Stability in linear problems

Mihai Sorin Stupariu

Abstract

Using analytic stability we construct quotients associated to linear actions and we give a necessary and sufficient condition for their compactness. These results are applied for quiver factorisation problems, deducing that the compactness of the corresponding quotients is related to a combinatorial property. As an example, we discuss the case of linear systems, constructing compactifications of the spaces of accessible and of minimal linear systems.

0 Introduction

In the present note we will be concerned with linear problems, i.e. we will be interested in the study of the action of a connected reductive complex Lie group G on a finite dimensional complex vector space W through a representation ρ and in the construction of associated spaces of orbits.

In the first of the two sections of the paper we describe general results concerning such linear problems. After a preliminary part we recall the corresponding definitions for analytic, symplectic and GIT-stability. Our main tool in the study of linear problems will be analytic stability, as introduced and studied in [6] and [10]. In the general set-up the analytic stability depends on a symplectisation of the G-action and we point out that, for linear problems, it depends only on a parameter τ varying in a finite dimensional real vector space. We also give the explicit description of the (semi)stable loci in a concrete example, which arises in control theory. The stability concepts make possible the construction of orbit spaces since the restriction of the action to the semistable locus yields a geometric quotient. Moreover, such a quotient admits an alternative description using tools from symplectic geometry. An important result of this section is stated in Proposition 1.9 and relates the compactness of these quotients to the study of the properness of a certain map, related to the symplectic approach (namely the canonical moment map μ_{can}).

In the second section we focus our attention to a special class of linear problems. We explain how, starting with a pair (Q, S) consisting of a quiver Q (i.e. a diagram containing points and arrows) and of a subset S of the set Q_{0} of points of Q, one obtains in a natural way a linear problem, which is called quiver factorisation problem (QFP for shorthand) associated to the combinatorial data (Q, S). The corresponding quotients are called QFP-quotients and their geometry depends on (Q, S), on a dimension vector α ∈ ℕ^{Q_{0}} and on a parameter τ ∈ ℝ^{S}. However, we show in Theorem 2.3 that the compactness of a QFP-quotient is related only to a combinatorial property of the pair (Q, S). In the last part of the paper we return to the example concerning linear systems, which can be represented as a quiver factorisation problem. The corresponding QFP-quotients are not compact and in Theorem 2.7 we prove that they admit natural compactifications which are themselves QFP-quotients. It remains an open question whether it is possible to construct analogous natural compactifications for arbitrary non-compact QFP-quotients.

1 General results

Let G be a connected reductive complex Lie group with Lie algebra 𝔤 (throughout this paper the Lie algebra of a Lie group will be denoted by the corresponding ‘german’ character). We denote by Z the center of G and by Z_{ℝ} its unique maximal compact subgroup. We also set:

T_{G} := {τ ∈ 𝔤^{∨} | τ_{|[𝔤,𝔤]} = 0, τ(𝔷_{ℝ}) ⊂ ℝ} ,

which is a real vector space, naturally isomorphic to (𝔷_{ℝ})^{∨} (the dual of the Lie algebra of Z_{ℝ}). Further let ρ : G → GL(W) be a representation of G on a finite dimensional complex vector space W; its kernel will be denoted by H.

1.1 Preliminaries

1.1.1 Elements of Hermitian type and weight spaces

An element s of 𝔤 is called of Hermitian type if there exists a compact subgroup K of G such that s ∈ i𝔨; the set of elements of Hermitian types will be denoted by H(G) . Other equivalent characterizations of the elements of H(G) can be found in [10, Definition 2.1]. In particular, if s ∈ H(G), the endomorphism ρ_{∗}(s) has only real eigenvalues and is diagonalizable. For an eigenvalue λ, we denote by W (λ) the corresponding subspace of eigenvectors and we set

W^{≤0}(s) := ⨁_{λ≤0} W (λ), W^{<0}(s) := ⨁_{λ<0} W (λ).

Every vector w ∈ W can be written uniquely as w = w_{−} + w_{0} + w_{+}, where w_{−} ∈ ⨁_{λ<0} W (λ), w_{0} ∈ W (0) and w_{+} ∈ ⨁_{λ>0} W (λ).

Remark 1.1 If W = ⨁_{i=1}^{p} W_{i} is the direct sum of the representations W_{i} of the group G, then for any s ∈ H(G) it holds

W^{≤0}(s) = ⨁_{i=1}^{p} W_{i}^{≤0}(s), W^{<0}(s) = ⨁_{i=1}^{p} W_{i}^{<0}(s).

1.1.2 The case of the general linear group

In this subsection we pay attention to the special case of the group GL(V) of general linear transformations of a finite dimensional complex vector space V and to some GL(V)-actions. Let s ∈ H(GL(V)) be an element of Hermitian type. We denote by λ_{1} < … < λ_{q} the (real) eigenvalues of s and by V (λ_{i}) the corresponding eigenspaces. For any i = 1, …, q we define

V_{i} := ⨁_{λ_{j}≤λ_{i}} V (λ_{j})

and we put V_{0} := {0}, obtaining in this way a filtration on V, denoted by F_{s},

{0} = V_{0} ⊂ V_{1} ⊂ … ⊂ V_{q} = V.

We also define

F_{s}^{≤0} := ⨁_{λ_{j}≤0} V (λ_{j}), F_{s}^{<0} := ⨁_{λ_{j}<0} V (λ_{j}).

In particular, there exist j_{1}, j_{2} such that F_{s}^{≤0} = V_{j_{1}}, F_{s}^{<0} = V_{j_{2}}.

Example 1.2 Let V, V_{1} and V_{2} be finite dimensional complex vector spaces and let s ∈ H(GL(V)) be a fixed element of Hermitian type.

(i) Take the GL(V)-action by conjugation on W := End_{ℂ}(V). Then φ ∈ W^{≤0}(s) if and only if the filtration F_{s} is φ-invariant.

(ii) Take the GL(V)-action on W_{1} := Hom_{ℂ}(V_{1}, V) given by (g, φ_{1}) ↦ g∘φ_{1}. Then φ_{1} ∈ W_{1}^{≤0}(s) if and only if im(φ_{1}) ⊂ F_{s}^{≤0}.

(iii) Take the GL(V)-action on W_{2} := Hom_{ℂ}(V, V_{2}) given by (g, φ_{2}) ↦ φ_{2} ∘ g^{−1}. Then φ_{2} ∈ W_{2}^{≤0}(s) if and only if F_{s}^{<0} ⊂ ker(φ_{2}).

Proof. We prove the assertion (i), the other ones follow using similar arguments. In the case of the action ρ given by conjugation, the eigenvalues of ρ_{∗}(s) are {λ_{a} − λ_{b}}_{a,b} and, according to our notations, let (W (λ_{a} − λ_{b}))_{a,b} be the corresponding eigenspaces. As noticed above, for an endomorphism φ ∈ W we have the decomposition

φ = φ_{−} + φ_{0} + φ_{+},

where φ_{−} ∈ ⨁_{ν<0} W (ν), φ_{0} ∈ W (0) and φ_{+} ∈ ⨁_{ν>0} W (ν).

We first remark that, if v ∈ V (λ) is a vector of the eigenspace V (λ) corresponding to the eigenvalue λ and ψ ∈ W (λ_{a} − λ_{b}), then ψ(v) ∈ V (λ_{a} − λ_{b} + λ), since one has

s(ψ(v)) = [s, ψ](v) + ψ(s(v)) = (λ_{a} − λ_{b} + λ)ψ(v).

