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Paolo Bravi, Bart Van Steirteghem, The Moduli Scheme of Affine Spherical Varieties with a Free Weight Monoid, International Mathematics Research Notices, Volume 2016, Issue 15, 2016, Pages 4544–4587, https://doi.org/10.1093/imrn/rnv281
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We study Alexeev and Brion's moduli scheme |${\text M}_{\Gamma }$| of affine spherical varieties with weight monoid |$\Gamma $| under the assumption that |$\Gamma $| is free. We describe the tangent space to |${\text M}_{\Gamma }$| at its “most degenerate point” in terms of the combinatorial invariants of spherical varieties and deduce that the irreducible components of |${\text M}_{\Gamma }$|, equipped with their reduced induced scheme structure, are affine spaces.
1 Introduction
As part of the classification problem of algebraic varieties equipped with a group action, spherical varieties, which include symmetric, toric, and flag varieties, have received considerable attention; see, for example, [9, 18, 20, 21]. In [2], Alexeev and Brion introduced an important new tool for the study of affine spherical varieties over an algebraically closed field |${\mathbb k}$| of characteristic 0. We recall that an affine variety |$X$| equipped with an action of a connected reductive group |$G$| is called spherical if it is normal and its coordinate ring |${\mathbb k}[X]$| is multiplicity-free as a |$G$|-module. For such a variety a natural invariant, which completely describes the |$G$|-module structure of |${\mathbb k}[X]$|, is its weight monoid|$\Gamma (X)$|. By definition, |$\Gamma (X)$| is the set of isomorphism classes of irreducible representations of |$G$| that occur in |${\mathbb k}[X]$|. In view of the classification problem, we have the following natural question: how “good” an invariant is |$\Gamma (X)$|, or more explicitly: to what extent does |$\Gamma (X)$| determine the multiplicative structure of |${\mathbb k}[X]$|?
In [11], Brion conjectured that the irreducible components of |${\text M}_{\Gamma }$| are affine spaces. A precise version of this conjecture is the following.
If |$\Gamma $| is a normal submonoid of |$\Lambda ^+$|, then the irreducible components of |${\text M}_{\Gamma }$|, equipped with their reduced induced scheme structure, are affine spaces.
This conjecture was verified for free and |$G$|-saturated monoids of dominant weights in [4]. In fact, Bravi and Cupit-Foutou proved that under these assumptions, |${\text M}_{\Gamma }$| is an affine space. In [23, 24], it is shown that |${\text M}_{\Gamma }$| is an affine space when |$\Gamma $| is the weight monoid of a spherical |$G$|-module. Luna provided the first non-irreducible example (unpublished): for |$G={\text SL}(2) \times {\text SL}(2)$| and |$\Gamma = \langle 2\omega , 4\omega + 2\omega '\rangle $|, where |$\omega $| and |$\omega '$| are the fundamental weights of the two copies of |${\text SL}(2)$|, the scheme |${\text M}_{\Gamma }$| is the union of two lines meeting in a point. In this paper, we verify that Conjecture 1.1 holds when |$\Gamma $| is free.
If |$\Gamma $| is a free submonoid of |$\Lambda ^+$|, then the irreducible components of |${\text M}_{\Gamma }$|, equipped with their reduced induced scheme structure, are affine spaces.
In their seminal paper [2], Alexeev and Brion equipped |${\text M}_{\Gamma }$| with an action of the maximal torus |$T$| of |$G$|. For this action, |${\text M}_{\Gamma }$| has a unique closed orbit, which is a fixed point |$X_0$|. Consequently, the tangent space |${\text T}_{X_0} {\text M}_{\Gamma }$| to |${\text M}_{\Gamma }$| at the point |$X_0$| is a finite-dimensional |$T$|-module. We describe this tangent space as follows.
If |$\Gamma $| is a free submonoid of |$\Lambda ^+$|, then |${\text T}_{X_0} {\text M}_{\Gamma }$| is a multiplicity-free |$T$|-module, and |$\gamma \in \Lambda $| occurs as a weight in |${\text T}_{X_0} {\text M}_{\Gamma }$| if and only if there exists an affine spherical |$G$|-variety |$X_{-\gamma }$| with weight monoid |$\Gamma $| and |$\Sigma ^N(X_{-\gamma }) = \{-\gamma \}$|.
To prove this, we first use the combinatorial theory of spherical varieties [17, 20, 21] to combinatorially characterize the weights |$\gamma $| for which such a variety |$X_{-\gamma }$| exists; see Corollary 2.17. Such a characterization was sketched by Luna in 2005 in an unpublished note.
To prove Theorem 1.2 we use Theorem 1.3: since it is known that the irreducible components of |${\text M}_{\Gamma }$|, equipped with their reduced induced scheme structure, are affine spaces after normalization (by [18, Theorem 1.3; 2, Corollary 2.14]), it is enough to show that they are smooth, and this follows from our description of the tangent space to |${\text M}_{\Gamma }$| at |$X_0$| (see Section 5).
Notation
Except if explicitly stated otherwise, |$\Gamma $| will be a free submonoid of |$\Lambda ^+$| with basis |$F = \{\lambda _1,\lambda _2, \ldots , \lambda _r\}.$| We will use |$S$| for the set of simple roots of |$G$| (associated with |$B$| and |$T$|) and |$R^+$| for the set of positive roots. The irreducible representation of |$G$| associated with the dominant weight |$\lambda \in \Lambda ^+$| is denoted by |$V(\lambda )$| and we use |$v_{\lambda }$| for a highest weight vector in |$V(\lambda )$|. We use |$\mathfrak {g}, \mathfrak {b}, \mathfrak {t}, \mathfrak {n},$| etc. for the Lie algebra of |$G,B,T, U,$| etc., respectively. When |$\alpha $| is a root, |$X_{\alpha } \in \mathfrak {g}_{\alpha }$| is a root operator and |$\alpha ^{\vee }$| the coroot. When |$\mathfrak {g}$| is simple, simple roots are denoted by |$\alpha _1,\ldots ,\alpha _n$| and numbered as by Bourbaki (see [7]), the corresponding fundamental weights are denoted by |$\omega _1,\ldots ,\omega _n$|.
2 Spherical Roots Adapted to |$\Gamma $|
In this section |$\Gamma $| denotes a normal, but not necessarily free, submonoid of |$\Lambda ^+$|. By combining results from [6, 17, 20, 21] we will describe when a set of spherical roots is “adapted” or “N-adapted” to |$\Gamma $|. In particular, in Corollaries 2.16 and 2.17 we give an explicit characterization for when an element |$\sigma $| of the root lattice is “adapted” or “N-adapted” to |$\Gamma $|.
We say that a subset |$\Sigma $| of |${\mathbb N}S$| is N-adapted to |$\Gamma $| if there exists an affine spherical |$G$|-variety |$X$| such that |$\Gamma (X) = \Gamma $| and |$\Sigma ^N(X) = \Sigma $|. By slight abuse of language, we say that an element |$\sigma $| of |${\mathbb N}S$| is N-adapted to |$\Gamma $| if |$\{\sigma \}$| is N-adapted to |$\Gamma $|.
We will give the definition of “adapted”, which requires some more notions from the theory of spherical varieties, in Definition 2.11. After recalling some basic definitions concerning spherical varieties, we briefly discuss, in Section 2.2, the notion of “spherically closed spherical systems”, and the role they play in classifying spherically closed spherical subgroups of |$G$|. We then, in Section 2.3 review Luna's “augmentations”. They classify the subgroups of |$G$| which have a given spherical closure |$K$|. Finally, after recalling some basic results from the Luna–Vust theory of spherical embeddings in Section 2.4, we deduce the combinatorial characterization of adapted and N-adapted spherical roots.
2.1 Basic definitions
In this section, we briefly recall the basic definitions of the theory of spherical varieties by freely quoting from [21]. For more details on these notions the reader can also consult [25, 27].
We recall that a (not necessarily affine) |$G$|-variety |$X$| is called spherical if it is normal and contains an open dense orbit for |$B$|. If |$X$| is affine, this is equivalent to the definition given before in terms of |${\mathbb k}[X]$|.
The complement of the open |$B$|-orbit in |$X$| consists of finitely many |$B$|-stable prime divisors. Among those, the ones that are not|$G$|-stable are called the colors of |$X$|. The set of colors of |$X$| is denoted by |$\Delta _X$|.
By the weight lattice|$\Lambda (X)$| of |$X$| we mean the subgroup of |$\Lambda $| made up of the |$B$|-weights in the field of rational functions |${\mathbb k}(X)$|. Since |$X$| has a dense |$B$|-orbit two rational |$B$|-eigenfunctions on |$X$| of the same weight are scalar multiples of one another.
Let |$P_X$| be the stabilizer of the open |$B$|-orbit and denote by |$S^p_X$| the subset of simple roots corresponding to |$P_X$|, which is a parabolic subgroup of |$G$| containing |$B$|.
Since |$X$| has a dense |$B$|-orbit, it has a dense |$G$|-orbit. Let |$H$| be the stabilizer of a point in this orbit, which we can then identify with |$G/H$|. The group |$H$| is called a spherical subgroup of |$G$| because |$G/H$| is a spherical |$G$|-variety. To |$H$|, we can associate a larger group |$\bar {H}$|, called the spherical closure of |$H$|: the normalizer of |$H$| in |$G$| acts by |$G$|-equivariant automorphisms on |$G/H$| and |$\bar {H}$| is the kernel of the induced action of this normalizer on |$\Delta _X$| (see [21, Section 6.1] or [5, Section 2.4.1]). We recall that it follows from [5, Lemma 2.4.2] that |$\bar {\bar {H}} = \bar {H}$| (see [26, Proposition 3.1] for a direct proof).
2.2 Spherical systems
Here, we briefly recall the definition of spherical system and its role in the classification of spherical varieties; see [5, 21].
Let |$K$| be a spherically closed spherical subgroup of |$G$|. The set |$\Sigma (G/K)$| of spherical roots of |$G/K$| is included in the root lattice |${\mathbb Z}S$| (because |$K$| contains the center of |$G$|) and it is a basis of |$\Lambda (G/K)$|. Let |${\bf A}_{G/K}$| be the set of colors that are not stable under some minimal parabolic containing |$B$| and corresponding to a simple root belonging to |$\Sigma (G/K)$|. The full Cartan pairing restricts to the |${\mathbb Z}$|-bilinear pairing |$c_{G/K}\colon {\mathbb Z} {\bf A}_{G/K}\times {\mathbb Z}\Sigma (G/K)\to {\mathbb Z}$|, also called restricted Cartan pairing.
It follows from [18, Theorem 1.3] that for |$X$| affine, |$\Sigma ^N(X)$| given in Definition 2.2 agrees with the description in Section 1 of the set of N-spherical roots of |$X$|.
Thanks to [20, Theorem 2] one can precisely describe the relationship between the three sets |$\Sigma (X)$|, |$\Sigma ^{sc}(X)$|, and |$\Sigma ^N(X)$|; see Proposition 2.9 and [28] for more information.
While |$\Sigma ^{sc}(X)$| and |$\Sigma ^N(X)$| are subsets of |${\mathbb N}S$|, there exist wonderful varieties |$X$| such that |$\Sigma (X) \not \subset {\mathbb Z}S$| (see [30]).
|$\Sigma (X)$| is not always a basis of |$\Lambda (X)$|, but it is when |$X$| is wonderful.
The weight lattice, valuation cone and spherical roots are birational invariants of the spherical variety |$X$| since they only depend on its open |$G$|-orbit |$G/H$|. The same is true of the colors and the Cartan pairing once we (naturally) identify the colors of |$G/H$| with their closures in |$X$|.
The set |$\Sigma ^{sc}(G)$| is finite. More precisely, there is the next proposition, which follows from the classification of spherically closed spherical subgroups |$K$| of |$G$| with |$\Lambda (G/K)$| of rank 1 [1, 20]; see also [5, Sections 1.1.6 and 2.4.1]. We recall that the support|${\text supp}(\sigma )$| of |$\sigma \in {\mathbb N}S$| is the set of simple roots which have a nonzero coefficient in the unique expression of |$\sigma $| as a linear combination of the simple roots.
An element |$\sigma $| of |${\mathbb N}S$| belongs to |$\Sigma ^{sc}(G)$| if and only if after numbering the simple roots in |${\text supp}(\sigma )$| like Bourbaki (see [7]) |$\sigma $| is listed in Table 1.
