Quasi-homogeneity of the moduli space of stable maps to homogeneous spaces

Let $G$ be a connected, simply connected, simple, complex, linear algebraic group. Let $P$ be an arbitrary parabolic subgroup of $G$. Let $X=G/P$ be the $G$-homogeneous projective space attached to this situation. Let $d\in H_2(X)$ be a degree. Let $\overline{M}_{0,3}(X,d)$ be the (coarse) moduli space of three pointed genus zero stable maps to $X$ of degree $d$. We prove under reasonable assumptions on $d$ that $\overline{M}_{0,3}(X,d)$ is quasi-homogeneous under the action of $G$. The essential assumption on $d$ is that $d$ is a minimal degree, i.e. that $d$ is a degree which is minimal with the property that $q^d$ occurs with non-zero coefficient in the quantum product $\sigma_u\star\sigma_v$ of two Schubert cycles $\sigma_u$ and $\sigma_v$, where $\star$ denotes the product in the (small) quantum cohomology ring $QH^*(X)$ attached to $X$. We prove our main result on quasi-homogeneity by constructing an explicit morphism which has a dense open $G$-orbit in $\overline{M}_{0,3}(X,d)$. To carry out the construction of this morphism, we develop a combinatorial theory of generalized cascades of orthogonal roots which is interesting in its own right.


Introduction
Let G be a connected, simply connected, simple, complex, linear algebraic group. Let P be a fixed but arbitrary parabolic subgroup of G. Let X = G/P be the G-homogeneous projective space attached to this situation. We select once and for all a maximal torus T and a Borel subgroup B of G such that T ⊆ B ⊆ P ⊆ G .
We call an effective homology class in H 2 (X) a degree. Let d be a degree. Let M 0,3 (X, d) be the (coarse) moduli space of three pointed genus zero stable maps to X of degree d. By definition, the moduli space M 0,3 (X, d) parametrizes isomorphism classes [C, p 1 , p 2 , p 3 , µ : C → X] where: • C is a complex, projective, connected, reduced, (at worst) nodal curve of arithmetic genus zero. • The marked points p i ∈ C are distinct and lie in the nonsingular locus.
• The pointed map µ has no infinitesimal automorphisms. Basic properties of the moduli space M 0,3 (X, d) can be found in [5]. It is a consequence of more general results in [5,9], namely of [5, Theorem 2(i)] and [9,Corollary 1], that M 0,3 (X, d) is a normal projective irreducible variety.
In this work, we ask the question if it is possible to prove stronger properties of M 0,3 (X, d) than irreducibility. Note that the group G acts on M 0,3 (X, d) by translation. Hence, the following question makes sense. To give an affirmative answer to Question 1.2 for specific degrees, it turns out that the class of all minimal degrees is a good starting point. Minimal degrees and their properties were studied in [1]. We build on this work in various ways. Definition 1.3. Let (QH * (X), ⋆) be the (small) quantum cohomology ring attached to X as defined in [5,Section 10]. For a Weyl group element w, we denote by σ w the Schubert cycle attached to w. We say that a degree d is a minimal degree if there exist Weyl group elements u and v such that d is a minimal degree in σ u ⋆ σ v , i.e. if the power q d appears with non-zero coefficient in the expression σ u ⋆ σ v and if d is minimal with this property. Remark 1.4. Using the theory of curve neighborhoods [4] reviewed in Subsection 3.1, one can give a quantum cohomology independent description of minimal degrees (cf. Definition 3.25). We will mostly choose the approach to minimal degrees through curve neighborhoods because it is more explicit and more computable in terms of combinatorics. In this introduction, we gave the definition of minimal degrees in terms of quantum cohomology because it is short and it shows where the ideas originate. Nevertheless, the rest of the text is basically independent from the context of quantum cohomology and does not assume familiarity with it. Notation 1.5. We denote by Π P the set of all minimal degrees. We use this notation to make clear relative to which P the minimal degrees are computed. In particular, the set of all minimal degrees in H 2 (G/B) is denoted by Π B . The sets Π P and Π B will be often simultaneously in use. Example 1.6. By Corollary 6.8, Remark 6.11, there exists a unique minimal degree in the quantum product pt⋆pt of two point classes. We denote this degree by d X ∈ Π P . In particular, we have d G/B ∈ Π B . By Corollary 6.12, we have an inclusion Π P ⊆ {0 ≤ d ≤ d X }. Note that this inclusion is possibly strict unless P is maximal (cf. [1,Example 10.14]). For more informations on the minimal degree d X , we refer to [1].
Starting from the class of minimal degrees, we impose one further assumption on a minimal degree, namely Assumption 7.13, to give an affirmative answer to Question 1.2. For this introduction, it suffices to know that Assumption 7.13 is satisfied for all minimal degrees if the root system R associated to G and T is simply laced or if P = B, i.e. if X is a generalized complete flag variety. We are able to obtain the following theorem. We prove Theorem 1.7 by constructing for every minimal degree d an explicit morphism f P,d . Then, we show that f P,d has a dense open orbit in M 0,3 (X, d) under the action of G if d satisfies Assumption 7.13. We do so by comparing the dimension of the orbit of f P,d with the dimension of the moduli space. For the convenience of the reader, we sketch the construction of f P,d in this introduction. Construction 1.8. Let d ∈ Π P . In Section 6, we uniquely associate to d a minimal degree e ∈ Π B -the so-called lifting of d. The precise definition of e is given in Definition 6.2. For the moment, it suffices to know that the image of e under the natural map H 2 (G/B) → H 2 (X) is d (cf. Fact 6.5(3)). In this way, the lifting helps us to transport the situation from X to G/B.
In the next step, we associate to the degree e ∈ Π B a set of (strongly) orthogonal roots B R,e -a so-called generalized cascade of orthogonal roots. We systematically investigate the properties of these generalized cascades of orthogonal roots in Section 4. The reader finds the precise definition of B R,e in Definition 4.1.
Let R P be the root system associated to the Levi factor of P and T . Let R + P be the positive roots of R P induced by B. For a root α ∈ B R,e \ R + P , there is a unique irreducible T -invariant curve C α ⊆ X containing the T -fixed points associated to 1 and s α , where s α is the reflection along α (cf. [6,Lemma 4.2]). Each of the curves C α is isomorphic to P 1 (cf. [6,Lemma 4.2]).
With these preliminaries, we can now define the morphism f P,d as in Definition 7.3. We define the morphism f P,d by the assignment f P,d : P 1 ֒→ α∈B R,e \R + P C α ֒→ X where the first morphism is the diagonal embedding of P 1 into card(B R,e \ R + P ) isomorphic copies of P 1 and the second morphism is the embedding into X which is well-defined due to the fact that two distinct elements of B R,e are (strongly) orthogonal (cf. Theorem 4.5(3)). Also, by the very same fact, the definition of f P,d is independent of the ordering of the product α∈B R,e \R + P . Hence, the morphism f P,d is well-defined. We call the image f P,d (P 1 ) the diagonal curve (associated to d).
One can show that f P,d has degree (f P,d ) * [P 1 ] = d (cf. Fact 7.4). Hence, the morphism f P,d gives indeed a candidate for an element of M 0,3 (X, d) which is interesting for an answer of Question 1.2.
We give an example for a class of degrees for which the diagonal curve turns out to be particularly simple and explicit. Example 1.9. Let α 1 , . . . , α k be P -cosmall roots (cf. Definition 3.3) such that the supports of α 1 , . . . , α k are pairwise totally disjoint (cf. Notation 3.8, Definition 3.20).
where C α i is the curve associated to α i as in Construction 1.8. Then we have d ∈ Π P (cf. Theorem 6.16). Let e be the lifting of d. By Theorem 6.16, we then have B R,e \ R + P = {α 1 , . . . , α k }. We see directly from the assumption on α i that two distinct elements of B R,e \R + P are (strongly) orthogonal (cf. Fact 3.21). Hence, in this case, we can verify directly that the diagonal curve (associated to d) is well-defined. The morphism f P,d is given by where the two arrows are defined as in Construction 1.8. Remark 1. 10. The proof of Theorem 1.7 for degrees as in Example 1.9 (i.e. degrees which satisfy Assumption 7.13(3)) is very simple and can be done directly without any further considerations. (As an exercise, the reader may provide the arguments that the morphism described in Example 1.9 has a dense open G-orbit.) The point is not that we get an affirmative answer to Question 1.2 for degrees which satisfy Assumption 7.13 (3), but rather that the general construction of f P,d subsumes this case as a trivial sub-case. In this way, we gain confidence that Construction 1.8 is reasonable even in a more general setting.