We deduce that for v ∈ V (λ) it holds

φ_{−}(v) ∈ ⨁_{μ<λ} V (μ), φ_{0}(v) ∈ V (λ), φ_{+}(v) ∈ ⨁_{μ>λ} V (μ).

Suppose first that φ ∈ W^{≤0}(s), that is φ_{+} = 0. In particular, for a vector v ∈ V (λ) one has φ(v) ∈ ⨁_{μ≤λ} V (μ). We conclude that for any i = 1, …, q the following inclusion hold:

φ(V_{i}) = φ( ⨁_{λ≤λ_{i}} V (λ)) ⊂ ⨁_{λ≤λ_{i}} V (λ) = V_{i},

that is the filtration F_{s} is φ-invariant.

Conversely, we suppose that F_{s} is φ-invariant and we claim that φ_{+} = 0. In fact, it is enough to show that for any eigenvalue λ it holds φ_{+}|_{V (λ)} = 0. We first notice that, for i suitable chosen one has V_{i} = ⨁_{μ≤λ} V (μ) ⊃ V (λ). Let now v ∈ V (λ) be a vector of the space V (λ). On one hand, φ(v) ∈ V_{i} (by assumption) and φ_{−}(v) ∈ V_{i} (by the arguments above), hence φ_{+}(v) ∈ V_{i}. On the other hand, φ_{+}(v) ∈ ⨁_{μ>λ} V (μ) and we conclude that φ_{+}(v) = 0.

1.1.3 Symplectisations

We return now to the general set-up of a linear action of a connected complex reductive group G on a finite dimensional complex vector space W. Let K be a maximal compact subgroup of G. Then its Lie algebra 𝔨 can be decomposed as 𝔨 = 𝔷_{ℝ} ⊕ [𝔨, 𝔨] (recall that 𝔷_{ℝ} is the Lie algebra of the torus Z_{ℝ}, which is the unique maximal compact subgroup of the center Z of G). In particular, since K is connected, the following relations hold

{τ ∈ 𝔨^{∨}|⟨τ, ad_{g}(ξ)⟩ = ⟨τ, ξ⟩, ∀g ∈ K, ξ ∈ 𝔨} = {τ ∈ 𝔨^{∨}|τ_{|[𝔨,𝔨]} = 0} ≃ T_{G}. (1)

In the sequel we will tacitly use this identification.

Let further h_{W} be a Hermitian inner product on W such that ρ(K) ⊂ U(W, h_{W}). In this way K acts in a symplectic fashion with respect to the symplectic structure induced by h_{W}. Moreover, there exists a standard moment map for the K-action on W, namely

μ_{can} : W → 𝔨^{∨}, μ_{can}(w) := −ρ_{∗}^{∨} (\frac{i}{2}(w ⊗ w̄)) .

This map depends on the representation, but we skipped the index ρ in order to simplify the notation. Here we denoted by (w⊗w̄) the Hermitian endomorphism given by w ⊗ w̄ = h_{W} (·, w)w, whereas ρ_{∗}^{∨} is the dual of ρ_{∗} : 𝔨 → u(W). We used the identification between u(W) and u(W)^{∨} given by the inner product (A, B) ↦ −Tr(AB). Moreover, from the equality (1) it follows that for any τ ∈ T_{G} the map given by

μ_{τ} : W → 𝔨^{∨}, μ_{τ} (w) := μ_{can}(w) − τ

is also a moment map for the K-action on W.

The triple (K, h_{W} , μ_{τ}) yields a symplectisation σ_{τ} of the G-action on W given by ρ. According to the definition given in [6], σ_{τ} is the equivalence class of (K, h_{W} , μ_{τ}) with respect to the following equivalence relation: let (K_{i}, h_{i}, μ_{i})

(i = 1, 2) be two triples consisting of a maximal compact subgroup K_{i} of G, a K_{i}-invariant Hermitian metric h_{i} on W and a moment map μ_{i} : W → 𝔨_{i}^{∨} for the K_{i}-operation on W with respect to the symplectic structure induced by h_{i}. They are considered to be equivalent if there exists g ∈ G such that

K_{2} = Ad_{g}(K_{1}), h_{2} = (g^{−1})^{∗}h_{1}, μ_{2} = ad_{g^{−1}}^{t} ∘ μ_{1} ∘ g^{−1}.

1.2 Analytic, symplectic and GIT-stability

In this section we briefly recall the corresponding definitions and we describe the relationship between analytic, symplectic and GIT-stability. For further details we refer the reader to the papers [5], [6] and [10].

I. Our main tool in the study of linear problems will be the analytic stability, as introduced and studied in [6] and [10]. For a symplectisation σ of the G-action on W, one constructs a well-defined map

λ_{σ} : H(G) × W → ℝ ∪ {∞}, (s, w) ↦ λ_{σ}^{s}(w).

A point w ∈ W is called analytically σ-semistable if and only if λ_{σ}^{s}(w) ≥ 0, for any s ∈ H(G). It is analytically σ-stable if and only if it is σ-semistable and λ_{σ}^{s}(w) > 0 for any s ∈ H(G) \ 𝔥.

In the sequel we will take a closer look at analytic stability in the case of linear problems. We begin by recalling the construction of the map λ. Take an element of Hermitian type s ∈ H(G) and choose a representant (K, h_{W} , μ_{τ}) of σ such that s ∈ i𝔨. Then, for any w ∈ W and t ∈ ℝ one defines

λ_{σ}^{s,t} : W → ℝ, λ_{σ}^{s,t}(w) := ⟨μ_{τ} (e^{ts} · w), −is⟩.

The map λ_{σ}, which is independent on the choice of the representant of the symplectisation, is given by

λ_{σ}^{s} : W → ℝ ∪ {∞}, λ_{σ}^{s}(w) := lim_{t→∞} λ_{σ}^{s,t}(w).

The following Lemma gives an explicit description of this map in the case of linear actions:

Lemma 1.3 Let σ be a symplectisation of the G-action on W, s ∈ H(G) an element of Hermitian type and (K, h_{W} , μ_{τ}) a representant of σ such that s ∈ i𝔨. Then for any w ∈ W it holds

λ_{σ}^{s}(w) = \{ \begin{matrix}⟨τ, is⟩, & if w ∈ W^{≤0}(s) \\ ∞, & otherwise. \end{matrix}

Proof. A straightforward computation shows that for any t ∈ ℝ one has λ_{σ}^{s,t}(w) = λ^{s,t}(w) + ⟨τ, is⟩, where

λ^{s,t}(w) := \frac{1}{2}h_{W} (e^{ts} · w, ρ_{∗}(s)(e^{ts} · w)).

We choose a h_{W} -orthonormal basis b_{1}, …, b_{m} of W consisting of eigenvectors of ρ_{∗}(s), such that any b_{i} corresponds to an eigenvalue λ_{i}. Writing w = ∑_{i} w_{i}b_{i}, we have

e^{ts} · w = ∑_{i=1}^{m}(e^{tλ_{i}}w_{i})b_{i}, ρ_{∗}(s)(e^{ts} · w) = ∑_{i=1}^{m}(λ_{i}e^{tλ_{i}}w_{i})b_{i},

and hence

λ^{s,t}(w) = ∑_{i=1}^{m} λ_{i}|e^{tλ_{i}}|^{2}|w_{i}|^{2}b_{i}.

