  d̃_{j} = d̃_{j}.

The verification of all the conditions is now straightforward.

The inclusion Z_{jij} ⊂ Z_{iji} can be proved similarly and equation (19) is clear by definition. So Proposition 32 is proved.

4. A representation of the Weyl group

In this section, following Nakajima [13], we show how to use the above action to construct an action of the Weyl group on the homology of quiver varieties. Maybe this action is related with the one constructed by Slodowy in the case of flag varieties ([16], ch.4).

First we recall some general about the action of the Weyl group. Let Z^{∨} = Q^{∨}⊗_{ℤ}ℂ and Z = Q ⊗_{ℤ} P. On Z, Z^{∨} there is a natural action of W.

Lemma 50. For all u ∈ Z^{∨} the set W u is discrete.

Lemma 51. Consider the action of W on ℙ(Z^{∨}). If p ∈ ℙ(Z^{∨}) then

\overline{W p} is countable.

If p ∈ ℙ(Z^{∨}) we define H_{p} = {x ∈ P ⊗_{ℤ} ℂ : <x, p>= 0}

Lemma 52. If p ∈ ℙ(Z^{∨}) then

\overline{W H_{p}} = ⋃_{q∈\overline{W p}} H_{q}.

We define H = \overline{W H_{v} ∪ H_{U}} and R = 𝔷 − H. By the previous lemmas H is the union of a countable number of real codimension 3 subspaces in ℨ and in particular R is simply connected. We need also the following definition

K = {u ∈ ℤ^{I} : − ∑ u_{i}α_{i} is dominant and supp u is connected }
P_{0} = {p ∈ P : p is dominant and <u^{∨}, p>≥ 2 for all u ∈ K}.

Now we choose d, v such that d̄= v̄.

Lemma 53. If d̄∈ P_{0} then μ̃ is surjective and is a locally trivial bundle over R.

Proof. By Proposition 10.5 and Corollary 10.6 in [14] there exists a closed orbit Gs in Λ_{0}(d, v) with trivial stabilizer. Then by Proposition 9 there exists t ∈ Gs such that μ̃(t) = 0 and by lemma 5 and 3 dμ̃_{t} is surjective. Now the surjectivity follows by homogeneity.

The local triviality over R follows also from lemma 5 and 3.

Now consider

R = {λ ∈ Z : (0, λ) ∉ R},
Λ(d, v) = {(λ, s) ∈ Z × S : s ∈ Λ_{λ}},
M(d, v) = Λ(d, v)//G_{v} and p : M(d, v) ⟶ Z the projection
𝔏(d, v) = {(ζ, s) ∈ ℨ × S : s ∈ 𝔏_{ζ}},
𝔐(d, v) = 𝔏(d, v)/U(V) and p̃: 𝔐(d, v) ⟶ ℨ the projection

We have the following commutative diagram

[figure: (λ, s); ∈; M(d, v); p; ı_{M}; Z; λ; ∋; (0, λ, s); ∈; 𝔐(d, v); p̃; ℨ; (0, λ); ∋]

By Proposition 10 the diagram is a pull back and by lemma 53 p and p̃ are locally trivial over R, R. We call M_{R} = p^{−1}(R) and 𝔐_{R} = p̃^{−1}(R).

Now consider the complex F = Rp_{∗}(ℤ_{M_{R}}) = ı_{M}^{−1}Rp̃_{∗}(ℤ_{𝔐_{R}}) which is cohomologically a locally constant complex. We observe now that Π_{1}(R) is trivial so Rp̃_{∗}(ℤ_{𝔐_{R}}) is isomorphic to cohomologically constant complex on R so it is F on R. In particular for any x, y ∈ R we have a canonically isomorphism

ψ_{x,y}^{i} : H^{i}(F_{x}) ⟶ H^{i}(F_{y}).

Now observe that by Proposition 32 there is an action on W on R, M_{R} and that p is equivariant with respect to this action. So we can define a W action on H^{i}(M_{0,λ}(d, v), ℤ) by

σ(c) = ψ_{σλ,λ}^{i}∘H^{i}(Φ_{σ,0,λ}^{d,v})(c)

for any σ ∈ W. To verify that this is an action we have only to verify that

ψ_{σλ^{2},λ^{1}}^{i} ∘H^{i}(Φ_{σ,0,λ^{2}}^{d,v})ψ_{λ^{1},λ^{2}}^{i}(c) = ψ_{σλ^{1},λ^{1}}^{i} ∘H^{i}(Φ_{σ,0,λ^{1}}^{d,v})(c).

Since R is connected and H^{i}(M, ℤ) is discrete this is clear. S o we have proved the following corollary.

Corollary 54. If d = v and (0, λ) ∈ R then there is an action of W on H^{i}(M_{0,λ}(d, v), ℤ).