Type of support . | |$\sigma $| . |
---|---|
|${\mathsf {A}}_1$| | |$\alpha $| |
|${\mathsf {A}}_1$| | |$2\alpha $| |
|${\mathsf {A}}_1 \times {\mathsf {A}}_1$| | |$\alpha +\alpha '$| |
|${\mathsf {A}}_n$|, |$n\geq 2$| | |$\alpha _1+\cdots +\alpha _n$| |
|${\mathsf {A}}_3$| | |$\alpha _1+2\alpha _2+\alpha _3$| |
|$\mathsf {B}_n$|, |$n\geq 2$| | |$\alpha _1+\cdots +\alpha _n$| |
|$2(\alpha _1+\cdots +\alpha _n)$| | |
|$\mathsf {B}_3$| | |$\alpha _1+2\alpha _2+3\alpha _3$| |
|$\mathsf {C}_n$|, |$n\geq 3$| | |$\alpha _1+2(\alpha _2+\cdots +\alpha _{n-1})+\alpha _n$| |
|$\mathsf {D}_n$|, |$n\geq 4$| | |$2(\alpha _1+\cdots +\alpha _{n-2})+\alpha _{n-1}+\alpha _n$| |
|$\mathsf {F}_4$| | |$\alpha _1+2\alpha _2+3\alpha _3+2\alpha _4$| |
|$\mathsf {G}_2$| | |$4\alpha _1+2\alpha _2$| |
|$\alpha _1+\alpha _2$| |
Type of support . | |$\sigma $| . |
---|---|
|${\mathsf {A}}_1$| | |$\alpha $| |
|${\mathsf {A}}_1$| | |$2\alpha $| |
|${\mathsf {A}}_1 \times {\mathsf {A}}_1$| | |$\alpha +\alpha '$| |
|${\mathsf {A}}_n$|, |$n\geq 2$| | |$\alpha _1+\cdots +\alpha _n$| |
|${\mathsf {A}}_3$| | |$\alpha _1+2\alpha _2+\alpha _3$| |
|$\mathsf {B}_n$|, |$n\geq 2$| | |$\alpha _1+\cdots +\alpha _n$| |
|$2(\alpha _1+\cdots +\alpha _n)$| | |
|$\mathsf {B}_3$| | |$\alpha _1+2\alpha _2+3\alpha _3$| |
|$\mathsf {C}_n$|, |$n\geq 3$| | |$\alpha _1+2(\alpha _2+\cdots +\alpha _{n-1})+\alpha _n$| |
|$\mathsf {D}_n$|, |$n\geq 4$| | |$2(\alpha _1+\cdots +\alpha _{n-2})+\alpha _{n-1}+\alpha _n$| |
|$\mathsf {F}_4$| | |$\alpha _1+2\alpha _2+3\alpha _3+2\alpha _4$| |
|$\mathsf {G}_2$| | |$4\alpha _1+2\alpha _2$| |
|$\alpha _1+\alpha _2$| |
Type of support . | |$\sigma $| . |
---|---|
|${\mathsf {A}}_1$| | |$\alpha $| |
|${\mathsf {A}}_1$| | |$2\alpha $| |
|${\mathsf {A}}_1 \times {\mathsf {A}}_1$| | |$\alpha +\alpha '$| |
|${\mathsf {A}}_n$|, |$n\geq 2$| | |$\alpha _1+\cdots +\alpha _n$| |
|${\mathsf {A}}_3$| | |$\alpha _1+2\alpha _2+\alpha _3$| |
|$\mathsf {B}_n$|, |$n\geq 2$| | |$\alpha _1+\cdots +\alpha _n$| |
|$2(\alpha _1+\cdots +\alpha _n)$| | |
|$\mathsf {B}_3$| | |$\alpha _1+2\alpha _2+3\alpha _3$| |
|$\mathsf {C}_n$|, |$n\geq 3$| | |$\alpha _1+2(\alpha _2+\cdots +\alpha _{n-1})+\alpha _n$| |
|$\mathsf {D}_n$|, |$n\geq 4$| | |$2(\alpha _1+\cdots +\alpha _{n-2})+\alpha _{n-1}+\alpha _n$| |
|$\mathsf {F}_4$| | |$\alpha _1+2\alpha _2+3\alpha _3+2\alpha _4$| |
|$\mathsf {G}_2$| | |$4\alpha _1+2\alpha _2$| |
|$\alpha _1+\alpha _2$| |
Type of support . | |$\sigma $| . |
---|---|
|${\mathsf {A}}_1$| | |$\alpha $| |
|${\mathsf {A}}_1$| | |$2\alpha $| |
|${\mathsf {A}}_1 \times {\mathsf {A}}_1$| | |$\alpha +\alpha '$| |
|${\mathsf {A}}_n$|, |$n\geq 2$| | |$\alpha _1+\cdots +\alpha _n$| |
|${\mathsf {A}}_3$| | |$\alpha _1+2\alpha _2+\alpha _3$| |
|$\mathsf {B}_n$|, |$n\geq 2$| | |$\alpha _1+\cdots +\alpha _n$| |
|$2(\alpha _1+\cdots +\alpha _n)$| | |
|$\mathsf {B}_3$| | |$\alpha _1+2\alpha _2+3\alpha _3$| |
|$\mathsf {C}_n$|, |$n\geq 3$| | |$\alpha _1+2(\alpha _2+\cdots +\alpha _{n-1})+\alpha _n$| |
|$\mathsf {D}_n$|, |$n\geq 4$| | |$2(\alpha _1+\cdots +\alpha _{n-2})+\alpha _{n-1}+\alpha _n$| |
|$\mathsf {F}_4$| | |$\alpha _1+2\alpha _2+3\alpha _3+2\alpha _4$| |
|$\mathsf {G}_2$| | |$4\alpha _1+2\alpha _2$| |
|$\alpha _1+\alpha _2$| |
Recall that |$K$| is a spherically closed spherical subgroup of |$G$|. Therefore, see [21, Section 7.1], the triple |$\mathscr {S}_{G/K}=(S^p_{G/K}, \Sigma (G/K), {\bf A}_{G/K})$| is a spherically closed Luna spherical system in the following sense.
Let |$(S^p,\Sigma ,{\bf A})$| be a triple where |$S^p$| is a subset of |$S$|, |$\Sigma $| is a subset of |$\Sigma ^{sc}(G)$| and |${\bf A}$| is a finite set endowed with a |${\mathbb Z}$|-bilinear pairing |$c\colon {\mathbb Z}{\bf A}\times {\mathbb Z}\Sigma \to {\mathbb Z}$|. For every |$\alpha \in \Sigma \cap S$|, let |${\bf A} (\alpha )$| denote the set |$\{D \in {\bf A} : c(D,\alpha )=1 \}$|. Such a triple is called a spherically closed spherical |$G$|-system if all the following axioms hold:
(A1) for every |$D \in {\bf A}$| and every |$\sigma \in \Sigma $|, we have that |$c(D,\sigma )\leq 1$| and that if |$c(D, \sigma )=1$|, then |$\sigma \in S$|;
(A2) for every |$\alpha \in \Sigma \cap S$|, |${\bf A}(\alpha )$| contains two elements, which we denote by |$D_\alpha ^+$| and |$D_\alpha ^-$|, and for all |$\sigma \in \Sigma $| we have |$c(D_\alpha ^+,\sigma ) + c(D_\alpha ^-,\sigma ) = \langle \alpha ^\vee , \sigma \rangle $|;
(A3) the set |${\bf A}$| is the union of |${\bf A}(\alpha )$| for all |$\alpha \in \Sigma \cap S$|;
(Σ1) if |$2\alpha \in \Sigma \cap 2S$|, then |$\frac {1}{2}\langle \alpha ^\vee , \sigma \rangle $| is a non-positive integer for all |$\sigma \in \Sigma \setminus \{ 2\alpha \}$|;
(Σ2) if |$\alpha , \beta \in S$| are orthogonal and |$\alpha + \beta $| belongs to |$\Sigma $|, then |$\langle \alpha ^\vee , \sigma \rangle = \langle \beta ^\vee , \sigma \rangle $| for all |$\sigma \in \Sigma $|;
(S) every |$\sigma \in \Sigma $| is compatible with |$S^p$|, that is, for every |$\sigma \in \Sigma $| there exists a spherically closed spherical subgroup |$K$| of |$G$| with |$S^p_{G/K}=S^p$| and |$\Sigma (G/K)=\{\sigma \}$|.
Condition (S) of Definition 2.5 can be stated in purely combinatorial terms as follows (see [5, Section 1.1.6]). A spherically closed spherical root |$\sigma $| is compatible with |$S^p$| if and only if:
- in case |$\sigma =\alpha _1+\cdots +\alpha _n$| with support of type |$\mathsf {B}_n$|\[ \left\{\alpha\in{\text supp}\sigma\colon\langle\alpha^\vee,\sigma\rangle=0\right\}\setminus\left\{\alpha_n\right\} \subseteq S^p\subseteq\left\{\alpha\in S\colon\langle\alpha^\vee,\sigma\rangle=0\right\}\setminus\left\{\alpha_n\right\}, \]
- in case |$\sigma =\alpha _1+2(\alpha _2+\cdots +\alpha _{n-1})+\alpha _n$| with support of type |$\mathsf {C}_n$|\[ \left\{\alpha\in{\text supp}\sigma\colon\langle\alpha^\vee,\sigma\rangle=0\right\}\setminus\left\{\alpha_1\right\} \subseteq S^p\subseteq\left\{\alpha\in S\colon\langle\alpha^\vee,\sigma\rangle=0\right\}, \]
- in the other cases\[ \left\{\alpha\in{\text supp}\sigma\colon\langle\alpha^\vee,\sigma\rangle=0\right\} \subseteq S^p\subseteq\left\{\alpha\in S\colon\langle\alpha^\vee,\sigma\rangle=0\right\}. \]
Definition 2.5 combines the standard definition of spherical system, see [21, Section 2], with the requirement that it be spherically closed, see [21, Section 7.1] and [5, Section 2.4].
As shown in [21], the set |$\Delta _{G/K}$| of colors and the Cartan pairing |$c$| of |$G/K$| are uniquely determined by |$\mathscr {S}_{G/K}$|, in the sense that they can be naturally identified with the set of colors of and the full Cartan pairing of |$\mathscr {S}_{G/K}$|, defined as follows. Let |$\mathscr {S}=(S^p,\Sigma ,{\bf A})$| be a (spherically closed) spherical |$G$|-system. The set of colors of |$\mathscr {S}$| is the finite set |$\Delta $| obtained as the disjoint union |$\Delta =\Delta ^a\cup \Delta ^{2a}\cap \Delta ^b$| where
|$\Delta ^a={\bf A}$|,
|$\Delta ^{2a}=\big \{D_\alpha : \alpha \in S\cap {\frac 1 2}\Sigma \big \}$|,
|$\Delta ^b=\big \{D_\alpha : \alpha \in S\setminus (S^p\cup \Sigma \cup {\frac 1 2}\Sigma )\big \}/\sim $|, where |$D_\alpha \sim D_\beta $| if |$\alpha $| and |$\beta $| are orthogonal and |$\alpha +\beta \in \Sigma $|.
2.3 Augmentations
We continue to use |$K$| for a spherically closed spherical subgroup of |$G$|. By [21, Proposition 6.4] spherical homogeneous spaces |$G/H$| such that |$\bar H$|, the spherical closure of |$H$|, is equal to |$K$| are classified by their weight lattice, which is an augmentation of |$\mathscr {S}_{G/K}$| .
Let |$\mathscr {S}=(S^p,\Sigma ,{\bf A})$| be a spherically closed spherical |$G$|-system with Cartan pairing |$c: {\mathbb Z} \mathbf {A} \times {\mathbb Z}\Sigma \to {\mathbb Z}$|. An augmentation of |$\mathscr {S}$| is a lattice |$\Lambda '\subset \Lambda $| endowed with a pairing |$c'\colon {\mathbb Z}{\bf A}\times \Lambda '\to {\mathbb Z}$| such that |$\Lambda '\supset \Sigma $| and
(a1) |$c'$| extends |$c$|;
(a2) if |$\alpha \in S\cap \Sigma $|, then |$c'(D_\alpha ^+,\xi )+c'(D_\alpha ^-,\xi )=\langle \alpha ^\vee ,\xi \rangle $| for all |$\xi \in \Lambda '$|;
(σ1) if |$2\alpha \in 2S\cap \Sigma $|, then |$\alpha \notin \Lambda '$| and |$\langle \alpha ^\vee ,\xi \rangle \in 2{\mathbb Z}$| for all |$\xi \in \Lambda '$|;
(σ2) if |$\alpha $| and |$\beta $| are orthogonal elements of |$S$| with |$\alpha +\beta \in \Sigma $|, then |$\langle \alpha ^\vee ,\xi \rangle =\langle \beta ^\vee ,\xi \rangle $| for all |$\xi \in \Lambda '$|; and
(s) if |$\alpha \in S^p$|, then |$\langle \alpha ^\vee ,\xi \rangle =0$| for all |$\xi \in \Lambda '$|.
By the definition of spherical closure, |$\Delta _{G/H}$| and |$\Delta _{G/\bar H}$| are naturally identified and the full Cartan pairing |${\mathbb Z} \Delta _{G/H} \times \Lambda (G/H) \to {\mathbb Z}$| on |$G/H$| is the full Cartan pairing of the augmentation corresponding to |$H$| (see Proposition 6.4 and the proof of Theorem 3 in [21]).
We state here, for future reference, the following consequence of [20, Theorem 2].
This follows immediately from comparing [20, Theorem 2], which describes the relationship between |$\Sigma (G/H)$| and |$\Sigma ^N(G/H)$| with [21, Lemma 7.1], which describes the relationship between |$\Sigma (G/H)$| and |$\Sigma ^{sc}(G/H)$|. Note that [21, Lemma 7.1] can be deduced from [20] without appealing to Luna's conjecture.
2.4 Strictly convex colored cones and weight monoids of affine spherical varieties
An equivariant embedding of a spherical homogeneous space |$G/H$| as a dense orbit in a spherical |$G$|-variety (an embedding of |$G/H$|, for short) is called simple if it has only one closed orbit. Affine spherical varieties are simple.