Taking the limit we obtain

lim_{t→∞} λ^{s,t}(w) = \{ \begin{matrix}0, & if w ∈ W^{≤0}(s) \\ ∞, & otherwise \end{matrix}

and our assertion follows.

In particular, Lemma 1.3 shows that the map λ_{σ}^{s} and hence the stability concept do not depend on K and h_{W}, but only on τ. More precisely, if we consider symplectisations σ = [(K, h_{W} , μ_{τ})] and σ̃ = [(K̃, h̃_{W} , μ_{τ})], corresponding to the pairs (K, h_{W}), respectively (K̃, h̃_{W}) and to the same τ, then a point w is σ-(semi)stable if and only if it is σ̃-(semi)stable. In this case we will say that w is τ-(semi)stable. In fact, we obtained the following explicit description of the τ-(semi)stable locus:

Proposition 1.4 a) The following assertions are equivalent:

(i) w is τ-semistable,

(ii) for any s ∈ H(G) such that w ∈ W^{≤0}(s), it holds ⟨τ, is⟩ ≥ 0.

b) The following assertions are equivalent:

(i) w is τ-stable,

(ii) w is τ-semistable and for any s ∈ H(G) \ 𝔥 such that w ∈ W^{≤0}(s) it holds ⟨τ, is⟩ > 0.

Remark 1.5 Let a > 0 be a positive constant. Then a point w is τ-(semi)stable if and only if it is aτ-(semi)stable.

II. The second concept of stability is that one of symplectic stability.

Let σ = [(K, h_{W} , μ_{τ})] be a symplectisation of the G-action on W. According to [6] a point w ∈ W is called symplectically σ-semistable if \overline{G · w} ∩ μ_{τ}^{−1}(0) ≠ ∅. It is called symplectically σ-stable if G · w ∩ μ_{τ}^{−1}(0) ≠ ∅ and dim G_{w} = dim H, where G_{w} denotes the stabilizer of w.

III. Last but no least, we briefly recall some definitions related to GITstability. As noticed in [5], a character χ : G → ℂ^{∗} of G yields a linearisation of the trivial complex line bundle L over W. On the total space of L^{−1} = W × ℂ, the group G acts by g · (w, λ) := (ρ(g)w, χ^{−1}(g)λ). In this context, Mumford’s definitions for (semi)stability with respect to this linearisation can be phrased as follows [5]. A point w ∈ W is called χ-semistable if there exist n ≥ 1 and a χ^{n}-equivariant polynomial f ∈ ℂ[W]^{G,χ^{n}} such that f(w) ≠ 0. If moreover, the G-operation on {w|f(w) ≠ 0} is closed and the dimension of the stabilizer G_{w} of w is equal to the dimension of H, then w will be called χ-stable.

These three stability concepts are, essentially, equivalent:

Proposition 1.6 Let σ = [(K, h_{W} , μ_{τ})] be a symplectisation of the G-action on W.

i) [6], [10] A point w in W is analytically τ-(semi)stable if and only if it is symplectically σ-(semi)stable.

ii) [5] Suppose that τ is provided by the derivative of a character χ of G. Then a point is symplectically σ-(semi)stable if and only if it is GIT χ-(semi)stable.

Notation For a fixed τ we will denote by W^{s,τ} and W^{ss,τ} the τ-stable, respectively τ-semistable locus.

1.2.1 An example from control theory: linear systems

In this subsection we will give an explicite description of the (semi)stable locus in a particular case. More precisely, we consider three complex vector spaces V, V_{1}, V_{2} of dimensions r > 0, r_{1}, respectively r_{2} such that (r_{1}, r_{2}) ≠ (0, 0) and we set

W := End_{ℂ}(V) ⊕ Hom_{ℂ}(V_{1}, V) ⊕ Hom_{ℂ}(V, V_{2}).

The group G := GL(V) acts in a natural fashion on W by

g · (φ, φ_{1}, φ_{2}) := (g ∘ φ ∘ g^{−1}, g ∘ φ_{1}, φ_{2} ∘ g^{−1}).

A triple (φ, φ_{1}, φ_{2}) will be called a linear system. We notice that, after fixing basis in the spaces V, V_{1}, V_{2}, one can represent a linear system by a triple of matrices. Such triples of matrices arise in control theory and one can find more results concerning these systems, for instance, in the monography [9].

We notice that the kernel of the representation described above is trivial, by the assumption that (r_{1}, r_{2}) ≠ (0, 0). Since Z(GL(V)) ≃ ℂ^{∗}, the real vector space T_{GL(V)} is one-dimensional and, hence, according to Remark 1.5, there are three different stability concepts, corresponding to the following elements of T_{GL(V)}:

τ_{0} := 0,  τ_{1} := iTr,  τ_{−1} := −iTr.

In order to describe explicitely the corresponding (semi)stable loci, we first fix some notations: for a triple (φ, φ_{1}, φ_{2}) ∈ W we put

V_{φ,φ_{1}} := im(φ_{1}) + im(φ ∘ φ_{1}) + … + im(φ^{r−1} ∘ φ_{1}) ⊂ V,
V^{φ,φ_{2}} := ker(φ_{2}) ∩ ker(φ_{2} ∘ φ) ∩ … ∩ ker(φ_{2} ∘ φ^{r−1}) ⊂ V.

Proposition 1.7 It holds:

(i) W^{s,τ_{0}} = {(φ, φ_{1}, φ_{2}) | V_{φ,φ_{1}} = V, V^{φ,φ_{2}} = 0}, W^{ss,τ_{0}} = W;
(ii) W^{s,τ_{1}} = W^{ss,τ_{1}} = {(φ, φ_{1}, φ_{2}) | V_{φ,φ_{1}} = V };
(iii) W^{s,τ_{−1}} = W^{ss,τ_{−1}} = {(φ, φ_{1}, φ_{2}) | V^{φ,φ_{2}} = 0}.

Proof. Let s ∈ H(G) be of Hermitian type. According to Remark 1.1 and to Example 1.2, a triple (φ, φ_{1}, φ_{2}) is an element of W^{≤0}(s) if and only if the filtration F_{s} is φ-invariant, im(φ_{1}) ⊂ F_{s}^{≤0} and ker(φ_{2}) ⊃ F_{s}^{<0}.

(i) The fact that W^{ss,τ_{0}} = W follows at once from Proposition 1.4 a). Let now (φ, φ_{1}, φ_{2}) be τ_{0}-stable and suppose that V_{φ,φ_{1}} ≠ V. We consider the φ-invariant filtration

{0} ⊂ V_{φ,φ_{1}} ⊆_{/} V.

One can find s ∈ H(G) \ {0} such that for the associated filtration F_{s} it holds F_{s}^{≤0} = V_{φ,φ_{1}} and F_{s}^{<0} = {0}. In particular, this means that the triple (φ, φ_{1}, φ_{2}) is an element of W^{≤0}(s). Since ⟨τ_{0}, is⟩ = 0, it follows that (φ, φ_{1}, φ_{2}) cannot be τ_{0}-stable. If V^{φ,φ_{2}} ≠ 0, one obtains an analogous contradiction by considering the filtration

{0} ⊆_{/} V^{φ,φ_{2}} ⊂ V.