Remark 55. If m_{+} = (1, …, 1) and λ = 0 it is easy to see that dμ_{s} is surjetive for all s ∈ Λ_{m_{+},0}(d, v). Then by lemma 53 there is a canonical isomorphism H_{∗}(M_{m_{+},0}(d, v)) ≃ H_{∗}(M_{0,λ}(d, v)) if (0, λ) ∈ R. So by Nakajima’s Theorem (Theorem 10.2 [14]) it is natural to make the following conjecture:

Conjecture 56. Let top = \frac{1}{2} dim H_{∗}(M_{0,λ}(d, v)) then

H^{top}(M_{(0,λ)}(d, v), ℂ) ≃ (L_{d})_{0}

where (L_{d})_{0} is the 0-weight space of the Kac-Moody algebra associated to the quiver of highest weight ∑_{i} d_{i}ω̄_{i}.

5. Reduction to the dominant case

As a consequence of Proposition 32 we see that if (m, λ) ∈ G_{v} then there exists σ ∈ W and v′ = σ · v such that d − v′ is dominant and M_{σm,σλ}(d, v′) ≃ M_{m,λ}(d, v). We generalize now this result to arbitrary λ.

On Q we consider the following order: v′ ≤ v if and only if v′_{i} ≤ v_{i}.

We consider now the following construction: let v′ ≤ v and fix an embedding V′_{i} ↪ V_{i} and a complement U_{i} of V′ in V_{i}, then we can define a map j̃: S(d, v′) ⟶ S(d, v) through:

j̃(B′, γ′, δ′) = ((\begin{matrix}B′ & 0 \\ 0 & 0 \end{matrix}) , (\begin{matrix}γ & ′ \\ 0 \end{matrix}) , (δ′ 0))  (23)

where the matrices of the new triple represents the maps described through the decomposition V_{i} = V′_{i} ⊕ U_{i}.

Suppose now that (m_{i}, λ_{i}) = 0 for all i such that v′_{i} ≠ v_{i}. Then it is easy to see that this map restrict to a map ȷ_{r} : Λ_{m,λ}(d, v′) ⟶ Λ_{m,λ}(d, v) and so induces a map ȷ_{v}^{v′} = ȷ: M_{0,λ}(d, v′) ⟶ M_{0,λ}(d, v).

Lemma 57. ȷ is a closed immersion

Proof. We prove that the map ȷ^{♯} : ℂ[Λ_{λ}(d, v)]^{G(v)} ⟶ ℂ[Λ_{λ}(d, v′)]^{G(v′)} is surjective. By proposition 18 this follows by the following two identities:

Tr (α (ȷ(s))) = Tr (α(s)) and β (ȷ(s)) = β(s)

for each B-path α and for each admissible path β.

Lemma 58. If 2v_{i} > d_{i} + ∑_{j∈I} a_{ij}v_{j} and v′ = v − α_{i} then ȷ is an isomorphism of algebraic varieties

Proof. It’s enough to prove that ȷ is surjective. Let s = (B, γ, δ) ∈ Λ_{0}(d, v) and consider the sequence (see (4) for the notation) :

T_{i} ⟶^{b_{i}} V_{i} ⟶^{a_{i}} T_{i}.

Since b_{i}a_{i} = 0 and 2 dim V_{i} > dim T_{i} we have that b_{i} is not surjective or that a_{i} is not injective.

Suppose that b_{i} is not surjective, then up to the action of G_{v} we can assume that Im b_{i} ⊂ v′_{i}. Then, for t ∈ ℂ^{∗} consider g_{t} = (g_{j,t}) ∈ G_{v} with

g_{i} = (\begin{matrix}Id_{v′_{i}} & 0 \\ 0 & t^{−1} \end{matrix}) and g_{j} = Id_{V_{j}} for j ≠ i.

Then 1. g_{i,t}B_{h} = B_{h} if h_{1} = i and g_{i}γ_{i} = γ_{i}, since Im B_{h}, Im γ_{i} ⊂ Im b_{i} ⊂ v′_{i},

2. ∃ lim_{t→0} B_{h}g_{i,t}^{−1} = B_{h} if h_{0} = i and δ_{i}g_{i}^{−1} = δ_{i} So ∃ lim_{t→0} g_{t}s = s′ and it is clear that s′ ∈ j̃(Λ_{0}(d, v′)) and that p_{0}(s) = p_{0}(s′) ∈ Im ȷ.

If b_{j} is surjective and a_{i} is not injective the argument is similar.

Proposition 59. For all λ and for all d ≥ 0, v ≥ 0 there exists v′ and σ ∈ W such that d̄− v̄′ is dominant and

M_{0,σλ}(d, v′) ≃ M_{0,λ}(d, v).

Proof. We prove this proposition by induction on the order ≤ on Q. First step: v = 0. If v = 0 we can take v′ = v and σ = 1. Inductive step. If d−v is not dominant then there exists i such that 2v_{i} > d_{i}+∑ a_{ij}v_{j}.

If λ_{i} ≠ 0 we observe that s_{i}v = v′ < v (that is v′ ≤ v and v′ ≠ v) and that M_{s_{i}m,s_{i}λ}(d, v′) ≃ M_{m,λ}(d, v) and so we can apply the inductive hypothesis.

If λ_{i} = 0 we apply the previous lemma and the inductive hypothesis.