If |$X$| is a simple embedding of the spherical homogeneous space |$G/H$|, let |$\mathcal {F}(X)$| be the set of colors of |$X$| containing the closed orbit (identified with elements of |$\Delta _{G/H}$|), and let |$\mathcal {C}(X)$| be the cone in |${\text Hom}(\Lambda (G/H), {\mathbb Q})$| generated by the valuations associated with the |$G$|-stable divisors of |$X$| (identified with elements of |$\mathcal {V}_{G/H}$|) and by |$c(\mathcal F(X),\cdot )$|. The couple |$(\mathcal {C}(X),\mathcal {F}(X))$| is a strictly convex colored cone in the sense of the following definition.
A strictly convex colored cone is a couple |$(\mathcal {C},\mathcal {F})$| where
– |$\mathcal {F}$| is a subset of |$\Delta _{G/H}$| such that the subset |$c(\mathcal {F},\cdot )$| of |${\text Hom}(\Lambda (G/H),{\mathbb Q})$| does not contain 0,
– |$\mathcal {C}$| is a strictly convex polyhedral cone in |${\text Hom}(\Lambda (G/H), {\mathbb Q})$| which is generated by |$c(\mathcal {F},\cdot )$| and finitely many elements of |$\mathcal {V}_{G/H}$| and whose relative interior intersects |$\mathcal {V}_{G/H}$|.
We recall from [17, Theorem 3.1] that simple embeddings |$X$| of the spherical homogeneous space |$G/H$| are classified by their strictly convex colored cones. By [17, Theorem 6.7], the simple embedding |$X$| is affine if and only if there exists a character |$\chi \in \Lambda (G/H)$| that is non-positive on |$\mathcal {V}_{G/H}$|, zero on |$\mathcal {C}(X)$| and |$c(\cdot ,\chi )$| is strictly positive on |$\Delta _{G/H}\setminus \mathcal {F}(X)$|.
We gather some known results about the weight monoid of affine spherical varieties.
If |$X$| is an affine spherical |$G$|-variety with weight monoid |$\Gamma (X)$| and open orbit |$G/H$|, then
the weight lattice of |$X$| (or of |$G/H$|) is |${\mathbb Z}\Gamma (X)$|;
the set |$S^p_X$| (which is the same as |$S^p_{G/H}$|) is equal to |$\{\alpha \in S \colon \langle \alpha ^{\vee },\gamma \rangle =0$| for all |$\gamma \in \Gamma (X)\}$|;
the dual cone |$\Gamma ^{\vee }(X):=\{v\in {\text Hom}({\mathbb Z}\Gamma (X),{\mathbb Q}): \langle v,\gamma \rangle \geq 0\mbox { for all }\gamma \in \Gamma (X)\}$| to |$\Gamma (X)$| is a strictly convex polyhedral cone;
every ray of |$\Gamma ^{\vee }(X)$| contains an element of |$c(\Delta _{G/H},\cdot )$| or of |$\mathcal V_{G/H}$|;
|$\Gamma ^{\vee }(X)$| contains |$c(\Delta _{G/H},\cdot )$|.
These statements are well known to experts and can be extracted from the results summarized in [27, Section 15.1]. For the reader's convenience, we provide a proof. Assertion (a) follows from the fact that a rational |$B$|-eigenfunction on |$X$| is necessarily equal to the quotient of two regular |$B$|-eigenfunctions; see for example [10, Proposition 2.8(i)]. Assertion (b) is [12, Lemme 10.2]. It follows from the fact that |$P_X$| is the common stabilizer of the |$B$|-stable lines in |${\mathbb k}[X]$|. This is the case because |$P_X$| is the common stabilizer of the |$B$|-stable prime divisors of |$X$| and the union of these divisors is the zero set of some |$B$|-eigenvector in |${\mathbb k}[X]$|. Assertion (c) is a standard fact in convex geometry. Parts (d) and (e) follow from the fact that a rational |$B$|-eigenfunction on |$X$| is regular if and only if it does not have poles along the colors or |$G$|-stable prime divisors of |$X$|. This, in turn, is so because |$X$| is normal.
2.5 Adapted spherical roots
Recall that |$\Gamma $| is a normal submonoid of |$\Lambda ^+$|. Combining the results recalled above, one derives the condition on a set of spherical roots |$\Sigma $| for being adapted to |$\Gamma $|.
We say that a subset |$\Sigma $| of |$\Sigma ^{sc}(G)$| is adapted (or N-adapted) to |$\Gamma $| if there exists an affine spherical |$G$|-variety |$X$| such that |$\Gamma (X) = \Gamma $| and |$\Sigma ^{sc}(X) = \Sigma $| (respectively, |$\Sigma ^N(X) = \Sigma $|).
Let |$\Sigma $| be a subset of |$\Sigma ^{sc}(G)$|. Losev's Theorem [19, Theorem 1.2] asserts that there is at most one affine spherical |$G$|-variety |$X$| with |$\Gamma (X) = \Gamma $| and |$\Sigma ^N(X)=\Sigma $|. Because |$\Sigma ^{sc}(X)$| determines |$\Sigma ^N(X)$| (see Proposition 2.9) there is also at most one affine spherical |$G$|-variety |$Y$| with |$\Sigma ^{sc}(Y)= \Sigma $| and |$\Gamma (Y) = \Gamma $|.
Let |$\Gamma $| be a normal submonoid of |$\Lambda ^+$|. A subset |$\Sigma $| of |$\Sigma ^{sc}(G)$| is adapted to |$\Gamma $| if and only if there exists a spherically closed spherical system |$\mathscr {S}=(S^p, \Sigma , {\bf A})$| such that
|$S^p=S^p(\Gamma )$|; and
|${\mathbb Z}\Gamma $| is an augmentation of |${\mathbb Z}\Sigma $|; and
if |$\delta \in E(\Gamma )$|, then |$\langle \delta , \sigma \rangle \leq 0$| for all |$\sigma \in \Sigma $| or there exists |$D\in \Delta $| such that |$c(D, \cdot )$| is a positive multiple of |$\delta $|; where |$\Delta $| is the set of colors of |$\mathscr {S}$| and |$c: {\mathbb Z}\Delta \times {\mathbb Z}\Gamma \to {\mathbb Z}$| is the full Cartan pairing of the augmentation; and
|$ c(D,\cdot ) \in \Gamma ^\vee $| for all |$D \in \Delta $|.
This is a consequence of the results we reviewed in Sections 2.2–2.4. We begin with the necessity of the conditions. Let |$X$| be an affine spherical |$G$|-variety with |$\Sigma ^{sc}(X) = \Sigma $| and |$\Gamma (X) = \Gamma $|. Let |$G/H$| be the open orbit of |$X$| and let |$\bar {H}$| be the spherical closure of |$H$|. Then |$\Sigma ^{sc}(X) = \Sigma (G/\bar {H})$| by definition, and |$S^p_{G/H} = S^{p}(\Gamma )$| by Proposition 2.10(b). Moreover, |$S^p_{G/H} = S^p_{G/\bar {H}}$|. It follows from Section 5.1 and Lemma 7.1 in [21] that |$(S^p(\Gamma ), \Sigma , {\bf A}_{G/\bar {H}})$| is a spherically closed spherical system. Since |$H$| has spherical closure |$\bar {H}$|, (2) follows from [21, Proposition 6.4]. Conditions (3) and (4) follow from (d) and (e) of Proposition 2.10.
We now show that the conditions are sufficient for |$\Sigma $| to be adapted to |$\Gamma $|. By [6] there exists a spherically closed spherical subgroup |$K$| of |$G$| with spherical system |$\mathscr {S}$|. Condition (2) implies by [21, Proposition 6.4] that there exists a spherical subgroup |$H$| of |$G$| with |$\bar {H} = K$| and |$\Lambda (G/H) = {\mathbb Z}\Gamma $|. What remains is to prove that |$G/H$| has an affine embedding |$X$| with weight monoid |$\Gamma $|. That is, by [17, Theorems 3.1 and 6.7] we have to show that there exists a strictly convex colored cone |$(\mathcal {C},\mathcal {F})$| in |${\text Hom}({\mathbb Z}\Gamma ,{\mathbb Q})$|, with respect to |$\mathcal V=\{v\in {\text Hom}({\mathbb Z}\Gamma ,{\mathbb Q}): \langle v,\sigma \rangle \leq 0\mbox { for all }\sigma \in \Sigma \}$| and the set of colors |$\Delta $| of |$\mathscr {S}$|, such that
there exists |$\chi \in {\mathbb Z}\Gamma $| that is non-positive on |$\mathcal {V}$|, zero on |$\mathcal {C}$| and strictly positive on |$\Delta \setminus \mathcal {F}$|; and
|$\Gamma =\{\gamma \in {\mathbb Z}\Gamma :\langle v,\gamma \rangle \geq 0\mbox { for all }v\in \mathcal {C}\cup \Delta \}$|.
We claim that if (1), (3), and (4) hold, then the desired strictly convex colored cone exists. Indeed, take |$\mathcal {C}$| to be the maximal face of |$\Gamma ^\vee $| whose relative interior meets |$\mathcal {V}$| with |$\mathcal {F}$| the set of colors contained in |$\mathcal {C}$| (such a maximal face exists since the zero face actually meets |$\mathcal {V}$|). The set |$c(\mathcal {F},\cdot )$| does not contain 0. Indeed, a color |$D$| with |$c(D,\cdot )=0$| necessarily belongs to |$\Delta ^b$|, whence |$c(D,\cdot )=\langle \alpha ^\vee ,\cdot \rangle $| for some |$\alpha \in S$| but by (1) this implies |$\alpha \in S^p$|. Moreover, |$\mathcal {C}$| is contained in a hyperplane that separates |$\mathcal {V}$| and |$\Delta \setminus \mathcal {F}$|. This yields |$\chi $|. The inclusion “|$\subset $|” of (ii) holds because |$\mathcal {C} \subset \Gamma ^\vee $| and because |$c(\Delta ,\cdot ) \subset \Gamma ^\vee $| by (4). The other inclusion follows from (3) and the maximality of |$\mathcal {C}$|.
It follows from equation (2.4) below that the spherical system |$\mathscr {S}$| and the Cartan pairing of the augmentation in Proposition 2.13 are uniquely determined by |$\Gamma $| and |$\Sigma $|.
Let |$\Gamma $| be a normal submonoid of |$\Lambda ^+$|. A subset |$\Sigma $| of |$\Sigma ^{sc}(G)$| is N-adapted to |$\Gamma $| if and only if there exists a subset |$\tilde {\Sigma }$| of |$\Sigma ^{sc}(G)$| which is adapted to |$\Gamma $| and such that |$\Sigma = (\tilde {\Sigma }\setminus \tilde {\Sigma }_l) \cup 2\tilde {\Sigma }_l$|, where |$\tilde {\Sigma }_l=\{\alpha \in \tilde {\Sigma } \cap S \colon a(\alpha ) \text { has one element}\}.$|
As the next two corollaries show, one can characterize very explicitly whether a single spherical root is adapted (Corollary 2.16) or N-adapted (Corollary 2.17) to |$\Gamma $|. In a 2005 working document, Luna had proposed a statement like Corollary 2.16. We remark that while Proposition 2.13 and Corollary 2.15 depend on the full classification of wonderful varieties by spherical systems (Luna's conjecture), the next two results only use the combinatorial classification of rank 1 wonderful varieties, which was obtained in [8] and also in [1].
Let |$\Gamma $| be a normal submonoid of |$\Lambda ^+$|. If |$\sigma \in \Sigma ^{sc}(G)$|, then |$\sigma $| is adapted to |$\Gamma $| if and only if all of the following conditions hold:
|$\sigma \in {\mathbb Z}\Gamma $|;
|$\sigma $| is compatible with |$S^p(\Gamma )$|;
if |$\sigma \notin S$| and |$\delta \in E(\Gamma )$| such that |$\langle \delta , \sigma \rangle > 0$|, then there exists |$\beta \in S\,{\setminus }\,S^p(\Gamma )$| such that |$\beta ^\vee $| is a positive multiple of |$\delta $|;
if |$\sigma \in S$|, then
|$a(\sigma )$| has one or two elements; and
|$\langle \delta , \gamma \rangle \ge 0$| for all |$\delta \in a(\sigma )$| and all |$\gamma \in \Gamma $|; and
|$\langle \delta , \sigma \rangle \le 1$| for all |$\delta \in E(\Gamma )$|;
if |$\sigma = 2\alpha \in 2S$|, then |$\alpha \notin {\mathbb Z}\Gamma $| and |$\langle \alpha ^{\vee }, \gamma \rangle \in 2{\mathbb Z}$| for all |$\gamma \in \Gamma $|;
if |$\sigma = \alpha + \beta $| with |$\alpha ,\beta \in S$| and |$\alpha \perp \beta $|, then |$\alpha ^{\vee } = \beta ^{\vee }$| on |$\Gamma $|.
We begin with the case |$\sigma \notin S$|. Then we have that |$\mathscr {S}$| is a spherically closed spherical |$G$|-system if and only if (2) holds. Then |$c$| gives an augmentation of |$\mathscr {S}$| if and only if (1), (5) and (6) hold. Condition (4) of Proposition 2.13 is vacuous since |$\Gamma \subset \Lambda ^+$| and every |$c(D,\cdot )$| is a positive multiple of a coroot. Condition (3) in the corollary is the same as condition (3) of Proposition 2.13 by the definition of |$c$|.
We proceed to the case |$\sigma \in S$|. Now |$\mathscr {S}$| is a spherically closed spherical |$G$|-system if and only if (2) and (a) hold. Next, by construction, |$c$| gives an augmentation of |$\mathscr {S}$| if and only if we have (1). Condition (4) of Proposition 2.13 is equivalent to (b). Finally, condition (3) of Proposition 2.13 is equivalent to (c), again by the definition of |$c$|.