Conversely, let’s suppose that it holds V_{φ,φ_{1}} = V and V^{φ,φ_{2}} = 0. We claim that there exists no element s ∈ H(G) \ {0} such that (φ, φ_{1}, φ_{2}) ∈ W^{≤0}(s). Indeed, let s ≠ 0 be an element of Hermitian type such that F_{s} is φ-invariant, im(φ_{1}) ⊂ F_{s}^{≤0} and ker(φ_{2}) ⊃ F_{s}^{<0}. Since s ≠ 0, then either F_{s}^{≤0} ≠ V or F_{s}^{<0} ≠ 0. In the first case, it follows that V_{φ,φ_{1}} ≠ V, whereas in the second situation it holds V^{φ,φ_{2}} ≠ 0 and this yields a contradiction.

(ii) For simplicity, we denote by M the set of triples (φ, φ_{1}, φ_{2}) such that V_{φ,φ_{1}} = V.

Let (φ, φ_{1}, φ_{2}) be an element of M. If s is an element of Hermitian type such that (φ, φ_{1}, φ_{2}) ∈ W^{≤0}(s), it follows that V_{φ,φ_{1}} = F_{s}^{≤0} = V. In particular all the eigenvalues of s are non-positive and hence ⟨τ_{1}, is⟩ = −Tr(s) ≥ 0. Moreover, if s ≠ 0, then ⟨τ_{1}, is⟩ > 0. This proves that M ⊂ W^{s,τ_{1}}.

Let now (φ, φ_{1}, φ_{2}) be a triple such that V_{φ,φ_{1}} ≠ V. Consider s with eigenvalues 0 and 1 and such that the associated filtration is

{0} ⊂ V_{φ,φ_{1}} ⊆_{/} V.

Then one has (φ, φ_{1}, φ_{2}) ∈ W^{≤0}(s) and it holds

⟨τ_{1}, is⟩ = −Tr(s) = − dim(V/V_{φ,φ_{1}}) < 0,

which shows that the given triple is not τ_{1}-semistable. Hence we proved that the semistable locus is included in M. Since the stable locus is included in the semistable one, the required equalities follow.

(iii) This assertion is the ‘dual’ of (ii) and can be proved using similar arguments.

Remark A triple (φ, φ_{1}, φ_{2}) such that V_{φ,φ_{1}} = V is called in the control theory accessible linear system, whereas a triple for which V_{φ,φ_{1}} = V and V^{φ,φ_{2}} = 0 is called minimal system. Hence, the first statement of Proposition 1.7 shows that the set of minimal systems can be described as the set of analytically stable points with respect to a suitable stability concept. The second statement of Proposition 1.7 together with Proposition 1.6 yield an alternative proof to a known result of Byrnes and Hurt [1, Theorem 4.1] which states that the set of accessible linear systems coincides with the GIT-(semi)stable locus corresponding to a suitable linearization.

1.3 Geometry of quotients

We return to the general set-up of an action of a connected reductive complex Lie group G on a complex vector space W. The following result follows from [10, Theorem 1.4] (compare also [2]) and from Proposition 1.6:

Proposition 1.8 Let τ ∈ T_{G} be fixed. The sets W^{ss,τ} , W^{s,τ} are open in W and there exists a good quotient

q_{τ} : W^{ss,τ} → W //(G, τ)

with the property that two G-orbits have the same image in W //(G, τ) if and only if their closures in W^{ss,τ} are not disjoint. Moreover, the natural map

μ_{τ}^{−1}(0)/K ⟶ W //(G, τ)

is a homeomorphism.

This result shows not only that the restriction to the semistable locus yields a good quotient, but also that this quotient admits, in the category of topological spaces, an alternative description obtained by using symplectic tools (i.e the moment map). Some topological properties of the quotient W //(G, τ) can be easier understood if one combines the two approaches. For instance, we will take in the sequel a closer look at the compactness of this quotient and we will show that it can be related to the properness of the moment map μ_{can} obtained by choosing a maximal compact subgroup K of G and a Hermitian inner product on W such that K acts on W by h_{W} -unitary transformations. Since the topology of W //(G, τ) depends only on the representation and on τ, the result is independent on the choice of the pair (K, h_{W}).

Proposition 1.9 (i) Suppose that μ_{can} is proper. Then for any τ the quotient W //(G, τ) is compact.

(ii) Suppose that μ_{can} is not proper. Then for any τ such that W //(G, τ) ≠ ∅, this quotient is not compact.

Proof. (i) If μ_{can} is proper, then for any τ the topological space

W //(G, τ) ≃ μ_{τ}^{−1}(0)/K = μ_{can}^{−1}(τ)/K

is obviously compact.

(ii) Suppose now that μ_{can} is not proper. By Lemmas 1.10 and 1.11 (stated below) one first deduces that the quotient

W //(G, τ_{0}) ≃ μ_{τ_{0}}^{−1}(0)/K = μ_{can}^{−1}(0)/K

is not compact. Moreover, since W^{ss,τ_{0}} = W, it is a non-empty connected topological space. For τ arbitrary one obtains, using the properties of the good quotients q_{τ} and q_{τ_{0}}, a surjective continuous map from W //(G, τ) to q_{τ_{0}}(W^{ss,τ}), which is an open subset of W //(G, τ_{0}) and our statement follows.

Lemma 1.10 [8, Lemma 1.1] The following assertions are equivalent: (i) The map μ_{can} is proper, (ii) μ_{can}^{−1}(0) = {0}, (iii) μ_{can}^{−1}(0) is bounded.

Lemma 1.11 Let (W, h_{W}) be a finite dimensional Hermitian vector space and let K be a compact group which acts on W by unitary transformations. If X ⊂ W is K-invariant and unbounded, the quotient space X/K cannot be compact.

Proof. We will denote by π : X → X/K the canonical projection, x ↦ K · x. Since X is not bounded, we can find a sequence (x_{n})_{n} ⊂ X for which it holds ∥x_{n}∥_{h_{W}} → ∞. Let’s suppose now that the topological space X/K is compact. Then, by passing at a subsequence, we can find an orbit π(y) ∈ X/K of an element y ∈ X such that π(x_{n}) → π(y). We claim that we can find a sequence (k_{n})_{n} ⊂ K and a subsequence (x_{m_{n}})_{n} such that k_{n} · x_{m_{n}} → y, hence, in particular, ∥k_{n} · x_{m_{n}}∥_{h_{W}} → ∥y∥_{h_{W}}. On the other hand, since K acts by unitary transformations, we have

∥k_{n} · x_{m_{n}}∥_{h_{W}} = ∥x_{m_{n}}∥_{h_{W}} → ∞,

which yields a contradiction. It remains to construct the sequence k_{n} indicated above. Let (B_{i})_{i∈ℕ} be a basis of neighborhouds of y such that B_{i+1} ⊂ B_{i}. Then (π(B_{i}))_{i∈ℕ} is a fundamental system of neighborhouds of π(y). We fix n ∈ ℕ. There exists i_{n} such that, for any i ≥ i_{n} one has π(x_{i}) ∈ π(B_{n}). Set m_{n} := max{n, i_{n}}; in particular π(x_{m_{n}}) ∈ π(B_{n}), i.e. we can find k_{n} such that k_{n} · x_{m_{n}} ∈ B_{n} and we conclude that k_{n} · x_{n_{m}} → y.