History. The problem to prove quasi-homogeneity of the moduli space of stable maps to homogeneous spaces was first posed in [12,Section 3.2]. A preliminary construction of the diagonal curve f P,d (P 1 ) for maximal parabolic subgroups P and degrees 0 ≤ d ≤ d X is given there and it is stated (without rigorous proof) that f P,d has a dense open orbit in M 0,3 (X, d) (cf. [12,Proposition 3.1]). Preliminary attempts to give a proof of this theorem for the degree d = d X were undertaken in [2,Section 9]. In this series of work, the present paper can be seen as a more rigorous and more final attempt to prove quasi-homogeneity for arbitrary parabolic subgroups and arbitrary minimal degrees.
The very idea of the construction of the diagonal curve f P,d (P 1 ) goes back to [12]. However, in this work, we make clear in a general context to which set of strongly orthogonal roots, namely B R,e where e is the lifting of a minimal degree d, one has to associate the morphism f P,d to -a question which is left open in [12].
In particular, the combinatorial aspects of minimal degrees which lead to the essential properties of generalized cascades of orthogonal roots and make our results eventually possible were already well prepared in [1]. The reader can consider this paper, in particular Section 3, 4, as a follow-up which completes some aspects of the theory developed in [1].
Organization. As being said in the history section, a reader only interested in the combinatorial aspects of minimal degrees and generalized cascades of orthogonal roots may read Section 3, 4 independently from the rest of the text. Moreover, most (or even all) of the results in Section 5, 6 are only interesting in the relative setting modulo P whenever P = B. The reader can check for each of those results that the statements turn out to be trivial if P = B.
Section 6 deals with the theory of liftings. The step of passing from a degree d ∈ Π P to its lifting e ∈ Π B is superfluous if P = B, in the sense that the lifting of e ∈ Π B is e itself. Therefore, the construction of f B,e for e ∈ Π B simplifies reasonably. Only the theory of generalized cascades of orthogonal roots is necessary to understand it.
All in all, a reader only interested in Theorem 1.7 for degrees which satisfy Assumption 7.13(2), i.e. in the theorem that M 0,3 (G/B, e) is quasi-homogeneous under the action of G for all minimal degrees e ∈ Π B , can skip Section 5, 6 and go directly from Section 4 to Section 7. For the case of a generalized complete flag variety G/B and minimal degrees e ∈ Π B , the proof of Theorem 1.7 simplifies a lot and relies only on our considerations in Section 4.

Notation and conventions
In this section, we set up basic notation and summarize well-known terminology concerning the theory of algebraic groups. The notation and the conventions are very similar to those in [1, Section 1]. We recall everything which is necessary to understand this paper. For more details, the reader may go to [1].
Let R be the root system associated to G and T . Let R + be the positive roots of R associated to B. Let ∆ be the set of simple roots associated to R + . Let W = N G (T )/T and W P = N P (T )/T be the Weyl group of G and P respectively. Let ∆ P = {β ∈ ∆ | s β ∈ W P }. We set R P = R ∩ Z∆ P and R + P = R P ∩ R + . The positive roots R + clearly induce a partial order "≤" on R. In turn, this partial order induces via restriction a partial order on R P which is still denotes by "≤" and coincides with the partial order induced by R + P . Notation 2.1. We denote by R − the set of negative roots of R. In other words, we have From now on, if we speak about a parabolic subgroup, we always mean what is usually called a standard parabolic subgroup (relative to the fixed B), i.e. a parabolic subgroup of G containing B. In other words, by convention, all parabolic subgroups are standard. Following this convention, parabolic subgroups of G correspond one to one to subsets of ∆ (cf. [7, 30.1]). In particular, the fixed parabolic subgroup P corresponds to ∆ P as defined above. For an arbitrary (standard) parabolic subgroup Q, we denote by ∆ Q the subset of ∆ associated to Q via this correspondence. Vice versa, for a given subset S of ∆, we often uniquely define a parabolic subgroup Q by the requirement ∆ Q = S.
On the Weyl group, we have a natural length function. For w ∈ W , the length of w, denoted by ℓ(w), is defined to be the number of simple reflections in a reduced expression of W . It is well-known that this number does not depend on the choice of the reduced expression. Each coset wW P ∈ W/W P has a unique minimal and maximal representative, i.e. contains a unique element of minimal and maximal length. The length function carries over from W to W/W P . The length of a coset wW P ∈ W/W P , denoted by ℓ(wW P ), is defined to be the length of the minimal representative in wW P . Notation 2.3. For each element w ∈ W , we denote by I(w) = {α ∈ R + | w(α) < 0} the inversion set of w. With this notation, we have the identities ℓ(w) = card(I(w)) and ℓ(wW P ) = card(I(w) \ R + P ) for all w ∈ W . A proof of these equalities can be found in [8, 5.6, Proposition (b)].
Notation 2.4. We denote by w o the longest element of W , i.e. the unique element of W with maximal length. Similarly, we denote by w P the longest element of W P , i.e. the unique element of W P with maximal length. Note that w o and w P are both involutions.
On the Weyl group, we have a natural partial order " " -the so-called Bruhat. This partial order can be defined in terms of the Bruhat graph as in [8, 5.9]. It has an equivalent characterization in terms of subexpressions as described in [8, 5.10]. The geometric meaning of the Bruhat order is given by inclusions of Schubert varieties in G/B. We explain this geometric meaning in more detail once we have recalled the notion of Schubert varieties in general (cf. Remark 2.6).
Convention 2.5. All homology and cohomology groups in this paper are taken with integral coefficients. By convention, we write H * (X) = H * (X, Z) and H * (X) = H * (X, Z). For a closed irreducible subvariety Z ⊆ X, we denote by [Z] ∈ H 2codim(Z) (X) the cohomology class of Z. By abuse of notation, we also denote with the same symbol [Z] ∈ H 2dim(Z) (X) the homology class of Z. Both definitions are Poincaré dual to each other.
We denote by X w = BwP/P the Schubert variety associated to w. We denote by Y w = B − wP/P the opposite Schubert variety associated to w. Note that X w and Y w depend only on wW P . We have the following equality for the dimension and codimension of Schubert and opposite Schubert varieties: Remark 2.6. Let w ∈ W . If we denote by (G/B) w the Schubert variety in G/B associated to w, the geometric meaning of the Bruhat order is given by the equivalence: Let w ∈ W . Using X w and Y w we can define Schubert cycles From the Bruhat decomposition of X (cf. [7,28.3,Theorem]) it follows easily that the cohomology of X decomposes as direct sums where each of the direct sums in the equation ranges over all minimal representatives w of the cosets in W/W P . Poincaré duality transforms one basis of Schubert cycles into the other basis of Schubert cycles and vice versa. Using Equation (1) and (2), we see that we have the following decompositions In this work, we will be very much concerned with elements of H 2 (X) and H 2 (X). Therefore, it is useful to use identifications as in [4,Section 2]. For a simple root β ∈ ∆ \ ∆ P , we will always identify the Schubert cycles σ(s β ) with β ∨ + Z∆ ∨ P ∈ Z∆ ∨ /Z∆ ∨ P and σ s β with the fundamental weight ω β . Using these identification, we will simply write Equation (3) as (4) H 2 (X) = Z∆ ∨ /Z∆ ∨ P and H 2 (X) = Z{ω β | β ∈ ∆ \ ∆ P } . Under these identifications, the Poincaré pairing H 2 (X) ⊗ H 2 (X) → Z simply becomes the restriction of the W -invariant scalar product (−, −) on R∆. Note that H 2 (X) and H 2 (X) are naturally endowed with a partial order "≤" which is given by comparing all the coefficients of the Z-bases pointed out in Equation (3).

Convention 2.7.
If we speak about a degree, without further specification, then we always mean an effective class in H 2 (X). If we speak about a degree in H 2 (G/B), we mean an effective class in H 2 (G/B). In the later case, where the lattice might be different from H 2 (X), we explicitly mention it in our terminology. We reserve the term degree without specification for degrees in H 2 (X).
Notation 2.8. Let α ∈ R + . One degree associated to α will be ubiquitous in our discussion. By definition, the degree d(α) is given by the equation . Note that the degree d(α) depends not only on α but also on P although P is not explicitly mentioned in the notation d(α). No confusion will arise from this sloppiness since we refer always to one and the same parabolic subgroup P which is fixed throughout the discussion. The geometric meaning of the degree d(α) will become clear later in the context of irreducible T -invariant curves (cf. Notation 7.2). Notation 2.9. We denote by c 1 (X) the first Chern class of the tangent bundle on X. The Chern class c 1 (X) is an element of H 2 (X) which will be often in use in this paper. For explicit computations, it is useful to have a description of c 1 (X) in terms of the root system. According to [6,Lemma 3.5] and the identifications made in Equation (4), we have c 1 (X) = γ∈R + \R + P γ.