The combinatorial conditions that characterize N-adapted spherical roots are exactly the same except for conditions (a) and (5). We report all of them again entirely in the next statement for later reference.
Let |$\Gamma $| be a normal submonoid of |$\Lambda ^+$|. If |$\sigma \in \Sigma ^{sc}(G)$|, then |$\sigma $| is N-adapted to |$\Gamma $| if and only if all of the following conditions hold:
|$\sigma \in {\mathbb Z}\Gamma $|;
|$\sigma $| is compatible with |$S^p(\Gamma )$|;
if |$\sigma \notin S$| and |$\delta \in E(\Gamma )$| such that |$\langle \delta , \sigma \rangle > 0$|, then there exists |$\beta \in S\,{\setminus }\,S^p(\Gamma )$| such that |$\beta ^\vee $| is a positive multiple of |$\delta $|;
if |$\sigma \in S$|, then
|$a(\sigma )$| has two elements; and
|$\langle \delta , \gamma \rangle \ge 0$| for all |$\delta \in a(\sigma )$| and all |$\gamma \in \Gamma $|; and
|$\langle \delta , \sigma \rangle \le 1$| for all |$\delta \in E(\Gamma )$|;
if |$\sigma = 2\alpha \in 2S$|, then |$\langle \alpha ^{\vee }, \gamma \rangle \in 2{\mathbb Z}$| for all |$\gamma \in \Gamma $|;
if |$\sigma = \alpha + \beta $| with |$\alpha ,\beta \in S$| and |$\alpha \perp \beta $|, then |$\alpha ^{\vee } = \beta ^{\vee }$| on |$\Gamma $|.
By Corollary 2.15, if |$\sigma \notin S \cup 2S$|, then |$\sigma $| is adapted to |$\Gamma $| if and only if it is N-adapted to |$\Gamma $|. From the same corollary it follows that |$\sigma \in S$| is N-adapted to |$\Gamma $| if and only if it is adapted to |$\Gamma $| and |$a(\sigma )$| has two elements. The only remaining case is |$\sigma = 2\alpha $| for some |$\alpha \in S$|. Again by Corollary 2.15, |$2\alpha $| is N-adapted to |$\Gamma $| if and only if either
|$2 \alpha $| is adapted to |$\Gamma $|; or
|$\alpha $| is adapted to |$\Gamma $| and |$a(\alpha )$| has one element.
We assume that (1) and (2) hold and claim that (3) and (5) hold if and only if (i) or (ii) is true. Indeed, it is clear from Corollary 2.16 that if |$2\alpha $| is adapted to |$\Gamma $|, then we have (3) and (5). On the other hand, if |$\alpha $| is adapted to |$\Gamma $| and |$a(\alpha )$| has one element, then that element is |$\frac {1}{2}\alpha ^{\vee }$| and so (5) holds. Moreover, condition (c) of Corollary 2.16 implies (3) of this corollary. Conversely, suppose that we have (3) and (5). Since the restriction of |$\alpha ^{\vee }$| to |${\mathbb Z}\Gamma $| belongs to |$\Gamma ^\vee $| and |$\langle \alpha ^{\vee }, 2\alpha \rangle > 0$|, there exists |$\delta \in E(\Gamma )$| such that |$\langle \delta , 2\alpha \rangle >0$|. It follows from (3) that |$\delta = q\beta ^{\vee }$| for some |$\beta \in S \setminus S^p(\Gamma )$| and |$q \in {\mathbb Q}_{>0}$|. Clearly, |$\beta = \alpha $|, which proves that |$\delta $| is the only element of |$E(\Gamma )$| that takes a positive value on |$2\alpha $|. Now, suppose that |$2\alpha $| is not adapted to |$\Gamma $|, that is, that (i) does not hold. Then |$\alpha $| must be an element of |${\mathbb Z}\Gamma $|. By (5), |$\frac {1}{2}\alpha ^{\vee }$| takes integer values on |${\mathbb Z}\Gamma $|, and since it takes value 1 on |$\alpha $|, it is primitive in |$({\mathbb Z}\Gamma )^*$| and therefore an element of |$E(\Gamma )$| and the only element of |$a(\alpha )$|. It follows from Corollary 2.16 that (ii) is true. This finishes the proof.
3 The |$T_{{\text ad}}$|-Weights in |$(V/\mathfrak {g}\cdot x_0)^{G_{x_0}}$|
For the remainder of the paper, |$\Gamma $| will be a free monoid with basis |$F \subset \Lambda ^+$|. In this section, we begin by recalling that the moduli scheme |${\text M}_{\Gamma }$| is an open subscheme of a certain invariant Hilbert scheme |${\text H}_{\Gamma }$|. This allows one to realize the tangent space |${\text T}_{X_0} {\text M}_{\Gamma }$| as a |$T$|-submodule of a certain vector space |$(V/\mathfrak {g}\cdot x_0)^{G_{x_0}}$|. In Section 3.2, we prove that if |$\gamma $| is a |$T$|-weight in |$(V/\mathfrak {g}\cdot x_0)^{G_{x_0}}$|, then it is a spherical root of spherically closed type. In Section 3.3, we further show that |$\gamma $| is compatible with |$S^p(\Gamma )$|. We also show that if |$\gamma \notin S$|, then the weight space |$(V/\mathfrak {g}\cdot x_0)^{G_{x_0}}_{(\gamma )}$| has dimension at most 1. For notational and computational convenience, we actually work with the opposite of Alexeev and Brion's |$T$|-action on |${\text M}_{\Gamma }$| and with a twist of their action on |${\text H}_{\Gamma }$| (see Section 3.1).
3.1 The invariant Hilbert scheme and its tangent space
The next proposition relates |${\text M}_{\Gamma }$| to |${\text H}_{\Gamma }$|.
This a matter of “formal bookkeeping.” Composing the action of |$G$| on |$V(\Gamma )$| with the Chevalley involution of |$G$| induces an isomorphism |${\text M}_{\Gamma } \simeq {\text M}_{\Gamma ^*}$|. Composing this isomorphism with the open |$T_{{\text ad}}$|-equivariant embedding |${\text M}_{\Gamma ^*} \hookrightarrow {\text H}_{\Gamma }$| chosen above gives an open embedding |${\text M}_{\Gamma } \hookrightarrow {\text H}_{\Gamma }$|. Comparing the definition of the action in [2] with that of the action in [23, Section 2.2] one shows that this open embedding is |$T_{{\text ad}}$|-equivariant for the actions as given in the proposition.
In what follows, |${\text M}_{\Gamma }$| and |${\text H}_{\Gamma }$| will always be equipped with the actions given in Proposition 3.1. The action Alexeev and Brion defined on |${\text M}_{\Gamma }$| is conceptually the most natural, while we find the action we are using on |${\text H}_{\Gamma }$| computationally more convenient.
3.2 The |$T_{{\text ad}}$|-weights in |$(V/\mathfrak {g}\cdot x_0)^{G_{x_0}}$| are spherical roots of |$G$|
In this section, we prove the following theorem.
If |$\gamma $| is a |$T_{{\text ad}}$|-weight in |$(V/\mathfrak {g}\cdot x_0)^{G_{x_0}}$|, then |$\gamma $| is a spherically closed spherical root of |$G$|.
Corollaries 3.8 and 3.14.
A basis of |$T_{{\text ad}}$|-eigenvectors of |$\mathfrak {g} \cdot x_0$| is given by |$\{v_{\lambda } \colon \lambda \in F\} \cup \{X_{-\beta }\cdot x_0\colon \beta \in R^+ \,{\setminus }\, F^{\perp }\}$|.
If |$[v]$| is a |$T_{{\text ad}}$|-eigenvector in |$V/\mathfrak {g}\cdot x_0$| of weight |$\gamma $|, then |$[v] \in (V/\mathfrak {g}\cdot x_0)^{G_{x_0}}$| if and only if |$\gamma \in {\mathbb Z}\Gamma $| and |$X_{\beta } \cdot v \in \mathfrak {g} \cdot x_0$| for all |$\beta \in S \cup -(S \cap F^{\perp })$|.
Assertion (a) follows from the fact that |$\mathfrak {g} \cdot x_0 = \mathfrak {b}^{-} \cdot x_0 = \mathfrak {t} \cdot x_0 + \mathfrak {n}^{-}\cdot x_0$| and that |$F$| is linearly independent. Assertion (b) follows from [23, Lemma 2.16] and the fact that |$\mathfrak {g}_{x_0}$| is generated as a Lie algebra by |$\mathfrak {t}_{x_0}$| and the root spaces |$\mathfrak {g}_\beta $| with |$\beta \in S \cup -(S \cap F^{\perp })$| (see, e.g., [15, Theorem 30.1]).
In the remainder of this section, |$\gamma $| is a |$T_{{\text ad}}$|-weight occurring in |$(V/\mathfrak {g}\cdot x_0)^{G_{x_0}}$| and |$v \in V$| a |$T_{{\text ad}}$|-eigenvector of weight |$\gamma $| such that |$[v]$| is a nonzero element of |$(V/\mathfrak {g}\cdot x_0)^{G_{x_0}}$|. By Proposition 3.4 (and the choice of our |$T_{{\text ad}}$|-action), the weight |$\gamma $| belongs to |${\mathbb N}S\cap {\mathbb Z}\Gamma $|.
There exists at least one simple root |$\alpha $| such that |$X_\alpha v\neq 0$|.
If |$\alpha $| is a simple root such that |$X_\alpha v\neq 0$| and |$\gamma \neq \alpha $|, then |$\gamma -\alpha $| is a positive root.
If |$\alpha $| is a simple root such that |$\gamma -\alpha $| is a root, then there exists |$z\in {\mathbb k}$| such that |$X_\alpha v=z\,X_{-\gamma +\alpha } x_0$|.
The vector |$v$| cannot be a linear combination of the highest weight vectors |$v_{\lambda _i}$|, otherwise (since the weights |$\lambda _i$| are linearly independent) it would belong to |$\mathfrak {t}\cdot x_0\subset \mathfrak {g}\cdot x_0$|. Moreover, since |$X_\alpha \in \mathfrak {g}_{x_0}$| for all |$\alpha \in S$|, |$X_\alpha v$| is a |$T_{{\text ad}}$|-eigenvector of weight |$\gamma -\alpha $| in |$\mathfrak {g}\cdot x_0$|.
We first deal with the case where |$\gamma $| is a root. Note that since |$\gamma \in {\mathbb N}S$|, it is then a positive root. As is well known, we then also have that |${\text supp}(\gamma )$| is a connected subset of the Dynkin diagram of |$G$|.
If |$\gamma $| is a root, which is not simple, then there exist at least two distinct simple roots |$\alpha $| such that |$\gamma -\alpha $| is a root.
Assume that there exists only one simple root |$\alpha $| such that |$\gamma -\alpha $| is a root. By Lemma 3.5, there exists |$z \in {\mathbb k}$| such that |$X_{\alpha } v=z\,X_{-\gamma +\alpha } x_0$|. Moreover, there exists |$z'\in {\mathbb k}^\times $| such that |$[X_\alpha ,X_{-\gamma }]=z'\,X_{-\gamma +\alpha }$|. Therefore, if we put |$z''=z/z'$|, then |$X_\alpha (v+z''\, X_{-\gamma } x_0)=0$|. Since |$[v] = [v+z''\, X_{-\gamma } x_0]$| in |$V/\mathfrak {g}\cdot x_0$| we can assume that |$X_{\alpha } v = 0$|. Since |$\gamma - \alpha '$| is not a positive root for all |$\alpha ' \in S\setminus \{\alpha \}$|, it then follows that |$X_\alpha v=0$| for all |$\alpha \in S$|, which contradicts Lemma 3.5(1).
If |$\gamma $| is a root, of which the support is not of type |$\mathsf {G}_2$|, then it is a locally dominant short root, that is, the dominant short root in the root subsystem generated by the simple roots of its support.
(III) Assume that there exist more than two simple roots, say |$\alpha _1,\ldots ,\alpha _k$|, such that |$\gamma -\alpha _j$| is a root for all |$j\in \{1,2,\ldots ,k\}$|. We claim that they can be reordered such that |$\alpha _j$| is orthogonal to |$\alpha _{j+1}$| for all |$j<k$| as in part II.
In types |$\mathsf {E}_6$|, |$\mathsf {E}_7$|, and |$\mathsf {E}_8$| all the roots have the same length so we would necessarily have |$\langle (\alpha _{j_m})^\vee ,\gamma \rangle =1$| for |$m \in \{1,2,3\}$|, but this is absurd since it would mean that |$\langle (\alpha _{j_1}+\alpha _{j_2}+\alpha _{j_3})^\vee ,\gamma \rangle =3$|. In type |$\mathsf {F}_4$|, the three simple roots would generate a root subsystem of type |$\mathsf {B}_3$| or of type |$\mathsf {C}_3$|. In the former case (type |$\mathsf {B}_3$|) we would necessarily have |$\langle (\alpha _{j_1})^\vee ,\gamma \rangle =\langle (\alpha _{j_2})^\vee ,\gamma \rangle =1$| assuming |$\alpha _{j_1}$| and |$\alpha _{j_2}$| are long, but this is absurd since it would mean |$\langle (\alpha _{j_1}+\alpha _{j_2}+\alpha _{j_3})^\vee ,\gamma \rangle \geq 4$|. In the latter case (type |$\mathsf {C}_3$|) we would necessarily have |$\langle (\alpha _{j_1})^\vee ,\gamma \rangle =1$| assuming |$\alpha _{j_1}$| is long. If |$\langle (\alpha _{j_3})^\vee ,\gamma \rangle $| is positive, then |$\langle (\alpha _{j_1}+\alpha _{j_2}+\alpha _{j_3})^\vee ,\gamma \rangle $| is greater than 2, which is not possible in type |$\mathsf {F}_4$|. If |$\langle (\alpha _{j_3})^\vee ,\gamma \rangle =0$|, then |$\gamma + \alpha _{j_3}$| is a root, and |$\langle (\alpha _{j_1}+\alpha _{j_2}+\alpha _{j_3})^\vee ,\gamma +\alpha _3\rangle $| is greater than 2, which is again absurd.
