Orbits of strongly solvable spherical subgroups on the flag variety

Article

Abstract

Let G be a connected reductive complex algebraic group and B a Borel subgroup of G. We consider a subgroup \(H \subset B\) which acts with finitely many orbits on the flag variety G / B, and we classify the H-orbits in G / B in terms of suitable root systems. As well, we study the Weyl group action defined by Knop on the set of H-orbits in G / B, and we give a combinatorial model for this action in terms of weight polytopes.

Keywords

Strongly solvable spherical subgroups Orbits of a Borel subgroup Weight polytopes 

Mathematics Subject Classification

14M10 14M27 

1 Introduction

Let G be a connected complex reductive algebraic group and let \(B \subset G\) be a Borel subgroup. A subgroup \(H \subset G\) is called spherical if it has an open orbit on the flag variety G / B. If this is the case, as independently proved by Brion [7] and Vinberg [35], then H acts with finitely many orbits on G / B.

The best-known example is that of B itself and more generally any parabolic subgroup of G. The B-orbits in G / B are indeed the Schubert cells, finitely many thanks to the Bruhat decomposition. Another well-studied case is that of the symmetric subgroups of G, i.e., when H is the subgroup of fixed points of some algebraic involution of G. Especially in this case, the study of the H-orbits on G / B, their classification and the geometry of their closures are important in representation theory (see, e.g., [33, 37]). An equivalent problem is the study of the B-orbits in G / H and the geometry of their closures: they are fundamental objects to understand the topology of G / H and of its embeddings (see [15]).

Spherical subgroups are classified in combinatorial terms, see [8, 22, 25, 26, 27] where several particular classes of subgroups are considered, and the more recent papers [6, 13, 24] where the classification is completed in full generality. Nevertheless, the set \(\mathscr {B}(G/H)\) of the B-orbits in G / H is still far from being understood, essentially except for the cases of the parabolic subgroups and of the symmetric subgroups of G (the latter especially thanks to the work of Richardson and Springer [31, 32]).

The goal of the present paper is to explicitly understand the set \(\mathscr {B}(G/H)\) in some other case and to produce some combinatorial model for it. More precisely, we consider the case of the strongly solvable spherical subgroups of G, that is, spherical subgroups of G which are contained in a Borel subgroup. As a consequence of a theorem of Brion (see [10, Theorem 6]), under these assumptions the H-orbit closures in G / B provide nice generalizations of the Schubert varieties: even though H might be not connected, they are always irreducible, and they have rational singularities, so that in particular they are normal and Cohen–Macaulay.

When H is a strongly solvable spherical subgroup, the set \(\mathscr {B}(G/H)\) has already been studied in the literature in some special cases: Timashev [34] treated the case where \(H = TU'\) (where T denotes a maximal torus of B and \(U'\) the derived subgroup of the unipotent radical of B), and Hashimoto [17] treated the case where \(G = \mathrm {SL}_n\) and H is a Borel subgroup of \(\mathrm {SL}_{n-1}\), regarded as a subgroup of \(\mathrm {SL}_n\).

Let \(\varPhi \) be the root system of G and let W be its Weyl group. Given a strongly solvable spherical subgroup \(H \subset G\), in our main result we give an explicit parametrization of the set \(\mathscr {B}(G/H)\) by attaching to every B-orbit an element of W and a root subsystem of \(\varPhi \). To do this, we build upon known results about the classification of strongly solvable subgroups, which is available in three different forms. The first one was given by Luna in 1993 (see [25]), the second one emerged in the framework of the general classification of spherical subgroups, and the third one, more explicit, has been given recently by Avdeev in [2] (see [3] for a comparison between the three approaches).

Our results on \(\mathscr {B}(G/H)\) also provide a nice description of the action of W on \(\mathscr {B}(G/H)\), defined by Knop in [20] for any spherical subgroup \(H\subset G\). While the simple reflections of W act in a rather explicit way, the resulting action of the entire W is quite difficult to study. When H is strongly solvable, we will see how this action becomes actually very simple: the fact that two B-orbits are in the same W-orbit will boil down to the fact that the associated root subsystems are W-conjugated. This will enable us to give a simple combinatorial model for \(\mathscr {B}(G/H)\) as a finite set endowed with an action of W in terms of “generalized faces” of weight polytopes.

We now explain our results in more detail. Fix a maximal torus \(T \subset B\) and denote by U the unipotent radical of B. We also denote by \(\varPhi \) the root system associated with T, by \(W = N_G(T)/T\) its Weyl group, by \(\varPhi ^+ \subset \varPhi \) the set of positive roots associated with B and by \(\Delta \subset \varPhi ^+\) the corresponding basis of \(\varPhi \). Let H be a strongly solvable spherical subgroup of G, up to conjugation we may assume \(H \subset B\). Up to conjugation by an element of B, we may assume as well that \(T \cap H\) is a maximal diagonalizable subgroup of H. Given \(\alpha \in \varPhi ^+\), let \(U_\alpha \subset U\) be the associated unipotent one-dimensional subgroup.