Preliminaries
In this section, we set up the preliminaries which are used throughout the rest of the text. In particular, this section completes some aspects of the theory developed in [1]. These aspects concern the properties of degrees with pairwise totally disjoint extended support (cf. Subsection 3.2). This situation was of no relevance for the analysis in [1] but becomes increasingly important for the purpose of this paper.
It should be clear that many notions and ideas we use originally go back to [4]. We rely in various ways on the theory of curve neighborhoods developed in [4]. We review everything we need from this theory in Subsection 3.1.
3.1. Curve neighborhoods. In this subsection, we review everything we need from the theory of curve neighborhood due to [4]. For examples, proofs and more informations, we encourage the reader to look at [4].   [4,Section 4]. This result plays an important role for the question of well-definedness of other objects described in terms of greedy decompositions. We will use it several times in this work. Although it is a non-trivial result due to [4], we will often use it without explicitly referring to the authors. The last sentence concerns in particular Section 4. Theorem 3.4. Let α be a P -cosmall root. Then we have (α, γ) = 0 for all γ ∈ R + P \ I(s α ). Proof. We first prove a seemingly weaker statement.
Let v = s β 1 · · · s β l be any reduced expression for v. Then we define the Hecke product of u and v by u · v = u · s β 1 · . . . · s β l .
That the expression u · v is well-defined (independent of the choice of the reduced expression for v) is proved in detail in [4, Section 3].
The reader finds a list of the most important properties of the Hecke product in [4, Proposition 3.1]. Among other properties, this list contains the associativity of the Hecke product ([1, Proposition 3.1(a)]). Mostly, we need the Hecke product to define for any degree d a Weyl group element z P d which captures the geometric properties of d. Definition 3.6 ([4, Section 4.2]). Let d be a degree. Let (α 1 , . . . , α r ) be a greedy decomposition of d. Then we define an element z P d ∈ W by the following equation z P d w P = s α 1 · . . . · s αr · w P . It is easy to see that z P d is the minimal representative in z P d W P (cf. Remark 3.7. Most or all intuitions concerning the element z P d for a degree d come from its geometric meaning which is illuminated in [4,Theorem 5.1]. This theorem says among other things that the degree d curve neighborhood of the zero dimensional Schubert variety X 1 is itself a Schubert variety parametrized by the element z P d . We will use [4, Theorem 5.1] in this weak form precisely once in this paper, namely in the proof of Theorem 8.2.

Supports.
This subsection is about various kinds of notions of supports. Most notably, we recapitulate the notion of the extended support of a degree which was first introduced in [1, Subsection 3.1]. It turns out that the extended support is the right way to extend the naive support with the proper amount of simple roots in ∆ P to get a useful notion. In this way, we can formulate a disjointness assumption (namely that the extended supports of two degrees are totally disjoint 2 ) which enables us to prove certain addition theorems in Subsection 3.6 (see for example Theorem 3.42).
Notation 3.8. Let α ∈ R + . Then we denote by ∆(α) the support of α, i.e. the set of simple roots β ∈ ∆ such that β ≤ α. Notation 3.9. Let α ∈ R + . Then we denote by ∆ • α the set of all simple roots which are orthogonal to α; in formulas ∆ • α = {β ∈ ∆ | (α, β) = 0}. Definition 3.10 ([1, Subsection 3.1]). Let d be a degree. We define the naive support of d to be the set ∆(d) of all simple roots β ∈ ∆ \ ∆ P such that (ω β , d) > 0. We define the extended support of d to be the set ∆ defined as the union ∆(d) = r i=1 ∆(α i ) where (α 1 , . . . , α r ) is a greedy decomposition of d. The extended support is clearly well-defined since the greedy decomposition is unique up to reordering.
Remark 3.11. Let d be a degree. We have the following trivial relations between the naive and the extended support: In particular, for a degree e ∈ H 2 (G/B), we have ∆(e) = ∆(e).  Convention 3.14. As we consider the empty set as a connected subset of the Dynkin diagram, the degree d = 0 is considered as a connected degree with empty greedy decomposition. In particular, there does not exist a unique first entry α(d) of the degree d = 0. We mentioned this exception once in Notation 3.13. To avoid trivial considerations, we will from now on tacitly assume that a connected degree d is nonzero whenever we speak about α(d). The reader can convince himself that the case d = 0 can be treated in a trivial way in all proofs we do: All statements about the connected degree d = 0 are obvious right in the beginning, although we will not explicitly say this each time.
Definition 3.15. Let w ∈ W . We define the support of w to be the set ∆(w) of all simple roots β ∈ ∆ such that s β w.
The reader finds a list of the most important properties of the support of a Weyl group element in [1,Proposition 3.17]. For later use, we add one further property which is subject to Lemma 3.16. Lemma 3.16. Let u, v ∈ W such that ∆(u) and ∆(v) are disjoint. Then we have u · v = uv.
Proof. Let Q be the parabolic subgroup of G such that ∆ Q = ∆(u). Let Q ′ be the parabolic subgroup of G such that ∆ Q ′ = ∆(v). By definition, we clearly have u ∈ W Q and v ∈ W Q ′ . We also have v −1 ∈ W Q ′ since ∆(v −1 ) = ∆(v). By [8, 5.5, Theorem (b)], it follows that The statement now follows from [4, Proposition 3.2: 3.3. Local notions. Local notions were studied intensively in [1,Section 6]. In particular, it was shown in [1, Theorem 6.10] that minimal degrees in quantum products behave well under "localization". In this work, we will only speak about locally high roots and about the root subsystem R(ϕ) of R associated to a positive root ϕ. We introduce these notions in this subsection.
Notation 3.17. Let ϕ be a positive root. We denote by R(ϕ) the root subsystem of R generated by ∆(ϕ). Since ∆(ϕ) is a connected subset of the Dynkin diagram, the root system R(ϕ) is always irreducible.    Proof. It is very easy to supply a proof of Fact 3.21. We leave the details to the reader. Proof. Let ϕ be a locally high root. Let α ∈ R + be a root such that ϕ ≤ α and α ∨ ≤ ϕ ∨ . In order to prove that ϕ is B-cosmall, we have to show that α = ϕ. The inequality α ∨ ≤ ϕ ∨ implies that ∆(α) ⊆ ∆(ϕ) and thus α ∈ R(ϕ). By definition, the root ϕ is the highest root of R(ϕ). Consequently, we necessarily have α ≤ ϕ. In total, this implies that α = ϕ -as required.
3.4. Connected components of a degree. As we already introduced a notion of connected (and disconnected) degrees, it is plausible also to introduce a notion of connected components of a degree. We do so in this subsection. The notion of connected components of a degree is useful because many characteristics of a degree, such as its set of maximal roots (cf. Theorem 3.36) or the number of its greedy decompositions (cf. Theorem 3.45), are already determined by its connected components.
Definition 3.23. Let d be a degree. Let ϕ 1 , . . . , ϕ k be locally high roots such that are the distinct connected components of ∆(d). Let (α 1 , . . . , α r ) be a greedy decomposition of d. Then we define the connected components of d to be the degrees Since the greedy decomposition of d is unique up to reordering, the connected components d 1 , . . . , d k of d are clearly well-defined -do not depend on the choice of the greedy decomposition but only on d.
Lemma 3.24. Let d be a degree. Let d 1 , . . . , d k be the connected components of d. Then . . , ϕ k be locally high roots such that ∆(ϕ), . . . , ∆(ϕ k ) are the distinct connected components of ∆(d). Let (α 1 , . . . , α r ) be a greedy decomposition of d. It is clear that (7)] and by definition of ∆(d i ), we have and consequently If one of the inclusions in Equation (5) is strict, it follows from Equation (6) that Therefore, we conclude that ∆(d i ) = ∆(ϕ i ) for all 1 ≤ i ≤ k, in other words that are the distinct connected components of ∆(d). Since each ∆(d i ) is in particular connected, it follows that each d i is a connected degree.

3.5.
Minimal degrees in quantum products. In this subsection we introduce the class of minimal degrees. We organize these degrees in a set Π P where the P indicates the parabolic subgroup relative to which they are computed. Minimal degrees are central because they feature many important properties (most prominently orthogonality relations, cf. [1, Section 8]) which eventually lead to the quasi-homogeneity result which is subject to this paper. We will elaborate on the consequences of the properties of a minimal degree e ∈ Π B for the greedy decomposition of e and the element z B e in Section 4. Minimal degrees naturally arise in the context of quantum cohomology as minimal degrees of quantum product (cf. [1]). In our exposition, we choose a completely combinatorial definition of minimal degrees which makes only use of the theory of curve neighborhoods surveyed in Subsection 3.1. We do so to keep the prerequisites as low as possible. Moreover, for all of our proofs the combinatorial approach is more suitable. The relation to quantum cohomology is mentioned only in a few remarks intended to the reader familiar with this theory and not needed to understand the purpose of this paper.