(IV) We now want to prove that |$\gamma $| is locally dominant (if the support of |$\gamma $| is not of type |$\mathsf {G}_2$|). The fact that |$\gamma $| is locally short then follows. Indeed, if the support of |$\gamma $| is not simply laced, then the highest root in the root system generated by that support does not satisfy Lemma 3.6:
– in type |$\mathsf {B}_n$|, |$n\geq 2$|, the highest root is |$\alpha _1+2(\alpha _2+\cdots +\alpha _n)=\omega _2$|;
– in type |$\mathsf {C}_n$|, |$n\geq 3$|, the highest root is |$2(\alpha _1+\cdots +\alpha _{n-1})+\alpha _n=2\omega _1$|;
– in type |$\mathsf {F}_4$| the highest root is |$2\alpha _1+3\alpha _2+4\alpha _3+2\alpha _4=\omega _1$|.
To obtain a contradiction we assume that |$\gamma $| is not locally dominant, that is, we assume that there exists |$\beta \in {\text supp}(\gamma )$| such that |$\langle \beta ^{\vee }, \gamma \rangle <0$|. Recall from Part III that in type different from |$\mathsf {G}_2$| if |$\gamma -\alpha $| is a root for a simple root |$\alpha $|, then |$\langle \alpha ^\vee ,\gamma \rangle \geq 0$|.
Suppose first that there are exactly |$k>2$| simple roots, say |$\alpha _1,\ldots ,\alpha _k$|, such that |$\gamma -\alpha _j$| is a root for all |$j\leq k$|. From the assumption that |$\gamma $| is not locally dominant, it follows that there exists |$\lambda \in F$| not orthogonal to |$\gamma -\alpha _1-\cdots -\alpha _k$|. By Parts II and III, we can then assume that |$X_{\alpha _j}v=0$| for all |$j\leq k$|. This contradicts Lemma 3.5(1).
If there are exactly two simple roots |$\alpha _1$| and |$\alpha _2$| such that |$\gamma -\alpha _1$| and |$\gamma -\alpha _2$| are roots, and |$\alpha _1$| and |$\alpha _2$| are orthogonal, then by Part I we get the same contradiction with Lemma 3.5(1).
Furthermore, if the support of |$\gamma $| has cardinality |$\leq 2$|, then the proposition follows by Lemma 3.6. Indeed, the only roots with support of cardinality |$\leq 2$| satisfying Lemma 3.6 are:
– with support of type |$\mathsf {A}_1$|, |$\alpha _1$|,
– with support of type |$\mathsf {A}_2$|, |$\alpha _1+\alpha _2$|,
– with support of type |$\mathsf {B}_2$|, |$\alpha _1+\alpha _2$|.
To conclude the proof of the proposition, we use once again an argument similar to that of Part I. Indeed, we will show in Part V that we can assume that |$X_{\alpha _1} v = X_{\alpha _2} v = 0$|, which contradicts Lemma 3.5(1).
(V) We finish by proving the following claim: if |$\alpha _1$| and |$\alpha _2$| are simple roots such that
– |$\alpha _1 + 2\alpha _2$| is not a root;
– |$\gamma -\alpha _1$|, |$\gamma -\alpha _2$|, |$\gamma -\alpha _1-\alpha _2$|, and |$\gamma -\alpha _1-2\alpha _2$| are roots; and
– |$\langle (\gamma -\alpha _1-2\alpha _2)^\vee , \lambda \rangle \neq 0$| for some |$\lambda \in F$|; then
we can assume that |$X_{\alpha _1} v = X_{\alpha _2} v = 0$|.
The following is Theorem 3.3 for the case that |$\gamma $| is a root.
Let |$\gamma $| be a |$T_{{\text ad}}$|-weight in |$(V/\mathfrak {g}\cdot x_0)^{G_{x_0}}$|. If |$\gamma $| is a root, then |$\gamma $| is a spherically closed spherical root of |$G$|.
If the support of |$\gamma $| is not of type |$\mathsf {G}_2$|, then by Proposition 3.7 we have only to check the locally dominant short roots. The following roots do not satisfy Lemma 3.6.
– With support of type |$\mathsf {D}_n$|, |$n\geq 4$|: |$\alpha _1+2(\alpha _2+\cdots +\alpha _{n-2})+\alpha _{n-1}+\alpha _n=\omega _2$|.
– With support of type |$\mathsf {E}_6$|: |$\alpha _1+2\alpha _2+2\alpha _3+3\alpha _4+2\alpha _5+\alpha _6=\omega _2$|.
– With support of type |$\mathsf {E}_7$|: |$2\alpha _1+2\alpha _2+3\alpha _3+4\alpha _4+3\alpha _5+2\alpha _6+\alpha _7=\omega _1$|.
– With support of type |$\mathsf {E}_8$|: |$2\alpha _1+3\alpha _2+4\alpha _3+6\alpha _4+5\alpha _5+4\alpha _6+3\alpha _7+2\alpha _8=\omega _8$|.
Therefore, we are left with all spherically closed spherical roots.
– With support of type |$\mathsf {A}_n$|, |$n\geq 1$|: |$\alpha _1+\cdots +\alpha _n $|.
– With support of type |$\mathsf {B}_n$|, |$n\geq 2$|: |$\alpha _1+\cdots +\alpha _n $|.
– With support of type |$\mathsf {C}_n$|, |$n\geq 3$|: |$\alpha _1+2(\alpha _2+\cdots +\alpha _{n-1})+\alpha _n $|.
– With support of type |$\mathsf {F}_4$|: |$\alpha _1+2\alpha _2+3\alpha _3+2\alpha _4 $|.
If the support of |$\gamma $| is of type |$\mathsf {G}_2$| the only positive root satisfying Lemma 3.6 is |$\alpha _1+\alpha _2$|, which is a spherically closed spherical root.
Let us now consider the case where |$\gamma $| is not a root. In contrast to the root case, here we notice the following general fact.
Let |$\alpha $| be a simple root and let |$\beta $| be a non-simple positive root such that |$\alpha +\beta $| is not a root. Then there exists no simple root |$\alpha '\neq \alpha $| such that |$(\alpha +\beta )-\alpha '$| is a root.
Assume that there exists a simple root |$\alpha '\neq \alpha $| such that |$\alpha +\beta -\alpha '$| is a root. Since |$\beta - \alpha '$| is nonzero, it is a root. This follows from the fact that |$\alpha + \beta $| is not a root, whence|$\langle \alpha ^\vee , \beta \rangle \geq 0$|, and so |$\langle \alpha ^\vee , \alpha +\beta - \alpha '\rangle >0$|. Finally, to deduce that |$\alpha +\beta $| is a root (i.e., a contradiction), one can use for example a saturation argument (see [14, Lemma 13.4.B]) as follows.
Restrict the adjoint representation to the Levi subalgebra associated with |$\alpha $| and |$\alpha '$|. Since |$\beta - \alpha '$| is a root, both |$\beta $| and |$\alpha +\beta - \alpha '$| occur as weights in the same irreducible summand, say of highest weight |$\lambda $|. From |$\langle \alpha ^\vee , \beta \rangle \geq 0$|, we get that |$\langle \alpha ^\vee , \alpha + \beta \rangle >0$|, and since |$\alpha + \beta $| is not a root, |$\langle (\alpha ')^\vee , \alpha + \beta - \alpha '\rangle \geq 0$|, and so |$\langle (\alpha ')^\vee , \alpha + \beta \rangle >0$|. Consequently, |$\alpha +\beta $| is dominant with respect to |$\alpha $| and |$\alpha '$|. Moreover, |$\lambda - \alpha -\beta $| is a sum of simple roots, because |$\lambda - \beta $| and |$\lambda - (\alpha + \beta - \alpha ')$| both belong to |${\text span}_{{\mathbb N}}\{\alpha , \alpha '\}$|. This implies that |$\alpha +\beta $| is a root.
Let |$\gamma $| be a |$T_{{\text ad}}$|-weight in |$(V/\mathfrak {g}\cdot x_0)^{G_{x_0}}$| which is not a root. Until Proposition 3.13, we assume that |$\gamma $| is not the sum of two orthogonal simple roots, so that we can speak of the unique simple root |$\alpha $| such that |$\gamma -\alpha $| is a root.
Let |$\alpha $| be the simple root such that |$\gamma -\alpha $| is a root. If |$\gamma \neq 2\alpha $|, then |$\alpha $| is orthogonal to |$\gamma -\alpha $|.
Let |$\alpha $| be the simple root such that |$\gamma -\alpha $| is a positive root. If |$\gamma -\alpha =\beta _1+\beta _2$| with |$\beta _1$| and |$\beta _2$| positive roots, then |$\alpha +\beta _1$| or |$\alpha +\beta _2$| is a root.
We can choose a basis as in the proof of Lemma 3.10 and, since |$X_\alpha v\neq 0$|, we can assume that |$X_\alpha v=X_{-\gamma +\alpha }x_0$|.
Suppose |$\gamma $| is not a root and let |$\alpha $| be a simple root such that |$\gamma -\alpha $| is a root. Then |$\gamma -\alpha $| is locally the highest root, that is, the highest root in the root subsystem generated by the simple roots of its support.
(I) First we want to prove that |$\gamma -\alpha $| is locally dominant. We can assume that |$\gamma -\alpha $| is not simple. Hence, by Lemma 3.10, |$\alpha $| is orthogonal to |$\gamma -\alpha $|.
There exists a simple root |$\delta $| (different from |$\alpha $|) such that |$\gamma -\alpha -\delta $| is a root. By Proposition 3.9 and Lemma 3.11 |$\alpha +\delta $| is a root.
Since |$\alpha +\delta $| is a root, |$\langle \alpha ^\vee , \delta \rangle < 0$|. Therefore, |$\langle \alpha ^\vee ,\gamma -\alpha -\delta \rangle >0$| hence |$\gamma -2\alpha -\delta $| is a root. If moreover |$2\alpha +\delta $| is a root, then by |$\mathfrak {sl}(2)$|-theory, |$\langle \alpha ^\vee , \alpha + \delta \rangle \leq 0$| and so |$\langle \alpha ^\vee ,\gamma -2\alpha -\delta \rangle \geq 0$|, whence |$\gamma -3\alpha -\delta $| is a root. If |$3\alpha +\delta $| is also a root, then |$\alpha $| and |$\delta $| span a root system of type |$\mathsf {G}_2$|. Consequently, |$\langle \alpha ^\vee ,\gamma -3\alpha -\delta \rangle =-1$| and |$\gamma -4\alpha -\delta $| is a root.
Therefore, we can apply Lemma 3.12 and obtain that, for some |$k\geq 1$|, |$\gamma -k\alpha $| is orthogonal to every |$\lambda \in F$|. This implies that |$\langle (\alpha ')^\vee , \gamma \rangle = 0$| for all |$\alpha ' \in {\text supp}(\gamma )\setminus \{\alpha \}$|, whence |$\langle (\alpha ')^\vee , \gamma -\alpha \rangle \geq 0$| for all such |$\alpha '$|. Since |$\alpha $| is orthogonal to |$\gamma -\alpha $|, it follows that |$\gamma -\alpha $| is locally dominant.
(II) To obtain a contradiction, we now assume that |$\gamma -\alpha $| is not locally the highest root, that is, a locally short dominant root with support of non-simply laced type:
– in type |$\mathsf {B}_n$|, |$n\geq 2$|, the short dominant root is |$\alpha _1+\cdots +\alpha _n=\omega _1$|;
– in type |$\mathsf {C}_n$|, |$n\geq 3$|, the short dominant root is |$\alpha _1+2(\alpha _2+\cdots +\alpha _{n-1})+\alpha _n=\omega _2$|;
– in type |$\mathsf {F}_4$| the short dominant root is |$\alpha _1+2\alpha _2+3\alpha _3+2\alpha _4=\omega _4$|;
– in type |$\mathsf {G}_2$| the short dominant root is |$2\alpha _1+\alpha _2=\omega _1$|.
By equation (3.13), |$\alpha $| is also short and |$k=2$|, in particular the support of |$\gamma $| is not of type |$\mathsf {G}_2$|. Moreover, by Lemma 3.10, |$\alpha $| is orthogonal to |$\gamma -\alpha $|. In type |$\mathsf {B}_n$| and in type |$\mathsf {F}_4$| this implies that |$\gamma $| is a root.
We are left with the case where the support of |$\gamma -\alpha $| is of type |$\mathsf {C}_n$|. Since |$\alpha $| is short, |$\alpha $| is orthogonal to |$\gamma -\alpha $|, |$\gamma $| is not a root, and moreover there exists a simple root |$\delta \neq \alpha $| satisfying the hypothesis of Lemma 3.12 for |$k=2$|, we have that |$n>3$|, |$\delta =\alpha _2$| and |$\alpha =\alpha _3$|. This contradicts Lemma 3.11, because |$\alpha _1$| and |$\gamma -\alpha -\alpha _1$| are roots, but neither |$\alpha _1+\alpha $| nor |$\gamma -\alpha _1$| is a root.