2 Quiver factorisation problems

2.1 Generalities. Compactness of QFP-quotients

The aim of this section is to discuss a special class of linear problems, called quiver factorisation problems. We begin by giving some definitions concerning quivers and their representations. A quiver is a diagram consisting of points and arrows. Formally, a quiver Q is a quartet (Q_{0}, Q_{1}, h, t) consisting of the set of vertices Q_{0}, the set of arrows Q_{1} and the maps h, t : Q_{1} → Q_{0}, which associate to every arrow a the head, respectively the tail of a. In the sequel we will consider only finite quivers, i.e. the sets Q_{0} and Q_{1} will be assumed to be finite. If a_{1}, …, a_{n} are arrows such that h(a_{i}) = t(a_{i+1}) for any i = 1, …, n−1, they give rise to the oriented path a_{n}·. . .·a_{1}. If, moreover, one has h(a_{n}) = t(a_{1}), this oriented path will be called a loop. A vertex v is called a sink (respectively a source) if all the arrows meeting v are directed to v (respectively to the other vertex). If Q contains no loops, then it has at least one source and one sink.

A representation (U, ψ) of a quiver Q (over the field of complex numbers) is given by a family of finite dimensional complex vector spaces (U_{v})_{v∈Q_{0}} and a family of linear maps (ψ_{a})_{a∈Q_{1}}, with ψ_{a} : U_{t(a)} → U_{h(a)}. The dimension vector α ∈ ℕ^{Q_{0}} of the representation (U, ψ) is defined by α_{v} := dim_{ℂ}(U_{v}) for any vertex v. A morphism of representations (U, ψ), (U′, ψ′), is given by linear maps (f_{a})_{a∈Q_{1}} such that for every a it holds f_{h(a)} ∘ ψ_{a} = ψ′_{a} ∘ f_{t(a)}.

For a fixed representation r = (U, ψ) of a quiver Q = (Q_{0}, Q_{1}, h, t) we consider the vector space

W_{r} := ⨁_{a∈Q_{1}} Hom(U_{t(a)}, U_{h(a)}).

The group ∏_{v∈Q_{0}} GL(U_{v}) acts on W_{r} in a natural fashion

(g_{v})_{v} · (ψ_{a})_{a} := (g_{h(a)} ∘ ψ_{a} ∘ g_{t(a)}^{−1})_{a}.

Since in some situations the action of a smaller symmetry group is needed, for a subset of vertices S ⊂ Q_{0} we define

G_{S} := ∏_{v∈S} GL(U_{v}) ⊂ ∏_{v∈Q_{0}} GL(U_{v}).

According to the terminology of [7], this is a quiver factorisation problem associated to the combinatorial data (Q, S). In the particular case when S = Q_{0}, it is called a standard quiver factorisation problem. We notice that the orbits in the space W_{r} of the symmetry group G_{S} are in bijective correspondence with the isomorphism classes of representations (Ũ, ψ̃) of Q having the same dimension vector as r = (U, ψ) and such that for any v ∈ Q_{0} \ S it holds Ũ_{v} = U_{v}.

Let now τ be an element of T_{G_{S}} ≃ ℝ^{S}. The categorical quotient W_{r}//(G_{S}, τ) will be called a QFP-quotient. According to the results described in the first section, such a quotient admits an alternative symplectic description. If (h_{U_{v}})_{v∈Q_{0}} are Hermitian inner products on the vector spaces (U_{v})_{v∈Q_{0}}, then we get in a natural fashion an induced Hermitian inner product h_{W_{r}} on the space W_{r}. Moreover, the group

K_{S} := ∏_{v∈S} U(U_{v}, h_{U_{v}})

acts on W_{r} by h_{W_{r}}-unitary transformations. The corresponding canonical moment map μ_{can} : W_{r} → ⨁_{v∈S} u(U_{v}, h_{U_{v}}) is given by

μ_{can}((ψ_{a})_{a}) = ⨁_{v∈S} (−\frac{i}{2} ∑_{v=h(a)} ψ_{a} ∘ ψ_{a}^{∗} + \frac{i}{2} ∑_{v=t(a)} ψ_{a}^{∗} ∘ ψ_{a})

and one has a homeomorphism between the symplectic quotient μ_{τ}^{−1}(0)/K_{S} and the QFP-quotient W_{r}//(G_{S}, τ).

We notice that if (U, ψ) and (Ũ, ψ̃) are representations of Q having the same dimension vector and if we consider the same subset of vertices S and parameters τ, τ̃ for which the associated semistable loci coincide, then the corresponding QFP-quotients will be isomorphic. We conclude that, up to isomorphism, a QFP-quotient depends on:

-the combinatorial data (Q, S),

-the dimension vector α ∈ ℕ^{Q_{0}},

-the parameter τ ∈ T_{G_{S}}. It is therefore a natural problem to understand how the geometry of the QFPquotients depends on these data. In the sequel we will give some results concerning the relationship between the compactness of these quotients and the following combinatorial property of a pair (Q, S):

Property (∗)

(i) Q has no loops,

(ii) any oriented path in Q contains at most one vertex lying in Q_{0} \ S.

Lemma 2.1 Let (U_{i}, h_{U_{i}})_{i=0,...,n} be a family of Hermitian vector spaces of positive finite dimension such that (U_{0}, h_{U_{0}}) = (U_{n}, h_{U_{n}}). Then one can find nonzero linear maps ψ_{i} : U_{i} → U_{i+1} (i = 0, …, n − 1), such that, putting ψ_{n} := ψ_{0}, one has for any i = 1, …, n

ψ_{i−1} ∘ ψ_{i−1}^{∗} = ψ_{i}^{∗} ∘ ψ_{i}.

Proof. For any i = 0, …, n − 1 we denote by l_{i} the dimension of the vector space U_{i} and we fix a h_{U_{i}}-orthonormal basis b^{i} = (b_{1}^{i}, …, b_{l_{i}}^{i}) in U_{i}; we also put b^{n} := b^{0}. The linear maps ψ_{i} (i = 0, …, n − 1) acting on the given basis by

ψ_{i}(b_{1}^{i}) = b_{1}^{i+1}, ψ_{i}(b_{j}^{i}) = 0 ∀ j = 2, …, l_{i}

satisfy the required property.

Proposition 2.2 Let Q = (Q_{0}, Q_{1}, h, t) be a quiver and S ⊂ Q_{0} a fixed subset of vertices. Let r = (U, ψ) be a representation of Q with Hermitian vector spaces (U_{v}, h_{U_{v}})_{v∈Q_{0}}.

(i) If the pair (Q, S) satisfies the Property (∗), then the canonical moment map associated to the corresponding quiver factorisation problem is proper.

(ii) If we assume that any vector space U_{v} has positive dimension, then the converse of (i) is also true.