Definition 3.25. Let d be a degree. We say that d is a minimal degree if d is a minimal element of the set Notation 3.26. We denote by Π P the set of all minimal degrees. In particular, the set of all minimal degrees in Remark 3.27 (For the reader familiar with quantum cohomology). Let (QH * (X), ⋆) be the (small) quantum cohomology ring attached to X as defined in [5,Section 10] . In terms of quantum cohomology, the set Π P can be described as follows: The definition of a minimal degree in the quantum product of two Schubert cycles is given in [1, Definition 5.14]. The inclusion "⊆" in Equation (7) follows since a degree d ∈ Π P is a minimal degree in σ z P d ⋆ pt (cf.  {d a degree such that w o W P = z P d W P } . This unique minimal element is denoted by d X . By definition, we clearly have d X ∈ Π P . In particular, we have d G/B ∈ Π B . Remark 3.29. The minimal degree d X is the main object of study of [1] and therefore very well understood. In particular, it can be explicitly computed (cf. [1, Corollary 7.12]). (1) Let α be a P -cosmall root.
This follows by repeated application of [1, Proposition 4.4 (9)]. (5) Let d 1 , . . . , d k ∈ Π P be minimal degrees such that ∆(d 1 ), . . . , ∆(d k ) are pairwise totally disjoint. Then we have k i=1 d i ∈ Π P . We will prove this statement later on (cf. Theorem 3.42). 3.6. Addition theorems. In this subsection we present various kinds of addition theorems. By an addition theorem, we mean a theorem which expresses properties of the sum d 1 + d 2 of two degrees (or more generally of k degrees) in terms of properties of the individual degrees d 1 and d 2 . In most cases, we do so by assuming from the beginning that ∆(d 1 ) and ∆(d 2 ) are totally disjoint. In this way, we are for example able to describe all greedy decomposition of d 1 + d 2 in terms of greedy decompositions of d 1 and d 2 whenever the extended supports are totally disjoint (cf. Theorem 3.34, 3.35). In particular, this gives us a way to reduce many problems, e.g. testing the minimality of a degree (cf. Theorem 3.42 and Corollary 3.43), from an arbitrary degree d to a connected degree by passing to the connected components of d.
Proof. We first reduce the theorem to the case of a connected degree d. Assume that the assertion is true for all connected degrees d. Let d 1 , . . . , d k be the connected components of d. By Lemma 3.24 we have Again, by Lemma 3.24, we then find that ∆(d) ⊆ ∆(d ′ ) (cf. Equation (6)).
Without loss of generality, we may assume that d is a connected degree. Let α = α(d).
Theorem 3.32 (Addition theorem for extended supports). Let d and d ′ be two degrees. Then Remark 3.33. The corresponding addition theorem for naive supports is absolutely trivial and also follows immediately from Theorem 3.32 by subtracting ∆ P . Therefore, we can think of Theorem 3.32 as a more refined statement.
Let α = µ∈∆(α) n µ µ be the expression of α as a linear combination of simple roots. Since α is a positive root (we even have α ∈ R + \ R + P ), we know that n µ > 0 for all µ ∈ ∆(α). By definition of α, we have β ′ ∈ ∆(α). Hence, we can write We already figured out that (β, β ′ ) < 0 and (α, β) ≥ 0. In view of these inequalities and the previous displayed equation, we find that Therefore, there must exist a µ ∈ ∆(α) \ {β ′ } such that (µ, β) > 0. Since all non-diagonal entries of a Cartan matrix are non-positive, it follows that µ = β and thus β ∈ ∆(α) ⊆ ∆(d) ⊆ ∆. This is a contradiction since by definition β ∈ S and β / ∈ ∆. This proves the claim. △ Let α be a root which occurs in a greedy decomposition of d + d ′ . By definition, we have Since ∆(α) is necessarily connected, the previous claim shows that either ∆ Since α was an arbitrary entry in a greedy decomposition of d + d ′ , the definition of the extended support shows that ∆(d + d ′ ) ⊆ ∆. This means that S = ∅ and ∆(d + d ′ ) = ∆ -as claimed.
. . , α r ) be a sequence of roots such that (α i 1 , . . . , α i r i ) is a subsequence of (α 1 , . . . , α r ) for all 1 ≤ i ≤ k. Then the sequence (α 1 , . . . , α r ) is a greedy decomposition of k i=1 d i . Proof. We first assume that k = 2 and prove the theorem for that case. We do so by induction on r. The case where r 1 = 0 or r 2 = 0 is obvious. Assume that r 1 > 0 and r 2 > 0. This means that r ≥ 2. Assume further that the statement is known for sequences of length strictly less than r. By renaming the indices if necessary (i ∈ {1, 2}), we may assume that Since ∆(d 1 ) and ∆(d 2 ) are totally disjoint and since ∆(α ′ ) is connected, it follows that ether which contradicts the assumption that ∆(d 1 ) and ∆(d 2 ) are totally disjoint. Therefore, we conclude that ∆(α ′ ) ⊆ ∆(d 1 ). In view of this inclusion, the inequality d(α ′ ) ≤ d implies that d(α ′ ) ≤ d 1 . By definition, α is a maximal root of d 1 . This means that we must have α = α ′ (since α ≤ α ′ and d(α ′ ) ≤ d 1 ). By the choice of α ′ , we have now proved that α is a maximal root of d -as claimed. △ In view of the claim, (α 1 , . . . , α r ) is a greedy decomposition of d if and only if (α 2 , . . . , α r ) is a greedy decomposition of d − d(α) = (d 1 − d(α)) + d 2 . We now apply the induction hypothesis to the sequence (α 2 , . . . , α r ) to see that the latter statement is true. (This is possible since (α 1 2 , . . . , α 1 r 1 ) is a greedy decomposition of d 1 − d(α), (α 2 1 , . . . , α 2 r 2 ) is a greedy decomposition of d 2 , (α 1 2 , . . . , α 1 r 1 ) and (α 2 1 , . . . , α 2 r 2 ) are subsequences of (α 2 , . . . , α r ), and since ∆( We now prove the statement for arbitrary k by induction. The case k = 1 is obvious. The case k = 2 was treated above. Assume that k > 2 and that the statement is known for all positive integers strictly less than k.
In other words, every greedy decomposition of k i=1 d i can be constructed in the way described in Theorem 3.34 and uniquely determines the greedy decompositions of d i for all By Theorem 3.32 and assumption, it is clear that we have Again by the totally disjointness assumption, it is therefore clear that for all 1 ≤ j ≤ r there exists a unique 1 ≤ i ≤ k such that ∆(α j ) ⊆ ∆(d i ). It follows that For all 1 ≤ i ≤ k, we are now forced to define (α i 1 , . . . , α i r i ) as the maximal subsequence of (α 1 , . . . , α r ) such that ∆(α i j ) ⊆ ∆(d i ) for all 1 ≤ j ≤ r i . This already proves the uniqueness part of the statement. By [1, Proposition 3.10 (7)] it also follows that (α i 1 , . . . , α i r i ) is indeed a greedy decomposition of d i for all 1 ≤ i ≤ k. By definition, we also have r = k i=1 r i . Theorem 3.36. Let d be a degree. Let d 1 , . . . , d k be the connected components of d. Then there exist precisely k maximal roots of d, namely α(d 1 ), . . . , α(d k ).
Proof. It is clear that α(d 1 ), . . . , α(d k ) are pairwise distinct, since their supports are even pairwise totally disjoint (cf. Lemma 3.24). Therefore, it suffices to show the equality of sets We first prove the inclusion "⊇". Let 1 ≤ i ≤ k. Each greedy decomposition of d i has as unique first entry α(d i ). Consequently, Theorem 3.34 applied to d 1 , . . . , d k shows that there exists a greedy decomposition of d which has as first entry α(d i ). (The assumption of Theorem 3.34 is satisfied in view of Lemma 3.24.) This means that α(d i ) is a maximal root of d. Next, we prove the inclusion "⊆". Let α be a maximal root of d. This means that there exists a greedy decomposition of d which has as first entry α. By Theorem 3.35 applied to d 1 , . . . , d k there exists an 1 ≤ i ≤ k and a greedy decomposition of d i such that the first entry of this greedy decomposition is α. (Again, the assumption of Theorem 3.35 is satisfied in view of Lemma 3.24.) In other words, we have α = α(d i ) for some 1 ≤ i ≤ k.