The following is Theorem 3.3 for the case that |$\gamma $| is not a root.
Let |$\gamma $| be a |$T_{{\text ad}}$|-weight in |$(V/\mathfrak {g}\cdot x_0)^{G_{x_0}}$|. If |$\gamma $| is not a root, then |$\gamma $| is a spherically closed spherical root of |$G$|.
We list all the locally highest roots |$\beta $| and deduce which are the only possible non-roots |$\gamma $| (obtained by adding to |$\beta $| a simple root) satisfying Lemmas 3.10, 3.11, and 3.12.
In general, |$\langle \alpha ^\vee ,\beta \rangle $| must be |$\geq 0$| otherwise |$\alpha +\beta \in R^+$|. If |$\alpha $| is not in the support of |$\beta $| it must be orthogonal to |$\beta $|, and in this case, by Lemma 3.11, |$\beta $| must necessarily be simple.
Let us now pass to |$\beta $| not simple and recall that |$\alpha $| must necessarily belong to the support of |$\beta $|, moreover by Lemma 3.10 |$\langle \alpha ^\vee ,\beta \rangle =0$| and by Lemma 3.12, for all |$\alpha '\in S\setminus \{\alpha \}$|, |$\langle (\alpha ')^\vee ,\alpha +\beta \rangle =0$|.
The remaining cases give no other possibilities:
– with support of type |$\mathsf {C}_n$|, |$n\geq 3$|, |$\beta =2(\alpha _1+\cdots +\alpha _{n-1})+\alpha _n=2\omega _1$|;
– with support of type |$\mathsf {E}_6$|, |$\beta =\alpha _1+2\alpha _2+2\alpha _3+3\alpha _4+2\alpha _5+\alpha _6=\omega _2$|;
– with support of type |$\mathsf {E}_7$|, |$\beta =2\alpha _1+2\alpha _2+3\alpha _3+4\alpha _4+3\alpha _5+2\alpha _6+\alpha _7=\omega _1$|;
– with support of type |$\mathsf {E}_8$|, |$\beta =2\alpha _1+3\alpha _2+4\alpha _3+6\alpha _4+5\alpha _5+4\alpha _6+3\alpha _7+ 2\alpha _8=\omega _8$|;
– with support of type |$\mathsf {F}_4$|, |$\beta =2\alpha _1+3\alpha _2+4\alpha _3+2\alpha _4=\omega _1$|.
3.3 Further properties of |$T_{{\text ad}}$|-weights in |$(V/\mathfrak {g}\cdot x_0)^{G_{x_0}}$|
After Theorem 3.3 the only possible |$T_{{\text ad}}$|-weights in |$(V/\mathfrak {g}\cdot x_0)^{G_{x_0}}$| are spherically closed spherical roots of |$G$|, but each of them occur only under special conditions which we are going to describe.
The first statement is indeed a refinement of Theorem 3.3. Recall the notion of compatibility with |$S^p$| (see axiom (S) of Definition 2.5 and Remark 2.6.1).
If |$\gamma $| is a |$T_{{\text ad}}$|-weight in |$(V/\mathfrak {g}\cdot x_0)^{G_{x_0}}$|, then |$\gamma $| is a spherically closed spherical root of |$G$| compatible with |$S^p(\Gamma )$|.
If |$\gamma = \alpha _1 + \alpha _2 + \cdots + \alpha _n$| with support of type |$\mathsf {A}_n$|, then |$\{\alpha _2, \alpha _3,\ldots , \alpha _{n-1}\} \subset S^p(\Gamma )$|. This follows from Part I of the proof of Proposition 3.7.
If |$\gamma = \alpha _1 + 2\alpha _2 + \alpha _3$| with support of type |$\mathsf {A}_3$|, then |$\{\alpha _1, \alpha _3\} \subset S^p(\Gamma )$|. This follows by Lemma 3.12 (|$\alpha =\alpha _2$|, |$\delta =\alpha _1$| and |$k=2$|).
If |$\gamma = \alpha _1 + \alpha _2 + \cdots + \alpha _n$| with support of type |$\mathsf {B}_n$|, then |$\{\alpha _2, \alpha _3,\ldots , \alpha _{n-1}\} \subset S^p(\Gamma )$| and |$\alpha _n\not \in S^p(\Gamma )$|. The former follows from Part I of the proof of Proposition 3.7. For the latter, we can assume that |$X_{\alpha _n}v=0$| and |$X_{\alpha _1}v=X_{-\gamma +\alpha _n}x_0$| nonzero, which implies |$\alpha _n\not \in S^p$|.
If |$\gamma = 2(\alpha _1 + \cdots + \alpha _n)$| with support of type |$\mathsf {B}_n$|, then |$\{\alpha _2, \ldots , \alpha _n\} \subset S^p(\Gamma )$|. This follows by Lemma 3.12 (|$\alpha =\alpha _1$|, |$\delta =\alpha _2$| and |$k=2$|).
If |$\gamma = \alpha _1 + 2\alpha _2 + 3\alpha _3$| with support of type |$\mathsf {B}_3$|, then |$\{\alpha _1, \alpha _2\} \subset S^p(\Gamma )$|. This follows by Lemma 3.12 (|$\alpha =\alpha _3$|, |$\delta =\alpha _2$| and |$k=3$|).
If |$\gamma = \alpha _1 + 2(\alpha _2+\cdots +\alpha _{n-1})+ \alpha _n$| with support of type |$\mathsf {C}_n$|, then |$\{\alpha _3, \alpha _4, \ldots , \alpha _n\} \subset S^p(\Gamma )$|. This follows from part V of the proof of Proposition 3.7.
If |$\gamma = 2(\alpha _1 + \cdots + \alpha _{n-2})+\alpha _{n-1}+\alpha _n$| with support of type |$\mathsf {D}_n$|, then |$\{\alpha _2, \ldots , \alpha _n\} \subset S^p(\Gamma )$|. This follows by Lemma 3.12 (|$\alpha =\alpha _1$|, |$\delta =\alpha _2$| and |$k=2$|).
If |$\gamma = \alpha _1 + 2\alpha _2 + 3\alpha _3 + 2\alpha _4$| with support of type |$\mathsf {F}_4$|, then |$\{\alpha _1, \alpha _2, \alpha _3\} \subset S^p(\Gamma )$|. This follows from part V of the proof of Proposition 3.7.
If |$\gamma = 4\alpha _1 + 2\alpha _2$| with support of type |$\mathsf {G}_2$|, then |$\alpha _2\in S^p(\Gamma )$|. This follows by Lemma 3.12 (|$\alpha =\alpha _1$|, |$\delta =\alpha _2$|, and |$k=4$|).
If |$\gamma $| is not a simple root then the |$T_{{\text ad}}$|-eigenspace |$(V/\mathfrak {g}\cdot x_0)^{G_{x_0}}_{(\gamma )}$| has dimension |$\leq 1$|.
If |$\gamma $| is a root (not simple), recall that there exist two simple roots, say |$\alpha _1$| and |$\alpha _2$|, such that |$\gamma -\alpha _1$| and |$\gamma -\alpha _2$| is a root, and |$\gamma -\alpha $| is not a root for all |$\alpha \in S\setminus \{\alpha _1,\alpha _2\}$|. In particular, for all |$\alpha \in S\setminus \{\alpha _1,\alpha _2\}$|, we necessarily have |$X_\alpha v=0$|. By adding to |$v$| a suitable scalar multiple of |$X_{-\gamma } x_0$|, we can assume that also |$X_{\alpha _2} v=0$|. Moreover, by choosing a suitable scalar multiple, we can assume that |$X_{\alpha _1} v=X_{-\gamma +\alpha _1} x_0$|.
If |$\gamma $| is neither a root nor the sum of two orthogonal simple roots, recall that there exists a simple root |$\alpha _1$| such that |$\gamma -\alpha _1$| is a root, and |$\gamma -\alpha $| is not a root for all |$\alpha \in S\setminus \{\alpha _1\}$|. In particular, for all |$\alpha \in S\setminus \{\alpha _1\}$|, we necessarily have |$X_\alpha v=0$|. Therefore, by choosing a suitable scalar multiple, we can assume that |$X_{\alpha _1} v=X_{-\gamma +\alpha _1} x_0$|.
In both cases, we claim that under the above assumptions |$v$| is uniquely determined. Indeed, if |$v_1$| and |$v_2$| are two vectors in |$V$| of |$T_{{\text ad}}$|-weight |$\gamma $| fulfilling the above conditions, then |$X_{\alpha }(v_1-v_2)=0$| for all |$\alpha \in S$|, which implies |$v_1=v_2$|.
We are left with only one case: the spherical root |$\gamma = \alpha +\alpha '$| with support of type |$\mathsf {A}_1\times \mathsf {A}_1$|. We can assume |$X_{\alpha }v=X_{-\alpha '}x_0$|. For all |$i\in \{1,\ldots ,r\}$|, |$\dim V(\lambda _i)_{(\gamma )}\leq 1$|, and the condition |$X_{\alpha } v=X_{-\alpha '} x_0$| uniquely determines every component |$v_i \in V(\lambda _i)$| of |$v$|.
4 The Weight Spaces of |${\text T}_{X_0} {\text H}_{\Gamma }$|
In this section, we prove the following theorem.
If |$\Gamma $| is a free monoid of dominant weights, then |${\text T}_{X_0}{\text H}_{\Gamma }$| is a multiplicity-free |$T_{{\text ad}}$|-module of which all the weights belong to |$\Sigma ^{sc}(G)$|. Moreover, if |$\gamma \in \Sigma ^{sc}(G)$| occurs as a |$T_{{\text ad}}$|-weight in |${\text T}_{X_0} {\text H}_{\Gamma }$|, then |$\gamma $| is N-adapted to |$\Gamma $|.
The assertion that all |$T_{{\text ad}}$|-weights of |${\text T}_{X_0}{\text H}_{\Gamma }$| belong to |$\Sigma ^{sc}(G)$| follows from the inclusion |${\text T}_{X_0}{\text H}_{\Gamma } \hookrightarrow (V/\mathfrak {g}\cdot x_0)^{G_{x_0}}$| and Theorem 3.3, while the assertion that the weight space |$({\text T}_{X_0}{\text H}_{\Gamma })_{(\gamma )}$| has dimension at most one follows from Proposition 3.16 if |$\gamma \notin S$|, and from Proposition 4.6 if |$\gamma \in S$|. The statement that if |$\gamma \in \Sigma ^{sc}(G)$| is a |$T_{{\text ad}}$|-weight in |${\text T}_{X_0}{\text H}_{\Gamma }$|, then |$\gamma $| is N-adapted to |$\Gamma $|, is contained in Proposition 4.6 for |$\gamma \in S$| and is shown in Section 4.3 for |$\gamma \notin S$|.
Recall from Proposition 3.1 that |${\text M}_{\Gamma }$| is |$T_{{\text ad}}$|-equivariantly isomorphic to an open subscheme of |${\text H}_{\Gamma }$|. Because every |$T_{{\text ad}}$|-weight in |${\text T}_{X_0}{\text M}_{\Gamma } \simeq {\text T}_{X_0}{\text H}_{\Gamma }$| is an element of |$\Sigma ^{sc}(G)$| (see Theorem 3.3) we obtain the following converse to the second statement in Theorem 4.1.
Let |$\Gamma $| be a free monoid of dominant weights and let |$\sigma \in \Sigma ^{sc}(G)$|. If |$\sigma $| is N-adapted to |$\Gamma $|, then |$\sigma $| is a |$T_{{\text ad}}$|-weight in |${\text T}_{X_0}{\text M}_{\Gamma }$|.
4.1 The extension criterion
We recall from [24] a criterion which allows to decide whether a |$T_{{\text ad}}$|-eigenvector |$[v] \in (V/\mathfrak {g}\cdot x_0)^{G_{x_0}} \simeq H^0(G\cdot X_0, \mathcal {N}_{X_0|V})^G$| belongs to the subspace |${\text T}_{X_0}{\text H}_{\Gamma } \simeq H^0(X_0, \mathcal {N}_{X_0|V})^G$|.
We denote by |$X_0^{\leq 1} \subset X_0$| the union of |$G \cdot x_0$| with all |$G$|-orbits of |$X_0$| that have codimension 1. By [10, Lemma 1.14] |$X_0^{\leq 1}$| is an open subset of |$X_0$|. The following proposition is a special case of [11, Lemma 3.9]. Together with Theorem 4.5 it gives the aforementioned criterion.
A section |$s \in H^0(G\cdot X_0,\mathcal {N}_{X_0|V})$| extends to |$X_0$| if and only if it extends to |$X_0^{\leq 1}$|.
We recall that the orbit structure of |$X_0$| is well understood [29, Theorem 8]. It is easy to describe the orbits of codimension 1 (see, e.g., [23, Proposition 3.1] for details).
Let |$v \in V$| be a |$T_{{\text ad}}$|-eigenvector of weight |$\gamma $| such that |$0 \not = [v] \in (V/\mathfrak {g}\cdot x_0)^{G_{x_0}}.$| Let |$\lambda \in F$|. Recall that |$z_{\lambda } = x_0 - v_{\lambda }$|. Assume that |$z_{\lambda } \in X_0^{\leq 1}$| and put |$Z:=G\cdot x_0 \cup G\cdot z_{\lambda }.$| Put |$a:=\langle \lambda ^\#, \gamma \rangle $|. Denote by |$s \in H^0 ( G \cdot x_0, \mathcal {N}_{X_0|V} )^G$| the |$G$|-equivariant section such that |$s(x_0) = [v]$|.