Proof. (i) We take ψ ∈ W_{r} such that μ_{can}(ψ) vanishes and we show that ψ is zero. We first notice that, since the pair (Q, S) verifies the Property (∗), the quiver Q has at least one sink or one source belonging to S. If Q has a source v ∈ S, then the projection of the canonical moment map on u(U_{v}, h_{U_{v}}) is (\frac{i}{2} ∑_{v=t(a)} ψ_{a}^{∗} ∘ ψ_{a}), whereas if Q has a sink v ∈ S, this pro-

jection is (−\frac{i}{2} ∑_{v=h(a)} ψ_{a} ∘ ψ_{a}^{∗}). In both cases the projection on u(U_{v}, h_{U_{v}}) vanishes if and only if ψ_{a} = 0 for any a which meets v. We consider now the quiver Q′ obtained from Q by eliminating v and those arrows which meet v. Then the pair (Q′, S \ {v}) also verifies the Property (∗). Applying the same reasoning, after a finite number of steps we eliminate all the vertices lying in S. Since any arrow meets at least one vertex in S, we conclude that ψ_{a} = 0 for any a ∈ Q_{1}, that is ψ = 0.

(ii) Suppose first that Q contains a loop with vertices v_{1}, …, v_{n} = v_{0} and with arrows a_{0}, a_{1}, …, a_{n−1}, a_{n} = a_{0} such that for any i = 0, …, n − 1 one has t(a_{i}) = v_{i}, h(a_{i}) = v_{i+1}. We construct non-zero linear maps ψ_{a_{i}} : U_{v_{i}} → U_{v_{i+1}} as in Lemma 2.1. For any a ∈ Q_{1} \ {a_{0}, …, a_{n−1}} we set ψ_{a} := 0, obtaining in this way a non-zero element ψ ∈ W_{r}. We claim that it holds μ_{can}(ψ) = 0. Indeed, if v = v_{i} (i = 1, …, n) one has

∑_{v=h(a)} ψ_{a} ∘ ψ_{a}^{∗} − ∑_{v=t(a)} ψ_{a}^{∗} ∘ ψ_{a} = ψ_{a_{i−1}} ∘ ψ_{a_{i−1}}^{∗} − ψ_{a_{i}}^{∗} ∘ ψ_{a_{i}} = 0

and for all other v this sum vanishes, since the corresponding linear maps are zero. Hence, by Lemma 1.10, it follows that μ_{can} is not a proper map.

Let’s now assume that Q contains an oriented path with vertices v_{1}, …, v_{n} such that v_{1} and v_{n} are in Q_{0} \ S and with arrows a_{1}, …, a_{n−1} such that t(a_{i}) = v_{i}, h(a_{i}) = v_{i+1} (i = 1, …, n − 1). Applying again Lemma 2.1, we construct non-zero linear maps ψ_{a_{i}} : U_{v_{i}} → U_{v_{i+1}} for i = 1, …, n − 1. Setting ψ_{a} := 0 for any a ∈ Q_{1} \ {a_{0}, …, a_{n−1}}, we get a non-zero element ψ ∈ W_{r}. Then:

∑_{v=h(a)} ψ_{a} ∘ ψ_{a}^{∗} − ∑_{v=t(a)} ψ_{a}^{∗} ∘ ψ_{a} = \{ \begin{matrix}−ψ_{a_{1}}^{∗} ∘ ψ_{a_{1}}, & if v = v_{1}, \\ ψ_{a_{n−1}} ∘ ψ_{a_{n−1}}^{∗}, & if v = v_{n}, \\ 0, & otherwise. \end{matrix}

But v_{1} and v_{n} are not elements of S and hence μ_{can}(ψ) = 0. Again by Lemma 1.10, it follows that μ_{can} cannot be a proper map.

Using Propositions 2.2 and 1.9 we deduce at once:

Theorem 2.3 (i) Any QF P -quotient associated to a pair (Q, S) satisfying the Property (∗) is compact.

(ii) If a non-empty QFP-quotient associated to a pair (Q, S) and corresponding to a dimension vector α ∈ ℕ_{+}^{Q_{0}} is compact, then the pair (Q, S) must fulfil the Property (∗).

2.2 Compactification of QFP-quotients

In the remaining of the paper we will focus our attention to the particular case concerning linear systems. As in section 1.2.1, we consider complex vector spaces of positive finite dimension V, V_{1}, V_{2} and the natural action of the group G := GL(V) on the space

W := End_{ℂ}(V) ⊕ Hom_{ℂ}(V_{1}, V) ⊕ Hom_{ℂ}(V, V_{2}).

This linear problem is, in fact, a quiver factorisation problem associated to the quiver Q

[figure: ∘; •; ∘]

and to the set S containing just one vertex, namely •. The pair (Q, S) does not verify the Property (∗) and hence, by Theorem 2.3, the corresponding quotients are not compact. This yields an alternative proof to the fact that the moduli space of accessible linear systems is not compact, as shown for instance in [9]. Our main aim is to construct natural compactifications for these spaces which are themselves QFP-quotients associated to a suitable pair (Q̃, S̃) and will be achieved in Theorem 2.7. Our construction is motivated by Helmke’s compactification of the space of accessible linear systems by using generalized linear systems, which is described in [3] and [4]. Hence, in the paper [3] one considers the group

G̃ := GL(V) × GL(V),

which acts on the space

W̃ := Hom_{ℂ}(V, V)^{⊕2} ⊕ Hom_{ℂ}(V_{1}, V) ⊕ Hom_{ℂ}(V, V_{2})

in a natural fashion

(g, h) · (ψ, φ, φ_{1}, φ_{2}) := (g ∘ ψ ∘ h^{−1}, g ∘ φ ∘ h^{−1}, g ∘ φ_{1}, φ_{2} ∘ h^{−1}).

The study of the G̃-action on W̃ can also be represented as a quiver factorisation problem associated to the quiver Q̃

\xymatrix{ & • \ar[d] \ar[d] \ar[dr] \\ ∘ \ar[r] & • & ∘}

and for which the set S̃ contains two vertices, both drawn with •. Since the pair (Q̃, S̃) verifies the Property (∗), it follows by Theorem 2.3 that the associated QFP-quotients are compact. In remains to get a natural relationship between a QFP-quotient corresponding to (Q, S) and a suitable QFP-quotient associated to (Q̃, S̃). We first notice that one has a natural map

i : W → W̃, i(φ, φ_{1}, φ_{2}) := (id, φ, φ_{1}, φ_{2}).

On the the other hand, the geometry of QFP-quotients also depends on a parameter τ, i.e. on a stability concept. In section 1.2.1, we explained that there are three stability concepts associated to this first quiver factorisation problem, which were denoted by τ_{ε} ∈ T_{G} (ε ∈ {0, ±1}). Our aim is to show that the images of the corresponding semistable loci under the map i are also included in some sets of semistable points, with respect to suitable elements in T_{G}. More precisely, we consider τ̃_{0}, τ̃_{1}, τ̃_{−1} ∈ T_{G} given by

⟨τ̃_{ε}, (s′, s′′)⟩ := \{ \begin{matrix}i(Tr(s′) − Tr(s′′)), & if ε = 0 \\ i((r + 1)Tr(s′) − rTr(s′′)), & if ε = 1 \\ i(−rTr(s′) + (r + 1)Tr(s′′)), & if ε = −1 \end{matrix}

and we claim that the following result holds

Proposition 2.4 Take any ε ∈ {0, ±1}. A triple (φ, φ_{1}, φ_{2}) ∈ W is τ_{ε}- (semi)stable if and only if i(φ, φ_{1}, φ_{2}) ∈ W̃ is τ̃_{ε}-(semi)stable.