Proof of Corollary 3.37 and Lemma 3.38. In order to prove Corollary 3.37, it clearly suffices to prove Lemma 3.38. This is because of Theorem 3.36 and the uniqueness of the greedy decomposition up to reordering. But Lemma 3.38 follows directly from Theorem 3.34 applied to d 1 , . . . , d k . The assumption of Theorem 3.34 is saitsfied in view of Lemma 3.24.
In particular, it follows from [4, Corollary 4.9] thatz P d is an involution. It can also be seen more directly thatz P d is an involution by using [4, Proposition 3.1(b)] and [1, Proposition 3.10(2)].
This means in particular that the Hecke and the ordinary product in Equation (9) is independent of the ordering of the factors.
Proof. Let (α i 1 , . . . , α i r i ) be a greedy decomposition of d i for all 1 ≤ i ≤ k. By Theorem 3.34, the sequence (α 1 1 , . . . , α 1 r 1 , . . . , α k 1 , . . . , α k r k ) is a greedy decomposition of d. Hence, the first equality in Equation (9) follows directly from the definition ofz P d . We prove the second equality of Equation (9) by induction on k. The case k = 1 is immediate. Assume that k > 1 and that the second equality of Equation (9) is known for all values strictly smaller than k. By induction hypothesis applied to d 2 , . . . , d k , it suffices to show thatz P  Proof. The implication from right to left follows from Theorem 3.35 and [1, Proposition 4.4 (9)]. Assume that d 1 , . . . , d k ∈ Π P . A simple induction on k shows that we may assume that k = 2. Let d ′ ∈ Π P be a minimal degree such that d ′ ≤ d and such that z P by the totally disjointness assumption. Therefore, we can write are two degrees such that d ′ By changing the order of the product in Equation (10) we also find the analogous equatioñ Using the last two displayed equations and the computation rules for the support of a Weyl group element (cf. [1, Proposition 3.17]) and the Hecke product (cf. [4, Proposition 3.1]), we see that Then we have the following recursive formula for the number of greedy decompositions of d which expresses N d in terms of N d ′ with d ′ < d: {(α 1 , . . . , α r ) | (α i 1 , . . . , α i r i ) is a subsequence of (α 1 , . . . , α r ) for all i} .
The disjoint union in this formula runs over N d 1 · · · N d k sets of equal cardinality and each of these sets has cardinality equal to the number of ways how to combine k disjoint sequences of length r 1 , . . . , r k respectively to a unique sequence of length r = k i=1 r i . But this number is precisely r! r 1 !···r k ! -a so-called multinomial coefficient. This directly leads to Equation (11) by taking cardinalities in the previous displayed equation.
Using the previous formula for r p and Theorem 3.45, a simple induction for the odd and the even case shows Indeed, we have the initial values N 3 = 1, N 4 = 3! = 6 and the recursive relation which results from Theorem 3.45.

Generalized cascades of orthogonal roots
In this section, we develop the theory of generalized cascades of orthogonal roots. This theory is a direct generalization of Kostant's cascade of orthogonal roots [11,Section 1] in the sense that Kostant's cascade is associated to the specific minimal degree d G/B ∈ Π B while the general construction works for an arbitrary minimal degree e ∈ Π B (for more details see Remark 4.2). Basically all properties of Kostant's cascade as they were investigated in [11, loc. cit.] carry over in a reasonable way to generalized cascades as the theorems in this section show. But the proofs are less evident. As a key feature of minimal degrees, we need the so-called orthogonality relations in greedy decompositions of minimal degrees which were first proved in [1,Theorem 8.1]. With the help of this property of minimal degrees, it is easy to see that two distinct elements of a generalized cascade of orthogonal roots are strongly orthogonal (Theorem 4.5 (3)). This kind of orthogonality is essential to make the diagonal curve (cf. Definition 7.3) well-defined and eventually to prove our main result on quasi-homogeneity. We call the set B R,e of positive roots a generalized cascade of orthogonal roots. Furthermore, we define C R,e (ϕ) = {α ∈ B R,e | α ≥ ϕ} . We call the subset C R,e (ϕ) of B R,e a generalized chain cascade. We define the length of a generalized chain cascade to be the cardinality of a generalized chain cascade.    (1) The generalized chain cascade C R,e (ϕ) is totally ordered.
(2) All elements of the generalized cascade of orthogonal roots B R,e are B-cosmall.
Then the sets R(α) and R(α ′ ) are totally disjoint. Proof of Item (1). We prove this statement by induction on the length of generalized chain cascades. For any generalized chain cascade of length one, the statement is trivially satisfied. Let n > 1 be an integer and suppose that any generalized chain cascade of length less than n is totally ordered. Let C R,e (ϕ) be an arbitrary generalized chain cascade of length n for some e ∈ Π B and some ϕ ∈ R + . Letê = α∈C R,e (ϕ) α ∨ .
Proof of Item (2). Let α ∈ B R,e . By definition, α occurs in a greedy decomposition of e. This means, there exists a degree e ′ ∈ H 2 (G/B) such that α ∨ ≤ e ′ ≤ e and such that α is a maximal root of e ′ . This implies in particular that α is B-cosmall. (3). The statement is a consequence of the assumption e ∈ Π B which leads to a phenomenon which we call "orthogonality relations in greedy decompositions" (cf. [1, Section 8]. By [1, Corollary 8.3], we know that two different entries of a greedy decomposition of e ∈ Π B are strongly orthogonal, in particular we know that two distinct elements of B R,e are strongly orthogonal.
The previous claim shows in particular that α, α ′ ∈ B R,ê .
Claim: We have C R,ê (α) ∩ C R,ê (α ′ ) = ∅. Suppose for a contradiction that By Equation (14), then there exists a ϕ ′ ∈ B R,e such that ϕ ′ < ϕ and such that ϕ ′ ≥ α and ϕ ′ ≥ α ′ . This means that ϕ ′ is an element of C which is strictly smaller than ϕ -contrary to the choice of ϕ as the unique minimal element of C. △ By the previous claim, we can apply Item (4) to α, α ′ ∈ B R,ê and get that R(α) and R(α ′ ) are totally disjoint -as desired.   Proof of Theorem 4.7. Let (α 1 , . . . , α r ) be a greedy decomposition of e. For brevity, let α = α 1 . By [1, Proposition 4.4(9)], we know that e − α ∨ ∈ Π B . By [1, Proposition 3.10 (7)], we know that B R,e−α ∨ = {α 2 , . . . , α r }. Moreover, since e ∈ Π B , we know that there are no repeated entries in a greedy decomposition of e ([1, Remark 8.4]). Hence, it follows that We now perform an induction on the length of the greedy decomposition of e. The statement of Theorem 4.7 is obvious whenever the greedy decomposition of e has length zero, i.e. if e = 0 and B R,e = ∅. Assume that r > 0 and that the statement is known for all minimal degrees in Π B whose greedy decomposition has length less than r. By [1, Proposition 3.10(7)], a greedy decomposition of e − α ∨ (e.g. (α 2 , . . . , α r )) has length r − 1. Hence, the induction hypothesis applies to e − α ∨ . In view of Equation (15), we find that This completes the proof of the claim. △ We are now able to prove Equation (18) which will complete the proof of Theorem 4.7. Let µ ∈ I(z B e−α ∨ ). By the previous claim, we have µ ∈ R + Q . By definition of Q, this means that α is orthogonal to µ. Thus, we have s α (µ) = µ > 0 and µ / ∈ I(s α ). This means that Equation (18)    Proof of Theorem 4.10. Let us first note that the second formula follows directly from the first by subtracting R + P . Indeed, we have I(s α ) ⊆ R + P for all α ∈ B R,e ∩ R + P by [8, 5.5, Theorem (b)]. We now prove the first formula.
Let (α 1 , . . . , α r ) be a greedy decomposition of e. For brevity, let α = α 1 . The situation is now precisely the same as in the proof of Theorem 4.7 and we can freely use the formulas we worked out there. By Equation (17) . We now perform an induction on the length of the greedy decomposition of e -the first formula of Theorem 4.10 being obvious whenever the greedy decomposition of e has length zero. By the induction hypothesis applied to e − α ∨ and Equation (15), we find that

Combined with the last equation, Inclusion (19) gives the inclusion
To prove that this inclusion is actually an equality (and hence Theorem 4.10), it suffices to show that the left and right set have the same cardinality. But this follows from Theorem 4.7 and the triangle inequality: Corollary 4.12 (Length additivity in generalized cascades of orthogonal roots). Let e ∈ Π B . Then we have the following formulas: Proof. This corollary follows directly from Theorem 4.10 by taking the cardinality of the involved sets.