If |$a \leq 0$|, then |$s$| extends to an element of |$H^0 (Z, \mathcal {N}_{X_0|V} )^G$|.
If |$a > 1$|, then |$s$| does not extend to an element of |$H^0 (Z, \mathcal {N}_{X_0|V})^G$|.
If |$a = 1$|, then the following are equivalent:
|$s$| extends to an element of |$H^0 (Z, \mathcal {N}_{X_0|V} )^G$|;
there exist |$\hat {v} \in V(\lambda )$| such that |$[v] = [\hat {v}]$| as elements of |$V/\mathfrak {g}\cdot x_0$|.
4.2 The spherical root |$\gamma = \alpha \in S$|
In this section, we discuss the |$T_{{\text ad}}$|-weight space |$({\text T}_{X_0}{\text H}_{\Gamma })_{(\alpha )}$|, where |$\alpha $| is a simple root. Specifically, we will prove the following proposition, which is a special case of Theorem 4.1.
If |$\alpha $| is a simple root, then |$\dim ({\text T}_{X_0}{\text H}_{\Gamma })_{(\alpha )} \leq 1$|. Moreover, if |$\dim ({\text T}_{X_0}{\text H}_{\Gamma })_{(\alpha )} = 1$|, then |$\alpha $| is |$N$|-adapted to |$\Gamma $|.
The proof of Proposition 4.6 will be given on p. . We first need a few lemmas and introduce notation we will use for the remainder of this section. Put |$F(\alpha ) := \{\lambda \in F \colon \langle \alpha ^\vee ,\lambda \rangle \neq 0\}$|. We order the elements of |$F$| such that for |$F(\alpha ) = \{\lambda _1, \lambda _2, \ldots , \lambda _p\}$| for some |$p \leq r$|. Then |$F \setminus F(\alpha ) = \{\lambda _{p+1}, \lambda _{p+2}, \ldots ,\lambda _r\}$|.
By elementary highest weight theory, the |$T_{{\text ad}}$|-weight space in |$V$| of weight |$\alpha $| is spanned by |$\{v_1,v_2,\ldots ,v_p\}$|, and the intersection of this weight space with |$\mathfrak {g} \cdot x_0$| is the line spanned by |$X_{-\alpha }x_0 = v_1 + v_2+ \cdots + v_p$|. A straightforward application of Proposition 3.4 shows that |$[v_i] \in (V/\mathfrak {g}\cdot x_0)^{G_{x_0}}$| for every |$i \in \{1,2,\ldots ,p-1\}$|.
Suppose |$\alpha \in {\mathbb Z}\Gamma $| and |$|F(\alpha )| \geq 2$|. Let |$\lambda \in F$|. If |$\langle \lambda ^\#,\alpha \rangle >0$|, then |$G\cdot z_{\lambda }$| has codimension 1 in |$X_0$|.
Let |$\alpha $| be a simple root. Recall that |$F(\alpha ) = \{\lambda \in F\colon \langle \alpha ^{\vee },\lambda \rangle \neq 0\}$| and put |$E(\alpha ) := \{ \delta \in E(\Gamma ) \colon \langle \delta ,\alpha \rangle = 1\}$|. Then |$\dim ({\text T}_{X_0}{\text H}_{\Gamma })_{(\alpha )} \leq 1$| and if |$\dim ({\text T}_{X_0}{\text H}_{\Gamma })_{(\alpha )} = 1$|, then
|$\alpha \in {\mathbb Z}\Gamma $|;
|$|F(\alpha )| \ge 2$|;
|$\langle \delta ,\alpha \rangle \le 1$| for all |$\delta \in E(\Gamma )$|;
|$|E(\alpha )| \le 2$|;
If |$|E(\alpha )| = 2$|, then |$E(\alpha ) = \{\lambda ^\# \in E(\Gamma ) \colon \lambda \in F(\alpha )\}.$|
Let us assume that |$\dim ({\text T}_{X_0}{\text H}_{\Gamma })_{(\alpha )} \geq 1$|. Let |$[v]$| be a nonzero element of |$(V/\mathfrak {g}\cdot x_0)^{G_{x_0}}_{(\alpha )}$| such that the |$G$|-equivariant section |$s \in H^0(G\cdot x_0,\mathcal {N})^G$| defined by |$s(x_0) = [v]$| extends to |$X_0$|. By Proposition 3.4 and Lemma 4.7, conditions (i) and (ii) hold. Lemma 4.8 and Theorem 4.5 then imply (iii). We now prove (iv). If |$|E(\alpha )|\ge 3$|, then by Theorem 4.5 and Lemma 4.8, there exist at least three elements |$\lambda , \mu $|, and |$\nu $| in |$F(\alpha )$| such that there exist |$y_\lambda \in V(\lambda )$|, |$y_\mu \in V(\mu )$| and |$y_\nu \in V(\nu )$| for which |$[v]=[y_{\lambda }]=[y_{\mu }]=[y_\nu ] \in V/\mathfrak {g}\cdot x_0$|. This is impossible by Lemma 4.7 and (iv) is proved. We turn to (v). Suppose |$E(\alpha )=\{\lambda ^\#, \mu ^\#\}$|. By Lemma 4.8 and Theorem 4.5, there exist |$y_\lambda \in V(\lambda )$| and |$y_\mu \in V(\mu )$| such that |$[v]=[y_{\lambda }]=[y_{\mu }] \in V/\mathfrak {g}\cdot x_0$|. Using Lemma 4.7 again, (v) follows.
Finally, we show that |$\dim ({\text T}_{X_0}{\text H}_{\Gamma })_{(\alpha )} \leq 1$|. Since |$\alpha \in {\mathbb Z}\Gamma $|, there is at least one |$\lambda \in E(\alpha )$|. Lemma 4.8 and Theorem 4.5 again imply that |$[v]=[y_{\lambda }]$| for some |$y_\lambda \in V(\lambda )$|, which finishes the proof.
By Corollary 4.2 and the proof of Proposition 4.9, the preceding lemma gives alternative conditions for |$\alpha $| to be N-adapted to |$\Gamma $| when |$\Gamma $| is free. We list them as a separate lemma, since they seem easier to check then those in Corollary 2.17.
Lemma 4.9 says that |$\dim ({\text T}_{X_0}{\text H}_{\Gamma })_{(\alpha )} \leq 1$|. We assume conditions (i)–(v) in Lemma 4.9 and deduce conditions (1), (2), (a), (b), and (c) in Corollary 2.17. For (1) and (c), there is nothing to show. For the spherical root |$\alpha $|, (2) follows from (1). To show (a), we first claim that |$E(\alpha )$| contains at least one element. Indeed, |$\alpha \in {\mathbb Z}\Gamma $| and |$\langle \lambda ^\#,\alpha \rangle >0$| for at least one |$\lambda \in F$|, for otherwise |$-\alpha $| would be a dominant weight. The claim now follows from (iii). Next, suppose |$\lambda ^\# \in E(\alpha )$|. Clearly |$\lambda ^\# \in a(\alpha )$|. We claim that |$\alpha ^{\vee }-\lambda ^\# \neq \lambda ^\#$|. Otherwise, we would have |$\lambda ^\# = \frac {1}{2}\alpha ^{\vee }$|, which would contradict (ii). This shows |$|a(\alpha )|\ge 2$|. Now, if |$a(\alpha )$| had a third element, then |$E(\alpha )$| would have two elements, say |$\lambda ^\#$| and |$\mu ^\#$|, with |$\alpha ^{\vee }-\lambda ^\# \neq \mu ^\#$|. But this yields a contradiction: by (v), we have that |$\langle \alpha ^{\vee }, \lambda \rangle = \langle \alpha ^{\vee },\mu \rangle =1$| and then that |$\alpha ^{\vee }-\lambda ^\#$| takes the same values as |$\mu ^\#$| on |$F$|. We have deduced (a). Finally, (b) is clear since |$a(\alpha )=\{\lambda ^\#, \alpha ^{\vee }-\lambda ^\#\}$| for some |$\lambda \in F(\alpha )$|.
4.3 The non-simple spherical roots
To complete the proof of Theorem 4.1, we show in this section that if |$\gamma $| is a spherically closed spherical root, which is not a simple root and which occurs as a |$T_{{\text ad}}$|-weight in |${\text T}_{X_0}{\text H}_{\Gamma }$|, then |$\gamma $| is N-adapted to |$\Gamma $|.
We recall that conditions (1) and (2) of Corollary 2.17 follow from Theorem 3.15. We now verify condition (3): if |$\delta \in E(\Gamma )$| such that |$\langle \delta , \gamma \rangle > 0$|, then there exists |$\beta \in S\setminus S^p(\Gamma )$| such that |$\beta ^\vee $| is a positive multiple of |$\delta $|. The argument is the same for all the non-simple spherical roots |$\gamma $|.
Let |$v \in V$| be a |$T_{{\text ad}}$|-eigenvector of weight |$\gamma $| such that |$0 \not = [v] \in (V/\mathfrak {g}\cdot x_0)^{G_{x_0}}.$| Let |$\lambda \in F$|. Recall that |$z_{\lambda } = x_0 - v_{\lambda }$| and put |$a=\langle \lambda ^\#, \gamma \rangle $|. Assume |$a>0$|.
We claim that under this assumption, |${\text codim}_{X_0}G\cdot z_{\lambda }\geq 2$|. Indeed, if |${\text codim}_{X_0}G\cdot z_{\lambda }$| were 1, then by Theorem 4.5(B) |$a=1$| and by Theorem 4.5(C) there would exist |$\hat {v} \in V(\lambda )$| such that |$[v] = [\hat {v}]$| as elements of |$V/\mathfrak {g}\cdot x_0$|. Therefore, there would exist |$\alpha \in S$| such that |$\gamma -\alpha \in R^+$|, and such that |$X_\alpha \hat {v}$| is nonzero and is equal to |$X_{-\gamma +\alpha }x_0$| up to a nonzero scalar multiple. This would imply |$X_{-\gamma +\alpha }v_\lambda \neq 0$| and |$X_{-\gamma +\alpha }v_\mu =0$| for all |$\mu \in F\setminus \{\lambda \}$|, and therefore that there exists |$\alpha '\in S$| such that |$\langle (\alpha ')^\vee ,\lambda \rangle >0$| and |$\langle (\alpha ')^\vee ,\mu \rangle =0$| for all |$\mu \in F\setminus \{\lambda \}$|, which gives a contradiction with Proposition 4.4 and proves the claim.
The fact that |${\text codim}_{X_0}G\cdot z_{\lambda }\geq 2$| means that there exists |$\beta \in S$| such that |$\langle \beta ^\vee ,\lambda \rangle >0$| and |$\langle \beta ^\vee ,\mu \rangle =0$| for all |$\mu \in F\setminus \{\lambda \}$|. This says exactly that the restriction of |$\beta ^{\vee }$| to |${\mathbb Z}\Gamma $| is a positive multiple of |$\lambda ^\#$|, which is condition (3).
We continue with the remaining conditions of Corollary 2.17. Condition (4) does not apply to non-simple spherical roots.
Condition (5) follows using the analysis of Section 3. Indeed, we have shown that if |$[v]$| is a nonzero |$T_{{\text ad}}$|-eigenvector of weight |$2\alpha $| in |$(V/\mathfrak {g}\cdot x_0)^{G_{x_0}}$|, with |$\alpha \in S$|, then |$X_\alpha v$| is a (nonzero) scalar multiple of |$X_{-\alpha }x_0$|. Since |$2\alpha \in {\mathbb Z}\Gamma $|, there exists |$\lambda \in F$| such that |$\langle \alpha ^{\vee }, \lambda \rangle >0$| and |$\langle \lambda ^\#, 2\alpha \rangle > 0$|. By the argument we used for condition (3), |$\lambda $| is the unique element of |$F$| which is non-orthogonal to |$\alpha $|. It follows that we actually have that |$X_\alpha v$| is a nonzero scalar multiple of |$X_{-\alpha }v_{\lambda }$|. This implies that the |$T$|-eigenspace of weight |$\lambda -2\alpha $| in |$V(\lambda )$| is nonzero, hence |$\langle \alpha ^\vee ,\lambda \rangle \geq 2$|. Consequently |$\langle \alpha ^\vee ,\lambda \rangle \in \{2,4\}$| and |$\langle \alpha ^\vee ,\mu \rangle =0$| for all |$\mu \in F\setminus \{\lambda \}$|, hence |$\alpha ^\vee $| takes an even value on every element of |${\mathbb Z}\Gamma $|.
Condition (6) follows analogously from Section 3. Indeed, we have shown that if |$[v]$| is a nonzero |$T_{{\text ad}}$|-eigenvector of weight |$\alpha +\alpha '$| in |$(V/\mathfrak {g}\cdot x_0)^{G_{x_0}}$|, with |$\alpha $| and |$\alpha '$| orthogonal simple roots, then |$X_\alpha v$|, if nonzero, is a scalar multiple of |$X_{-\alpha '}x_0$|, and |$X_{\alpha '} v$|, if nonzero, is a scalar multiple of |$X_{\alpha }x_0$|.