In order to prove the proposition, we need some preparations. Let (s′, s′′) be an element of H(G̃) (this space can be identified with H(GL(V)) × H(GL(V))). Let λ′_{1} < λ′_{2} < … < λ′_{a}, respectively λ′′_{1} < λ′′_{2} < … < λ′′_{b} be the eigenvalues of s′, respectively s′′ and let

{0} = V′_{0} ⊂ V′_{1} ⊂ … ⊂ V′_{a}, {0} = V′′_{0} ⊂ V′′_{1} ⊂ … ⊂ V′′_{b}

be the corresponding filtrations. We denote by d′_{i} := dim_{ℂ}(V′_{i}) (i = 1, …, a), respectively d′′_{i} := dim_{ℂ}(V′′_{i}) (i = 1, …, b) the dimensions of the vector spaces arising in the filtrations F_{s′}, respectively F_{s′′}. For any k = 1, …, b we put

j(k) := \{ \begin{matrix}max\{l | λ′_{l} ≤ λ′′_{k}\}, & if \{l | λ′_{l} ≤ λ′′_{k}\} ≠ ∅ \\ 0, & otherwise \end{matrix} .

We will need the following

Lemma 2.5 (i) Let ψ ∈ (Hom_{ℂ}(V, V))^{≤0}(s′, s′′). Then for any real number m one has

ψ( ⨁_{λ′′_{k}≤m} V (λ′′_{k})) ⊂ ⨁_{λ′_{j}≤m} V (λ′_{j}).

(ii) Suppose that id ∈ (Hom_{ℂ}(V, V))^{≤0}(s′, s′′). Then for any k = 1, …, b it holds d′′_{k} ≤ d′_{j(k)}. Moreover, if s′ and s′′ have the same eigenvalues with the same multiplicities, then s′ and s′′ have the same associated filtrations, i.e. F_{s′} = F_{s′′}.

Proof. (i) Since the eigenvalues of (s′, s′′) are {λ′_{p} − λ′′_{q}|p, q = 1, …, r}, we can decompose any ψ ∈ Hom_{ℂ}(V, V) as

ψ = ∑_{p,q} ψ_{λ′_{p}−λ′′_{q}},

where for any p, q it holds

s′ ∘ ψ_{λ′_{p}−λ′′_{q}} − ψ_{λ′_{p}−λ′′_{q}} ∘ s′′ = (λ′_{p} − λ′′_{q})ψ_{λ′_{p}−λ′′_{q}}.

Let now v ∈ V be an eigenvector of s′′ corresponding to an eigenvalue λ′′. Then for any component of ψ, the vector ψ_{λ′_{p}−λ′′_{q}}(v) is an eigenvector of s′, corresponding to the eigenvalue (λ′′ + λ′_{p} − λ′′_{q}). The assumption that ψ is an element of (Hom_{ℂ}(V, V))^{≤0}(s′, s′′) means that in the decomposition of ψ occur only components corresponding to nonpositive eigenvalues. In particular, we deduce that for an eigenvector v ∈ V (λ′′), we have ψ(v) ∈ ⊕_{λ′_{j}≤λ′′}V (λ′_{j}) and this yields our assertion.

(ii) Using (i), we deduce that for any k it holds

V′′_{k} = id( ⨁_{λ′′_{j}≤λ′′_{k}} V (λ′′_{j})) ⊂ ⨁_{λ′_{j}≤λ′′_{k}} V (λ′_{j}) = ⨁_{λ′_{j}≤λ′_{j(k)}} V (λ′_{j}) = V′_{j(k)}

and hence d′′_{k} ≤ d′_{j(k)}. We notice that the assertion remains true if we replace id with ψ ∈ Hom_{ℂ}(V, V) such that det ψ ≠ 0.

Let’s now suppose that a = b, that for any k = 1, …, a we have λ′_{k} = λ′′_{k} and that the corresponding multiplicities are equal. In particular we deduce that for any k one has j(k) = k and dim(V′_{k}) = dim(V′′_{k}). Since V′′_{k} ⊂ V′_{j(k)} = V′_{k}, we conclude that the two filtrations coincide.

Remark 2.6 For any l ∈ {1, …, r} we put

λ′(l) := λ′_{min{m | d′_{m}≥l}}, λ′′(l) := λ′′_{min{m | d′′_{m}≥l}}.

The sequence of real numbers (λ′(1), …, λ′(r)) is nothing else but the sequence of eigenvalues of s′ in which each eigenvalue occurs as many times as its multiplicity is; an analogous statement holds for the sequence (λ′′(1), …, λ′′(r)). Then the condition that for any k one has d′′_{k} ≤ d′_{j(k)} (stated in Lemma 2.5 ii)) is equivalent to the condition that for any l = 1, …, r it holds λ′(l) ≤ λ′′(l). Moreover, it holds d′′_{k} = d′_{j(k)} if and only if λ′′(d′′_{k}) < λ′(d′′_{k} + 1).

Proof of Proposition 2.4.

We first notice that for any ε ∈ {0, ±1} it holds: if (φ, φ_{1}, φ_{2}) ∈ W^{≤0}(s) and ⟨τ_{ε}, is⟩ < (≤)0, then (id, φ, φ_{1}, φ_{2}) ∈ W̃^{≤0}(s, s) and ⟨τ̃_{ε}, i(s, s)⟩ < (≤)0. This means that if (φ, φ_{1}, φ_{2}) is not τ_{ε}-(semi)stable, then (id, φ, φ_{1}, φ_{2}) cannot be τ̃_{ε}-(semi)stable.

We now prove the converse assertion: if (φ, φ_{1}, φ_{2}) is τ_{ε}-(semi)stable, then (id, φ, φ_{1}, φ_{2}) is τ̃_{ε}-(semi)stable.

• The case ε = 0. Our first claim is that every element (id, φ, φ_{1}, φ_{2}) is τ̃_{0}-semistable. Indeed, if (s′, s′′) is a pair such that (id, φ, φ_{1}, φ_{2}) ∈ W̃^{≤0}(s′, s′′), then, by Remark 2.6, we deduce that for any i = 1, …, r it holds λ′(i) ≤ λ′′(i) and hence

⟨τ̃_{0}, i(s′, s′′)⟩ = −(Tr(s′) − Tr(s′′)) = ∑_{i=1}^{r}(−λ′(i) + λ′′(i)) ≥ 0.

Suppose now that (id, φ, φ_{1}, φ_{2}) is not τ̃_{0}-stable. This means that we have (s′, s′′) such that (id, φ, φ_{1}, φ_{2}) ∈ W̃^{≤0}(s′, s′′) and that it holds

∑_{i=1}^{r}(−λ′(i) + λ′′(i)) = ⟨τ̃_{0}, i(s′, s′′)⟩ ≤ 0.

On the other hand, again by Remark 2.6, we have λ′(i) ≤ λ′′(i) for any i = 1, …, r. We conclude that λ′(i) = λ′′(i) for any i, that is that s′ and s′′ have the same eigenvalues with the same multiplicities. By Lemma 2.5 ii), we deduce that the filtrations F_{s′} and F_{s′′} coincide. In particular, it follows that F_{s′} is φ-invariant, im(φ_{1}) ⊂ F_{s′}^{≤0} and that ker(φ_{2}) ⊃ F_{s′′}^{<0} = F_{s′}^{<0}, that is one has (φ, φ_{1}, φ_{2}) ∈ W^{≤0}(s′). Since ⟨τ_{0}, is′⟩ = 0, we conclude that (φ, φ_{1}, φ_{2}) is not τ_{0}-stable.