Positivity in generalized cascades of orthogonal roots
In this section, we work out a positivity statement in generalized cascades of orthogonal roots (Theorem 5.1). This theorem closes the general properties of generalized chain cascades which we started to investigate in Section 4. We apply this theorem to degrees e ∈ Π B where z B e is the maximal representative in z B e W P (cf. Theorem 5.5). In Section 6, we set up a framework where such degrees naturally occur. Namely, they occur as the lifting of a degree d ∈ Π P (cf. Fact 6.5(2)). Finally, we give an application specific to the combinatorics in type A which might be useful in different contexts (Theorem 5.7). This section is not strictly necessary to understand the main result on quasi-homogeneity. The impatient reader can skip it.
Remark 5.2. Note that the choice of e ∈ Π B in Theorem 5.1 does not depend on P . Thus, for a fixed e ∈ Π B , we get a bunch of positivity results by varying P .
Remark 5.4. Lemma 5.3 says in particular that all or all except one inequality in Theorem 5.1 are actually equalities. This follows from the fact that I(z B e ) ⊆ R + P whereê depends on e and P as in Theorem 5.1. We saw this fact in the proof of Theorem 5.1. It is also easy to deduce more directly without invoking Theorem 4.10.
Proof of Lemma 5.3. Let e and γ be as in the statement. Suppose for a contradiction there exist two distinct roots α, α ′ ∈ B R,e \ R + P such that (α, γ) > 0 and (α ′ , γ) > 0. In view of the assumption, it is easy to see that γ ∈ I(s α ) ∩ I(s α ′ ). (Indeed, s α (γ) and s α ′ (γ) must contain a simple root in ∆ \ ∆ P with negative coefficient in their expression as linear combination of simple roots.) This contradicts the disjointness result in Theorem 4.10.
Theorem 5.5. Let e ∈ Π B . Suppose that z B e is the maximal representative in z B e W P . Let γ ∈ R + P . Suppose there exists an α ∈ B R,e \ R + P such that (α, γ) < 0. Then there exists an α ′ ∈ B R,e \ R + P such that (α ′ , γ) > 0. Remark 5.6. Lemma 5.3 shows that the α ′ in Theorem 5.5 is unique. {γ ∈ R + P | ∃α ∈ B R,e \ R + P : (α, γ) < 0} ⊆ R + P \ I(z B e ) . By assumption z B e is the maximal representative in z B e W P . Therefore we have R + P ⊆ I(z B e ). Theorem 4.10 and [1, Proposition 3.10(7), 4.4 (9)] show that I(z B e ) = I(z B e ) ∐ I(z B e ). Both facts together yield that R + P \ I(z B e ) = I(z B e ) ∩ R + P . Finally, it is easy to see that (25) I(z B e ) ∩ R + P = {γ ∈ R + P | ∃α ∈ B R,e \ R + P : (α, γ) > 0} . (This equation is a direct consequence of Theorem 4.10 applied toẽ. One only has to note that we have B R,ẽ = B R,e \ R + P by [1, loc. cit.]. For the proof of the inclusion "⊇" one may use similar arguments as in the proof of Lemma 5.3. Note also, as a consequence of Lemma 5.3, we can equally well write ∃!α instead of ∃α on the right side of Equation (25).) Inclusion (24) and Equation (25) together yield the inclusion {γ ∈ R + P | ∃α ∈ B R,e \ R + P : (α, γ) < 0} ⊆ {γ ∈ R + P | ∃α ∈ B R,e \ R + P : (α, γ) > 0} . This inclusion shows all we claimed in the statement.
Theorem 5.7. Assume that R is of type A. Let e ∈ Π B . Suppose that z B e is the maximal representative in z B e W P . Let γ ∈ R + P . Then there exists at most one α ∈ B R,e \ R + P such that (α, γ) < 0.
Remark 5.8. Note that Theorem 5.7 is specific to type A. In general, there may exist several α's as in the statement of Theorem 5.7. The reader can find examples for this behavior in type D 4 or type E 6 for the degree e = d G/B ∈ Π B .
Proof of Theorem 5.7. Assume that R is of type A n for some n ≥ 1. Let ∆ = {β 1 , . . . , β n } be the set of simple roots with the numbering as in [3, Plate I].
By Theorem 5.5 there exists an α ′′ ∈ B R,e \ R + P such that (α ′′ , γ) > 0. The root α ′′ is clearly distinct from α and α ′ . Write The inequality (α ′′ , γ) > 0 implies that we have either i * (α ′′ ) = i or i * (α ′′ ) = i. In the first case we have (α, α ′′ ) < 0. In the second case we have (α ′ , α ′′ ) < 0. Both conclusions are contrary to Theorem 4.5(3) according to which we have (α, α ′′ ) = (α ′ , α ′′ ) = 0. This shows that the conclusion of Theorem 5.7 holds for all γ ∈ ∆ P . △ Second case: Assume that γ ∈ R + P is arbitrary. By the first case, Lemma 5.3 and Theorem 5.5, we have α∈B R,e \R + P (β, α ∨ ) ≥ 0 for all β ∈ ∆ P . By summing over all β ∈ ∆(γ), we also obtain (26) Let l be the number of roots α ∈ B R,e \ R + P such that (α, γ) < 0. If l = 0, there is nothing to prove. Assume that l ≥ 1. We have to show that we even have l = 1. But by Lemma 5.3 and Theorem 5.5 we clearly have Together with Inequality (26) this gives l ≤ 1 and thus l = 1 -as desired. △ The second case shows that the conclusion of Theorem 5.7 holds for all γ ∈ R + P . Corollary 5.9. Assume that R is of type A. Let e ∈ Π B . Suppose that z B e is the maximal representative in z B e W P . Let γ ∈ R + P . Then we have Proof. This follows directly from Lemma 5.3 and Theorem 5.5, 5.7.

The lifting of a minimal degree
In this section, we introduce the notion of the lifting of a minimal degree. This notion basically prepares the construction of the diagonal curve in the relative setting modulo P by associating to a degree d ∈ Π P the generalized cascade of orthogonal roots B R,e where e ∈ Π B is the lifting of d. In case of a generalized complete flag variety G/B (i.e. P = B) the step of passing from a minimal degree to its lifting is superfluous. It is only necessary in a more general context.
The nontrivial input in this section is certainly [13,Corollary 3] which states that there exists a unique minimal degree in the quantum product of two Schubert classes in H * (G/B). This result guarantees in particular the uniqueness of the lifting of a minimal degree and also has other consequences concerning the uniqueness of minimal degrees in the relative setting modulo P which we investigate in the beginning of this section. Notation 6.1. Let d be a degree. Let Q be a parabolic subgroup of G containing P . We denote by d Q the image of d under the natural map H 2 (X) → H 2 (G/Q), i.e. we have d Q = d + Z∆ ∨ Q . We will mostly apply this notation for the relative situation B ⊆ P , i.e. if e ∈ H 2 (G/B) is a degree, we write e P = e + Z∆ ∨ P . 5 Definition 6.2. Let d ∈ Π P . By   (1) We have e ∈ Π B and z P d w P = z B e . (2) The element z B e is the maximal representative in z B e W P .  Corollary 6.7. Let d, d ′ ∈ Π P such that z P d = z P d ′ . Then we have d = d ′ . Proof. This is an immediate corollary of Theorem 6.6. Corollary 6.8. Let d ∈ Π P . Then d is the unique minimal element of the set Remark 6.9. That d ∈ Π P is a minimal element of the prescribed set is content of Definition 3.25. Corollary 6.8 makes a statement about the uniqueness which is nontrivial.
Remark 6.10. Corollary 6.8 shows in particular that d X is the unique minimal element of the set {d a degree such that w o W P = z P d W P } . This result was found independently from [13,Corollary 3] in [1, Definition 4.1, Theorem 4.7] (cf. Notation 3.28).
Remark 6.11 (For the reader familiar with quantum cohomology). By Remark 3.27, we know that a degree d ∈ Π P is a minimal degree in σ z P d ⋆ pt. In view of [1, Definition 4.1, Theorem 5.15], Corollary 6.8 says that a degree d ∈ Π P is even the unique minimal degree in σ z P d ⋆ pt. In particular, the degree d X is the unique minimal degree in pt ⋆ pt.
Proof of Corollary 6.8. Let d ∈ Π P . Let d ′′ be a minimal element of the set We have to show that d = d ′′ . By [1, Definition 4.1, Proposition 4.4(10)], we know that d ′′ ∈ Π P . By definition of d ′′ , we also have z P d z P d ′′ . Therefore, Theorem 6.6 implies that d ≤ d ′′ . By the minimality of d ′′ , this implies that d = d ′′ -as claimed. Proof of Corollary 6.12. Let d ∈ Π P . By definition, we have w o W P = z P d X W P and thus z P d z P d X . As d, d X ∈ Π P , Theorem 6.6 now implies that d ≤ d X . Theorem 6.14. Let d ∈ Π P . Let e be the lifting of d. Letê = α∈B R,e \R + P α ∨ . A root α is a maximal root of d if and only if α is a maximal root ofê.