The information given in this remark is not needed for our results. We include it because it gives explicit conditions on |$F$| for each spherically closed spherical root |$\gamma $|, which is not a simple root, to occur as a |$T_{{\text ad}}$|-weight in |${\text T}_{X_0}{\text H}_{\Gamma }$|, that is, to be N-adapted to |$\Gamma $|.
For each spherically closed spherical root |$\gamma $|, there exists |$\alpha \in S$| such that |$\langle \alpha ^\vee , \gamma \rangle > 0$|. If |$\gamma $| is a |$T_{{\text ad}}$|-weight in |${\text T}_{X_0}{\text H}_{\Gamma }$|, then |$\gamma \in {\mathbb Z}\Gamma $|, and so there exits |$\lambda \in F$| such that |$\langle \alpha ^\vee , \lambda \rangle > 0$| and |$\langle \lambda ^\#, \gamma \rangle >0$|. If |$\gamma $| is not a simple root, then by the argument above showing that |$\gamma $| satisfies condition (3) of Corollary 2.17, we have that |$\lambda $| is the only element of |$F$| which is not orthogonal to |$\alpha $|, that is, |$b\lambda ^\# = \alpha ^{\vee }$| on |${\mathbb Z}\Gamma $| for some positive integer |$b$|.
We now list, for each |$\gamma $|, the possibilities for |$\lambda ^\#$|.
If |$\gamma = 2\alpha $|, with |$\alpha $| a simple root, then locally |$\gamma = 4\omega $|. In this case |$\alpha ^\vee =b\lambda ^\#$| with |$b\in \{2,4\}$|.
If |$\gamma = \alpha + \alpha '$|, with |$\alpha $| and |$\alpha '$| two orthogonal simple roots, then locally |$\gamma =2\omega +2\omega '$|. In this case |$\alpha ^\vee =(\alpha ')^\vee =b\lambda ^\#$| with |$b\in \{1,2\}$|.
If |$\gamma = \alpha _1 + \alpha _2 + \cdots + \alpha _n$| with support of type |$\mathsf {A}_n$| with |$n\ge 2$|, then locally |$\gamma =\omega _1+\omega _n$|. In this case, |$\alpha ^\vee =\lambda ^\#$| with |$\alpha \in \{\alpha _1,\alpha _n\}$|.
If |$\gamma = \alpha _1 + 2\alpha _2 + \alpha _3$| with support of type |$\mathsf {A}_3$|, then locally |$\gamma =2\omega _2$|. In this case, we have |$\alpha _2^\vee = b\lambda ^\#$| with |$b\in \{1,2\}$|.
If |$\gamma = \alpha _1 + \cdots + \alpha _n$| with support of type |$\mathsf {B}_n$| with |$n \ge 2$|, then locally |$\gamma =\omega _1$|. Here |$\alpha _1^\vee =\lambda ^\#$|.
If |$\gamma =2 \alpha _1 + 2\alpha _2 + \cdots +2 \alpha _n$| with support of type |$\mathsf {B}_n$| with |$n \ge 2$|, then locally |$\gamma =2\omega _1$|. Here |$\alpha _1^\vee =b\lambda ^\#$|, with |$b\in \{1,2\}$|.
If |$\gamma = \alpha _1 + 2\alpha _2 + 3 \alpha _3$| with support of type |$\mathsf {B}_3$|, then locally |$\gamma =2\omega _3$|. Here |$\alpha _3^\vee =b\lambda ^\#$| with |$b\in \{1,2\}$|.
If |$\gamma = \alpha _1 + 2\alpha _2 + 2\alpha _3 + \cdots +2\alpha _{n-1} + \alpha _n$| with support of type |$\mathsf {C}_n$| with |$n\ge 3$|, then locally |$\gamma =\omega _2$|. Here |$\alpha _2^\vee =\lambda ^\#$|.
If |$\gamma = 2\alpha _1 + 2\alpha _2 + \cdots + 2\alpha _{n-2} + \alpha _{n-1}+\alpha _{n}$| with support of type |$\mathsf {D}_n$| with |$n \ge 4$|, then locally |$\gamma =2\omega _1$|. Here |$\alpha _1^\vee =b\lambda ^\#$| with |$b\in \{1,2\}$|.
If |$\gamma = \alpha _1 + 2\alpha _2 + 3\alpha _3 + 2\alpha _4$| with support of type |$\mathsf {F}_4$|, then locally |$\gamma =\omega _4$|. Here |$\alpha _4^\vee =\lambda ^\#$|.
If |$\gamma = 4\alpha _1 + 2\alpha _2$| with support of type |$\mathsf {G}_2$|, then locally |$\gamma =2\omega _1$|. Here |$\alpha _1^\vee =b\lambda ^\#$| with |$b\in \{1,2\}$|.
If |$\gamma = \alpha _1 + \alpha _2$| with support of type |$\mathsf {G}_2$|, then locally |$\gamma =-\omega _1+\omega _2$|. Here |$\alpha _2^\vee =\lambda ^\#$|.
5 The Irreducible Components of |${\text M}_{\Gamma }$|
In this section, we prove the following theorem.
Let |$\Gamma $| be a free monoid of dominant weights. Then the |$T_{{\text ad}}$|-orbit closures in |${\text M}_{\Gamma }$|, equipped with their reduced induced scheme structure, are affine spaces.
The proof is given below. By [2, Proposition 2.13] this theorem has the following formal consequence.
If |$X$| is an affine spherical variety with free weight monoid, then its root monoid |${\mathscr {M}}_X$| is free too.
Another consequence is that Conjecture 1.1 holds for free monoids.
If |$\Gamma $| is a free monoid of dominant weights, then the irreducible components of |${\text M}_{\Gamma }$|, equipped with their reduced induced scheme structure, are affine spaces.
Since the |$T_{{\text ad}}$|-orbits in |${\text M}_{\Gamma }$| are in bijection with isomorphism classes of affine spherical |$G$|-varieties, by [2, Theorem 1.12] and there are only finitely many such isomorphism classes, by [2, Corollary 3.4], we have that every irreducible component |$Z$| of |${\text M}_{\Gamma }$| contains a dense |$T_{{\text ad}}$|-orbit. It then follows from Theorem 5.1 that |$Z$|, equipped with its reduced induced scheme structure, is an affine space.
Recall that |$\Sigma ^N(X)$| is the basis of the monoid obtained by saturation of the root monoid |${\mathscr {M}}_X$|. To deduce (5.1) we make use of Theorem 4.1: the |$T_{{\text ad}}$|-weights in |${\text T}_{X_0}(\overline {T_{{\text ad}}\cdot X})\subseteq {\text T}_{X_0}{\text M}_{\Gamma }$| are spherical roots N-adapted to |$\Gamma $|, each one occurring with multiplicity 1. This, together with the fact that every |$T_{{\text ad}}$|-weight in |${\text T}_{X_0}(\overline {T_{{\text ad}}\cdot X})$| has to be an element of the root monoid |${\mathscr {M}}_X$|, and hence a nonnegative integer linear combination of elements of |$\Sigma ^N(X)$|, gives (5.1) once we prove Proposition 5.4. Indeed, applying this proposition with |$\Sigma = \Sigma ^N(X)$| yields that the |$T_{{\text ad}}$|-weights in |${\text T}_{X_0}(\overline {T_{{\text ad}}\cdot X})$| belong to |$\Sigma ^N(X)$|, while |$\dim \overline {T_{{\text ad}}\cdot X} = |\Sigma ^N(X)|$| by [2, Proposition 2.13].
Let |$\Sigma $| be a subset of |$\Sigma ^{sc}(G)$| such that every |$\gamma \in \Sigma $| is N-adapted to |$\Gamma $|. If |$\sigma \in \Sigma ^{sc}(G)\cap {\mathbb N}\Sigma $| is N-adapted to |$\Gamma $|, then |$\sigma \in \Sigma $|.
First of all, |$\sigma $| (of spherically closed type) must be compatible with |$S^p(\Gamma )$| and is a nonnegative integer linear combination of other elements of |$\Sigma ^{sc}(G)$| that satisfy the same compatibility condition. This gives strong restrictions. Indeed, |$\sigma $| can only be the sum of two simple roots (equal or not, orthogonal or not). All the other types of spherical roots have support that nontrivially intersects |$S^p(\Gamma )$|, and they can be excluded by a straightforward if somewhat lengthy case-by-case verification.
Moreover, |$\sigma $| cannot be the double of a simple root, say |$2\alpha $|, with |$\alpha \in \Sigma $|, since |$\alpha $| and |$2\alpha $| cannot both be N-adapted to |$\Gamma $|. Indeed, if |$2\alpha $| is N-adapted to |$\Gamma $| then, since |$\langle \alpha ^\vee , 2\alpha \rangle >0$| and |$\alpha ^\vee \in \Gamma ^\vee $|, there exists |$\delta \in E(\Gamma )$| such that |$\langle \delta , 2\alpha \rangle >0$|. Condition (3) of Corollary 2.17 tells us that |$\alpha ^\vee $| is a positive multiple of |$\delta $|. By condition (5) of the same corollary, |$\alpha ^\vee $| is not primitive in |$({\mathbb Z}\Gamma )^*$|. If now |$\alpha \in {\mathbb Z}\Gamma $|, then it follows from |$\langle \alpha ^\vee , \alpha \rangle =2$| that |$\alpha ^\vee =2\delta $| on |${\mathbb Z}\Gamma $|. Hence |$\delta $| is the only element of |$a(\alpha )$| and |$\alpha $| is not N-adapted to |$\Gamma $|.
Analogously, |$\sigma $| cannot be the sum of two orthogonal simple roots, say |$\alpha +\alpha '$|, with |$\alpha $| and |$\alpha '$| in |$\Sigma $|. Indeed, since |$\alpha +\alpha '$| is adapted to |$\Gamma $| and |$\langle \alpha ^\vee ,\alpha \rangle \neq \langle (\alpha ')^\vee ,\alpha \rangle $|, |$\alpha $| cannot belong to |${\mathbb Z}\Gamma $|.
Finally, let |$\sigma $| be the sum of two nonorthogonal simple roots, say |$\alpha _1+\alpha _2$|, with |$\alpha _1$| and |$\alpha _2$| in |$\Sigma $|. Take |$\delta \in E(\Gamma )$| with |$\langle \delta ,\sigma \rangle >0$|. Such a |$\delta $| exists because |$\langle \alpha _1^{\vee }, \sigma \rangle $| or |$\langle \alpha _2^\vee , \sigma \rangle $| is positive, |$\sigma \in {\mathbb Z}\Gamma $| and |$\Gamma \subset \Lambda ^+$|. Then |$\delta $| must be positive on at least one of the two simple roots |$\alpha _1$| or |$\alpha _2$|. Suppose it is positive on |$\alpha _1$|. Then |$\delta \in a(\alpha _1)$|, since |$\alpha _1$| is N-adapted to |$\Gamma $|, hence |$\delta $| takes the value 1 on |$\alpha _1$|. By condition (3) of Corollary 2.17 it follows that |$\alpha _1^\vee =2\delta $|, which is not possible if |$\alpha _1$| is N-adapted to |$\Gamma $|.
While the reduced induced scheme structure is the only natural scheme structure on the |$T_{{\text ad}}$|-orbit closures of Theorem 5.1, there is at least one other natural scheme structure on the irreducible components of |${\text M}_{\Gamma }$|, namely the one given by the primary ideals of |${\mathbb k}[{\text M}_{\Gamma }]$| associated with minimal primes. One can ask whether Conjecture 1.1 remains true for that scheme structure. Another natural question is whether or when |${\text M}_{\Gamma }$| is in fact a reduced scheme. We note that the tangent space |${\text T}_{X_0}{\text M}_{\Gamma }$| might fail to detect the “non-reducedness” of |${\text M}_{\Gamma }$|. For example, the two affine schemes |${\text Spec}({\mathbb k}[x,y]/\langle xy\rangle )$| and |${\text Spec}({\mathbb k}[x,y]/\langle x^2y\rangle )$| have the same tangent space at the point corresponding to the maximal ideal |$\langle x,y\rangle $|.
Funding
The second author received support from The City University of New York PSC-CUNY Research Award Program and from the National Science Foundation through grant DMS 1407394.
Note added during review
While this paper was under review, a second version of the preprint [3] was posted on the arXiv, in which Avdeev and Cupit-Foutou propose a proof of Conjecture 1.1 for all normal monoids |$\Gamma $| and an example of a non-reduced moduli scheme |${\text M}_{\Gamma }$| (cf. Remark 5.5).
Acknowledgements
The authors are grateful to the Institut Fourier for hosting them in the summer of 2011, when work on this project began. They also thank the Centro Internazionale per la Ricerca Matematica in Trento, as well as Friedrich Knop and the Emmy Noether Zentrum in Erlangen for their hospitality in the summers of 2012 and 2013, respectively.
The authors are grateful to Michel Brion for suggesting this problem and for helpful discussion. They also thank Domingo Luna for discussion and suggestions in the summer of 2011 and for sharing his working paper of 2005; they were particularly helpful for Section 2. They thank Jarod Alper for a clarifying exchange, summarized in Remark 5.5, about scheme structures on irreducible components of affine schemes. It alerted them to a mistake in an earlier version of this paper.
The authors thank the referee for several helpful suggestions, and in particular for providing an elementary proof of Proposition 2.10(b) and for correcting an error in an earlier version of the proof of (d) and (e) of the same proposition.
As this paper was being completed, R. Avdeev and S. Cupit-Foutou announced that they had independently obtained similar results [3].