6. On normality and connectedness of quiver variety in the finite type case

In this section we restrict our attention to the case of quiver varieties of finite type and to the case m = (1, …, 1) and λ = 0 and we fix d, v. By remark 2 we can assume without loss of generality that v_{i} > 0 for all i. We would like to prove the following conjecture:

Conjecture 60. M_{m,0}(d, v) is connected and M_{0,0}(d, v) is normal.

Remark 61. If m̃ = (m_{1}, …, m_{n}) ∈ ℕ_{+}^{I} it is easy to see that Λ_{m̃,0} = Λ_{m,0} ao in particular M_{m,0}(d, v) is smooth. Instead M_{0,0} is a cone so it is clearly connected.

By proposition 59it is enough to prove the theorem in the case d − v dominant. Unfortunately I’m not able to prove the conjecture only in the case d − v regular: <d − v, α_{i}>> 0 for all i. To prove the conjecture in this case we will use the following stratification introduced by Lusztig in [9].

Definition 62. For any s ∈ S and i ∈ I let

V_{i}^{+} = V_{i}^{+}(s) = ∑_{α a B−path : α_{1}=i} Im(α(s)γ_{α_{0}})

If v′ = (v′_{1}, …, v′_{n}) ∈ ℕ^{n} we define

Λ^{v′} = {s ∈ Λ_{0}(d, v) : dim V_{i}^{+}(s) = v′_{i}}.

Observe that Λ^{v} = Λ_{m_{+},0}(d, v). To prove our result we will use the following lemma of Lusztig.

Lemma 63 (Lusztig: [9] Proposition 4.5 and Proposition 5.3). If 0 ≤ v′_{i} ≤ v_{i} for each i then

dim Λ^{v′}(d, v) = dim S − ∑_{i∈I} dim gl(V_{i})− <(v − v′)^{∨}, d − v> −\frac{1}{2} <(v − v′)^{∨}, v − v′>

Our result follows trivially from the following lemma.

Lemma 64. 1) If d − v is dominant then Λ_{0,0}(d, v) is a complete intersection.

2) If d − v is regular then Λ_{0,0} is normal and irreducible and Λ_{m_{+},0}(d, v) is connected.

Proof. Observe that Λ_{0,0}(d, v) = μ^{−1}(0) so each irreducible component of Λ_{0,0}(d, v) must have dimension at least dim S − ∑_{i} dim gl(V_{i}) = δ_{V}.

Suppose now that d − v is dominant. By Nakajima’s theorem ([14] Theorem 10.2) M_{m_{+},0} is not empty. Observe also that by Proposition 14 Λ_{m_{+},0}(d, v) is a smooth subset of Λ_{0,0}(d, v) of dimension δ_{V}.

It is well known that Λ_{m_{+},0}(d, v) = Λ^{v}. Hence

Λ_{0,0}(d, v) − Λ_{m_{+},0}(d, v) = ⋃_{v′≤v and v′≠v} Λ^{v′}.

By the lemma above we have that if v′ ≤ v and v′ ≠ v dim Λ^{v′} < δ_{V}. So Λ_{m_{+},0}(d, v) must be dense in Λ_{0,0}(d, v) and Λ_{0,0}(d, v) is a complete intersection. Moreover if d−v is regular we have that dim Λ^{v′} < δ_{V} − 1 so the singular locus has codimension at least two and normality and irreducibility follows. Finally by our discussion it is clear that if Λ_{m_{+},0}(d, v) is disconnected then Λ_{0,0}(d, v) is not irreducible.

Remark 65. In the lemma we can substitute Λ_{m_{+},0}(d, v) with any other subset Reg of regular points in Λ(d, v). In this way is indeed possible to improve a little bit the theorem but Crawley-Boevey explained me that this strategy cannot work in general because there are cases where d − v is dominant and Λ_{0,0}(d, v) is not normal. It should be also pointed out that Crawley-Boevey proved the connectedness in complete generality ([1]). He said me that is also able to prove normality for a much bigger class of quiver varieties.

References

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[9] , Quiver varieties and Weyl group actions, preprint, 1999.

[10] A. Maffei, Quiver varieties, Ph.D. thesis, Università di Roma ”La Sapienza”, 1999.

[11] L. Migliorini, Stability of homogeneous vector bundles, Bollettino dell’Unione Matematica Italiana B(7) 10 (1996), 963–990.

[12] D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, third ed., Ergebenisse der Mathematik und ihrer Grenzgebiete, vol. 34, Springer, 1994.

[13] H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. Jour. 76 (1994), 365–416.

[14] , Quiver varieties and Kac-Moody algebras, Duke Math. Jour. 91 (1998), 515 – 560.

[15] P. Newstead, Introduction to moduli problems and orbit spaces, Tata Lectures, vol. 51, Springer, 1978.

[16] P. Slodowy, Four lecture on simple groups and singularities, Communications of the mathematical institute, Rijksuniversiteit Utrecht, 1980.

Mathematical Science Research Institute, Berkeley

  E-mail address: amaffei@mat.uniroma1.it