Proof. Let α be a maximal root of d. First, we show that α is also a maximal root ofê. By definition, α is P -cosmall and also B-cosmall. Hence, we have z P d(α) w P = s α · w P and Claim: α is a maximal root of e. Indeed, let α ′ be a maximal root of e such that α ≤ α ′ . (As α ∨ ≤ e, we can clearly choose such an α ′ .) By definition, we have α ′∨ ≤ e and thus by Fact 6.5(3) that d(α ′ ) ≤ d. Since α ≤ α ′ and α ∈ R + \ R + P by definition, it is also clear that α ′ ∈ R + \ R + P . Since α is a maximal root of d, it follows that α = α ′ . In other words, α is a maximal root of e. △ As α is a maximal root of e, there exists a greedy decomposition of e such that α occurs as its first entry. This means in particular that α ∈ B R,e and since α ∈ R + \ R + P also that α ∈ B R,e \ R + P . By [1, Proposition 3.10(7), 4.4 (9)], we haveê ∈ Π B and B R,ê = B R,e \ R + P . Thus, we have α ∈ B R,ê and consequently α ∨ ≤ê ≤ e. Since α is a maximal root of e by the claim, the last inequality implies that α is also a maximal root ofê. This proves the implication from left to right.
To prove the other implication, let α be a maximal root ofê. We have to show that α is also a maximal root of d. By definition, there exists a greedy decomposition ofê such that α occurs as its first entry. This means that α ∈ B R,ê = B R,e \ R + P . (The last equality was already justified in the paragraph before.) In particular, we have α ∈ R + \ R + P . Moreover, as α ∨ ≤ê by definition, it follows from Fact 6.5(3) that d(α) ≤ê P = e P = d. (To see that e P = e P , just note that e = α∈B R,e α ∨ in view of [1,Remark 8.4].) These facts show that we can choose a maximal root α ′ of d such that α ≤ α ′ . By the first implication already shown, α ′ is also a maximal root ofê. Since α is also a maximal root ofê by our initial choice, it follows that α = α ′ . In other words, α is a maximal root of d. This proves the implication from right to left. Proof. Let d 1 , . . . , d k be the connected components of d. Letê 1 , . . . ,ê k ′ be the connected components ofê. By Theorem 3.36, 6.14, it follows that k = k ′ and By Notation 3.13 and Lemma 3.24, we also have The result follows from these facts. Theorem 6.16. Let α 1 , . . . , α k be P -cosmall roots such that ∆(α 1 ), . . . , ∆(α k ) are pairwise totally disjoint. Let d = k i=1 d(α i ) for short. Then we have d ∈ Π P . Let e be the lifting of d. Then we have B R,e \ R + P = {α 1 , . . . , α k }. Proof. For all 1 ≤ i ≤ k, let d i = d(α i ) for short. We have ∆(d i ) = ∆(α i ) for all 1 ≤ i ≤ k since α i is P -cosmall. By Theorem 3.34 applied to d 1 , . . . , d k , we know that (α 1 , . . . , α k ) is a greedy decomposition of d. In particular, we have ∆(d) = k i=1 ∆(α i ) and that ∆(α 1 ), . . . , ∆(α k ) are the distinct connected components of ∆(d). By definition, we see that d 1 , . . . , d k are the connected components of d.
To prove that d ∈ Π P , it suffices to prove that d 1 , . . . , d k ∈ Π P (Corollary 3.43). But the later statement was proved in [1,Proposition 4.4(6)]. This proves the first claim of the theorem. The lifting e of d is now well-defined.
Claim: We have α k+1 , . . . , α r ∈ R + P . To prove the claim, it clearly suffices to show thatê − k i=1 α ∨ i ∈ Z∆ ∨ P or equivalent to show thatê P = d. But we already saw this equality in the proof of Theorem 6.14 △ Also, in the proof of Theorem 6.14, we saw that B R,ê = B R,e \ R + P , so that all members of every greedy decomposition ofê are elements of R + \ R + P . This fact is only compatible with the claim if k = r. In other words, we have shown that (α 1 , . . . , α k ) is a greedy decomposition ofê. We can reformulate this result by saying that B R,e \ R + P = B R,ê = {α 1 , . . . , α k }. This is all we wanted to prove. Notation 6.17. Let d ∈ Π P . Let e be the lifting of d. Then we denote by Σ P,d the sum defined by Definition 6.18. Let d ∈ Π P . We say that d is a P -admissible degree if Σ P,d ≥ 0.
(1) All minimal degrees in Π B are B-admissible.
Then all minimal degrees in Π P are P -admissible.
Proof of Item (1). This is immediately clear from the definition. For all e ∈ Π B , we have Σ B,e = 0 since R + B = ∅. Proof of Item (2). Let α 1 , . . . , α k be as in the statement. Let d = k i=1 d(α i ) for short. By Theorem 6.16, we have d ∈ Π P . To see that d is a P -admissible degree, we have to show that Σ P,d ≥ 0. In order to prove that Σ P,d ≥ 0, it suffices to prove that γ∈R + P (γ, α ∨ i ) ≥ 0 for all 1 ≤ i ≤ k (cf. Theorem 6.16). In other words, Theorem 6.16 helps us to reduce the assertion Σ P,d ≥ 0 to the case k = 1. We assume from now on that k = 1 and we write α = α 1 and d = d(α). As γ∈R + \R + P γ ∈ H 2 (X) = Z{ω β | β ∈ ∆ \ ∆ P } we also find that In view of the previous claim, these equations give Σ P,d = ℓ(s α ) −ℓ(s α W P ). But this quantity is by definition certainly nonnegative. Hence, we find Σ P,d ≥ 0 -as required.
Proof of Item (3). By Fact 6.5(2), the claim follows from the stronger statement that for all e ∈ Π B such that z B e is the maximal representative in z B e W P the inequality holds. In turn, this statement follows from the more stronger statement that for all e ∈ Π B such that z B e is the maximal representative in z B e W P and for all γ ∈ R + P the inequality holds. But this is clear in view of the even more precise statement in Corollary 5.9.
The following statements until the end of this section are meant to prepare the proof of the main result on quasi-homogeneity. They are less interesting on their own right. Lemma 6.20. Let d ∈ Π P . Let e be the lifting of d. Then we have the following formula: . Proof. Let d and e be as in the statement. By Fact 6.5(1), (2), we know that z P d is the minimal and that z B e is the maximal representative in z P d W P = z B e W P . Hence, we have ℓ(z P d ) = ℓ(z B e ) − card(R + P ). The result follows from this and Corollary 4.14. Lemma 6.21. Let d ∈ Π P . Let e be the lifting of d. Then we have the following equality: Proof. By arguments very similar to those in the proof of the claim in Proposition 6.19 (2) and as e P = d by The last sum can be split into two summands since e = α∈B R,e α ∨ by Fact 6.5(1) and [1,Remark 8.4].
Corollary 6.22. Let d be a P -admissible degree. Let e be the lifting of d. Then we have the following inequality: Proof. This follows directly from the assumption on d since Lemma 6.21 expresses the difference in question as a sum of three nonnegative summands.
Proof. By Lemma 6.20, 6.21 we have the identity Proof. We can split the sum Σ P,d into two summands as follows Let α ∈ B R,e \ R + P and γ ∈ I(s α ) ∩ R + P . By Theorem 4.5(2), the root α is B-cosmall. Hence, we have (γ, α ∨ ) = 1 since α = γ (cf. [4, Theorem 6.1: (a) ⇒ (c)]). This means that all terms in the second summand of the previous displayed equation are equal to one. In other words, we have in view of Theorem 4.10. The lemma now follows from the last two displayed equations and Lemma 6.24.

The diagonal curve
In this section, we introduce in full detail the diagonal curve associated to a minimal degree (cf. Definition 7.3). For this construction, we make use of generalized cascades of orthogonal roots and liftings of minimal degrees which we introduced in the sections before. We formulate an additional assumption on minimal degrees (Assumption 7.13). For minimal degrees satisfying this assumption, we prove in the next section that the morphism associated to the diagonal curve has a dense open orbit in the moduli space of stable maps. In the course of this proof, the set of tangent directions associated to a minimal degree plays a certain role. We introduce and investigate this set in this section (cf. Notation 7.11). It parametrizes certain tangent directions which result from the action of a suitable subgroup of G on the tangent direction of the diagonal curve.   where the first morphism is the diagonal embedding of P 1 into card(B R,e \ R + P ) isomorphic copies of P 1 and the second morphism is the embedding into X which is well-defined due to Theorem 4.5(3). Again by Theorem 4.5(3), the definition of f P,d is independent of the ordering of the product α∈B R,e \R + P . Hence, the morphism f P,d is well-defined. We call the image f P,d (P 1 ) the diagonal curve (associated to d).