• The case ε = 1. We will prove that if (id, φ, φ_{1}, φ_{2}) is not τ̃_{1}-stable, then (φ, φ_{1}, φ_{2}) is not τ_{1}-semistable. Let (s′, s′′) ∉ 𝔤̃_{0} be a pair such that (id, φ, φ_{1}, φ_{2}) ∈ W̃^{≤0}(s′, s′′) and such that ⟨τ̃_{1}, i(s′, s′′)⟩ ≤ 0. This inequality can be written as

∑_{i=1}^{r} λ′(i) + r(∑_{i=1}^{r}(λ′(i) − λ′′(i))) ≥ 0.  (2)

We claim that:

(a) s′ has at least one positive eigenvalue.

(b) let q be such that F_{s′}^{≤0} = V′_{q}; in particular, if q ≠ 0, λ′_{q} is the greatest nonpositive eigenvalue of s′. There exists t such that q ≤ j(t) < r and such that d′′_{t} = d′_{j(t)}.

Let’s suppose that we proved the assertions (a) and (b). We showed in the proof of Lemma 2.5 that V′′_{t} ⊂ V′_{j(t)} and since their dimensions are equal (d′′_{t} = d′_{j(t)}), it follows that these subspaces of V coincide. Moreover, by Lemma 2.5 i) we have

φ(V′_{j(t)}) = φ(V′′_{t}) = φ( ⨁_{λ′′_{i}≤λ′′_{t}} V (λ′′_{i})) ⊂ ⨁_{λ′_{i}≤λ′′_{t}} V (λ′_{i}) = ⨁_{λ′_{i}≤λ′_{j(t)}} V (λ′_{i}) = V′_{j(t)}

and it holds

im(φ_{1}) ⊂ F_{s′}^{≤0} = V′_{q} ⊂ V′_{j(t)}.

We now consider s with eigenvalues 0 and 1 and with associated filtration F_{s}

{0} ⊂ V′_{j(t)} ⊆_{/} V.

We already proved that F_{s} is φ-invariant and that im(φ_{1}) ⊂ F_{s}^{≤0}; we obviously have ker(φ_{2}) ⊃ F_{s}^{<0} = {0} and we conclude that (φ, φ_{1}, φ_{2}) ∈ W^{≤0}(s). On the other hand, one has

⟨τ_{1}, is⟩ = −Tr(s) = − dim(V/V′_{j(t)}) < 0

and we conclude that (φ, φ_{1}, φ_{2}) is not τ_{1}-stable.

We now prove the assertions (a) and (b). If all the eigenvalues of s′ were nonpositive, then we would have λ′(i) ≤ 0 for any i. On the other hand, by Remark 2.6, for any i we have λ′(i) − λ′′(i) ≤ 0. Using the inequality (2), we deduce that all the numbers λ′(i), λ′′(i) must be zero, which contradicts our assumption that (s′, s′′) ∉ 𝔤̃_{0}.

By Remark 2.6, it is enough to show that there exists d′_{q} ≤ l < r such that λ′′(l) < λ′(l + 1) (if d′_{q} = 0, we set λ′(0) = λ′′(0) := 0). Let’s suppose that this property is not fulfiled, that is for any d′_{q} ≤ l ≤ r − 1 we have λ′′(l) ≥ λ′(l + 1). By multiplicating each inequality with a suitable nonnegative number (more precisely me multiply the inequality between λ′′(l) and λ′(l + 1) with r − l + 1) and by using the fact that λ′(d′_{q}) ≤ 0 < λ′(r) ≤ λ′′(r), we obtain the following inequality

∑_{l=d′_{q}+1}^{r} λ′(l) + ∑_{l=d′_{q}}^{r}(r − l + 1)(λ′(l) − λ′′(l)) < 0.

Since λ′(1), …, λ′(d′_{q}), λ′(1)−λ′′(1), …, λ′(r)−λ′′(r) are nonpositive, this would imply

∑_{i=1}^{r} λ′(i) + r ∑_{i=1}^{r}(λ′(i) − λ′′(i)) < 0,

and this contradicts the relation (2).

• The case ε = −1. The proof of this case is analogous to the proof of the second one. Using similar arguments it can be shown that s′′ has at least one negative eigenvalue. Let q such that F_{s′′}^{<0} = V′′_{q}. One proves that there exists 0 < t ≤ q such that d′′_{t} = d′_{j(t)}. Then, considering s with eigenvalues −1 and 0 and with associated filtration

0 ⊆_{/} V′′_{t} ⊂ V,

one gets the desired conclusion.

We are now able to prove the main result of this part:

Theorem 2.7 For any τ ∈ T_{G}, there exists an element τ̃ ∈ T_{G}, such that the

QFP-quotient W̃//(G̃, τ̃) is a compactification of the QFP-quotient W //(G, τ).

Proof. We fix an arbitrary ε ∈ {0, ±1}. Then, according to Proposition 2.4, the image of the τ_{ε}-(semi)stable locus i(W^{(s)s,τ_{ε}}) is included in the τ̃_{ε}-(semi)stable locus W̃^{(s)s,_{τ_{ε}}}. In particular, this yields an induced map between the corresponding categorical quotients, denoted by ι_{ε} and one gets a commutative diagram

\xymatrix{W^{ss,τ_{ε}} \ar[d] \ar[r]^{i} & W̃^{ss,_{τ_{ε}}} \ar[d] \\ W //(G, τ_{ε}) \ar[r]^{ι_{ε}} & W̃//(G̃, τ̃_{ε})}

It is easy to check that ι_{ε} is an injective map and, by Proposition 2.4, its image is

im(ι_{ε}) = {[ψ, φ, φ_{1}, φ_{2}] ∈ W̃//(G̃, τ̃_{ε}) | det ψ ≠ 0},

which is an open subset of W̃//(G̃, τ̃_{ε}). We claim that this set is dense in W̃//(G̃, τ̃_{ε}). Indeed, if one considers an arbitrary τ̃_{ε}-semistable element in W̃, say x = (ψ, φ, φ_{1}, φ_{2}), then one can find a sequence (x_{n})_{n} of τ̃_{ε}-semistable elements converging to x and such that, writing x_{n} = (ψ_{n}, φ_{n}, φ_{1,n}, φ_{2,n}), one has det(ψ_{n}) ≠ 0. Then the sequence ([x_{n}])_{n} is included in im(ι_{ε}) and converges to [x]. We conclude that W̃//(G̃, τ̃_{ε}) is a compactification of W //(G, τ_{ε}), getting the desired result.

The following question, which would yield a generalization to Theorem 2.7, remains open: given a pair (Q, S) which does not fulfil the Property (∗), is it possible to find (Q̃, S̃) verifying (∗) such that any QFP-quotient corresponding to (Q, S) admits a compactification which is a QFP-quotient associated to (Q̃, S̃)?

Acknowledgements. This work was supported by a post-doctoral research fellowship at the Mathematical Institute of Leiden University. I am grateful to the members of the staff for their hospitality and for the stimulating mathematical environment. I would also like to thank Dr. M. Lübke for very useful discussions.

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