Fact 7.4. Let d ∈ Π P . The morphism f P,d is a closed immersion of degree (f P,d ) * [P 1 ] = d. Moreover, we have f P,d (0) = x(1) and f P,d (∞) = x(z P d ). Proof. The morphism f P,d is defined as the composition of two closed immersion. Hence, f P,d is itself a closed immersion. We now compute the degree of f P,d . Let e be the lifting of d.
By definition of f P,d and the result [6, Lemma 3.4] recalled in Notation 7.2, we have As in the proof of Theorem 6.14, by Fact 6.5(3) and [ This equation basically shows that f P,d (∞) = x(z P d ). (Note that ∞ is sent to x(s α ) under the isomorphism P 1 ∼ = C α for all α ∈ R + \ R + P .) Remark 7.5. Let d ∈ Π P . One may conjecture that the morphism f P,d has a dense open orbit in M 0,3 (X, d) under the action of the automorphism group Aut(X). In this generality, we are not able to prove a quasi-homogeneity result, mainly because we do not have a type independent description of Aut(X). In this paper, we will prove the sharper and more restrictive result that the morphism f P,d has a dense open orbit in M 0,3 (X, d) under the action of G as long as d satisfies Assumption 7.13. Note that all minimal degrees satisfy Assumption 7.13 if R is simply laced or if P = B. In this way, we obtain quasi-homogeneity of the moduli space M 0,3 (X, d) for a large class of degrees d.
Fact 7.6. Let d be a degree. Then we have (z P d ) −1 = w P z P d w P . Proof. Let e ∈ H 2 (G/B) be a sufficiently large degree such that e P = d and such that z P d w P = z B e . We can choose such a degree e ∈ H 2 (G/B) by [4,Corollary 4.12(d)]. By [4, Corollary 4.9], we know that z B e is an involution. Hence, we have (z P d w P ) −1 = z P d w P or equivalent (z P d ) −1 = w P z P d w P -as claimed. Notation 7.7. For a root α ∈ R, we denote by U α the associated root group as defined in [7, 26.3, Theorem (a)].
Notation 7.8. To simplify notation, we write R(P ) = R + ∪ R P for short. The set of roots R(P ) is precisely the set of roots α ∈ R such that U α ⊆ P . Notation 7.9. For a Weyl group element z ∈ W , we denote by P z the conjugate of P , i.e. P z = zP z −1 .
Lemma 7.10. Let d be a degree. Let z = z P d for short. Then we have U −γ ⊆ P ∩ P z for all γ ∈ R + P .
We have to show that this set contains R − P . Since R − P ⊆ R(P ), it clearly suffices to show that zw P (R − P ) ⊆ R(P ). Since w P is the longest element of the Weyl group W P , we know that I(w P ) = R + P . But this means that w P (R + P ) = R − P . Since w P is an involution, the equation w P (R + P ) = R − P is equivalent to w P (R − P ) = R + P . Therefore, the claim zw P (R − P ) ⊆ R(P ) is equivalent to the statement z(R + P ) ⊆ R(P ). We now show the latter statement. By definition, the element z is the minimal representative in zW P , so that we have I(z) ∩ R + P = ∅. In other words, we have z(R + P ) ⊆ R + ⊆ R(P ) which suffices to prove the lemma.
Notation 7.11. Let d ∈ Π P . Let e be the lifting of d. Then we denote by TD P,d the set of roots defined by We call this set of roots the set of tangent directions (associated to d).
(2) We have P = B, i.e. X is a generalized complete flag variety.
Proof. Assume first that Assumption 7.13(1) is satisfied. Let (α, γ) be a pair as it occurs in the index set of the double sum in the statement. By definition, we know that (γ, α ∨ ) ≤ 0 -otherwise γ ∈ I(s α ). (We use here that α ∈ R + \ R + P and γ ∈ R + P .) Since α is long by assumption, Lemma 7.14 applies and we find that (γ, α ∨ ) ∈ {−1, 0}. If we discard all summands from the double sum which are zero, we are counting precisely the pairs (α, γ) such that (γ, α ∨ ) = −1. In this counting, we can equally well allow γ to range over all of R + P (instead of γ ∈ R + P \ I(s α )), since all elements γ ∈ I(s α ) satisfy (γ, α ∨ ) = 1 by [4, Theorem 6.1: (a) ⇒ (c)]. (Here we use that α is B-cosmall by Theorem 4.5 (2).) This proves the equality in this case. If Assumption 7.13(2) is satisfied, both sides of the claimed equality are obviously zero. There is nothing to prove.
Finally, assume that Assumption 7.13(3) is satisfied. Let α 1 , . . . , α k be P -cosmall roots such that ∆(α 1 ), . . . , ∆(α k ) are pairwise totally disjoint and such that d = k i=1 d(α i ). By Theorem 6.16, we know that B R,e \ R + P = {α 1 , . . . , α k }, in particular all elements of B R,e \ R + P are P -cosmall. In view of this fact, Theorem 3.4 shows that the left side of the claimed equality is zero. But Theorem 3.4 also shows that the right side of the equality is zero. (For α ∈ B R,e \ R + P , all elements γ ∈ R + P \ I(s α ) satisfy (α, γ) = 0 and all elements γ ∈ I(s α ) satisfy (γ, α ∨ ) = 1 as it was explained in the first paragraph of this proof.) This shows the claimed equality in the last case.

Main result on quasi-homogeneity
This section is devoted to the proof of the main result on quasi-homogeneity. After all preparations, it remains to combine the results proved until now and to relate them to the geometric question of quasi-homogeneity of the moduli space of stable maps. Proof. We fix a minimal degree d ∈ Π P which satisfies Assumption 7.13. Let z = z P d for short. Let M = M 0,3 (X, d) for short. We denote by M(2) the fiber of the total evaluation map ev 1 × ev 2 : M → X × X over the point (x(1), x(z)). Let f = f P,d for short. Let T f be the tangent space at f of the orbit Lf ⊆ M (2) of f under the action of L. In order to prove that f has a dense open orbit in M (2) under the action of L, it suffices to prove that dim(M(2)) ≤ dim(T f ).
Claim: We have card(TD P,d ) ≤ dim(T f ). Let g be the Lie algebra of G, let p be the Lie algebra of P , and let t be the Lie algebra of T . For a root α ∈ R, we denote by g α the Lie algebra of U α . By [14, Proposition 1.1], the tangent space of X at the point x(1) can be identified with g/p. Hence, the tangent space T f can be identified with a vector subspace of g/p. We will now prove an inclusion of vector spaces as follows: The claim follows immediately from this by taking dimensions. (Note that TD P,d ⊆ R − \ R − P by definition.) Let e be the lifting of d. For a root α ∈ R, we denote by x α ∈ g α the associated root vector. For a root α ∈ R, we also write h α = [x α , x −α ] ∈ t. In order to prove the desired inclusion of vector spaces, we prove first that x −α + p ∈ T f for all α ∈ B R,e \ R + P . By definition of f , the tangent direction of f at the point x(1) is the image of x = α∈B R,e \R + P x −α in g/p. Hence, we have x + p ∈ T f . In view of Theorem 4.5(3), we see that [h α , x] = [h α , x −α ] = −2x −α ∈ g −α \ {0} for all α ∈ B R,e \ R + P . Since T ⊆ B ∩ B z ⊆ L and since t acts on T f , we see from this that x −α + p ∈ T f for all α ∈ B R,e \ R + P . Let µ ∈ TD P,d . The desired inclusion of vector spaces follows if we prove that (g µ + p)/p ⊆ T f . By the previous paragraph and the definition of TD P,d , we may assume that µ = −α − γ for some α ∈ B R,e \ R + P and some γ ∈ R + P . Since µ is a root, we have [g −γ , g −α ] = g µ . By Lemma 7.10, we know that g −γ acts on T f . Hence, the previous paragraph and the previous equation show that we have indeed (g µ + p)/p ⊆ T f . △ After all, the desired inequality dim(M (2)) ≤ dim(T f ) is now easy to deduce from our previous results. Indeed, we have dim(M (2)) = (c 1 (X), d) − ℓ(z) by the first claim ≤ card(TD P,d ) by Theorem 7.16 ≤ dim(T f ) by the second claim.
This completes the proof of the desired inequality and hence the proof of the theorem.