Two natural sets which are attached to H are the set of active roots
$$\begin{aligned} \varPsi = \{ \alpha \in \varPhi ^+ \, : \,U_\alpha \not \subset H\}, \end{aligned}$$
introduced by Avdeev in [2], and the corresponding set of restricted characters
$$\begin{aligned} \overline{\varPsi }= \{\alpha _{|{T \cap H}} \, : \,\alpha \in \varPsi \}. \end{aligned}$$
From a geometrical point of view, \(\overline{\varPsi }\) is canonically identified with the set of the B-stable prime divisors in G / H which map dominantly to G / B via the natural projection (see Sect. 3.1). Given \(I \subset \overline{\varPsi }\), let \(\varPsi _I \subset \varPsi \) be the subset of those roots \(\alpha \) such that \(\alpha _{|{T \cap H}} \in I\) and set
$$\begin{aligned} \varPhi _I= \mathbb Z\varPsi _I \cap \varPhi . \end{aligned}$$
This is a parabolic subsystem of \(\varPhi \) which is explicitly described once H is described in terms of active roots following Avdeev’s classification, and the intersections \(\varPhi ^\pm _I = \varPhi _I \cap \varPhi ^\pm \) define a subdivision of \(\varPhi _I\) into positive and negative roots.

Given \(w \in W\) and \(I \subset \overline{\varPsi }\), we say that (wI) is a reduced pair if \(w(\varPhi ^+_I) \subset \varPhi ^-\). Our first main result is the following.

Theorem 1

(Corollary 5.4) Let H be a spherical subgroup of G contained in B. There is a natural bijection between the set of reduced pairs and the set of B-orbits in G / H.

Given a reduced pair (wI) we denote by \(\mathscr {O}_{w,I}\) the corresponding B-orbit in G / H. Reduced pairs encode important properties of the corresponding B-orbit, and they can be used to encode combinatorial properties of the entire set \(\mathscr {B}(G/H)\) as well. Denote by \(W_I\) the Weyl group of \(\varPhi _I\), canonically embedded in W. Our next main result describes in these terms the action of W on \(\mathscr {B}(G/H)\).

Theorem 2

(Theorems 6.6 6.9 and Corollary 6.10) Let H be a spherical subgroup of G contained in B and let (wI) be a reduced pair, then \({{\mathrm{Stab}}}_W(\mathscr {O}_{w,I}) = w W_I w^{-1}\). Moreover, two orbits \(\mathscr {O}_{w,I}\) and \(\mathscr {O}_{v,J}\) are in the same W-orbit if and only if \(I = J\), in which case \(\mathscr {O}_{w,I} = wv^{-1} \cdot \mathscr {O}_{v,I}\). In particular, the following formula holds:
$$\begin{aligned} |\mathscr {B}(G/H)| = \sum _{I \subset \overline{\varPsi }} |W/W_I|. \end{aligned}$$

A very special example of a strongly spherical subgroup was treated by Timashev in [34], where the corresponding set of B-orbits is studied. More precisely, let \(U'\) be the derived subgroup of U, namely \(U' = \prod _{\varPhi ^+ {\smallsetminus }\Delta } U_\alpha \), and consider the subgroup \(TU' \subset G\): this is a spherical subgroup of G contained in B, and we have equalities \(\overline{\varPsi }= \varPsi = \Delta \). In this case, the parametrization of the B-orbits in terms of reduced pairs can be proved in a simple way by using the commutation relations among root subgroups, and the set of reduced pairs (wI) is easily seen to be in a W-equivariant bijection with the set of faces of the weight polytope associated with any given dominant regular weight of G. This elegant description generalizes to the case of any strongly solvable spherical subgroup as follows.

Let \(\lambda \) be a regular dominant weight and let \(P = {{\mathrm{conv}}}(W\lambda )\) be the associated weight polytope in \(\Lambda _\mathbb Q= \Lambda \otimes _\mathbb Z\mathbb Q\), where \(\Lambda \) denotes the weight lattice of G. Since \(\lambda \) is regular, the elements \(w\lambda \) with \(w \in W\) are all distinct and coincide with the vertices of P. By a subpolytope of P, we mean the convex hull of a subset of vertices of P. Notice that P is naturally endowed with an action of W which permutes the subpolytopes of P, and we denote by \(\mathscr {S}(P)\) the set of subpolytopes of P.

Given a spherical subgroup \(H \subset G\) contained in B, to any reduced pair (wI) we may associate a subpolytope of P by setting \(\mathscr {S}_{w,I} = {{\mathrm{conv}}}(wW_I\lambda )\). This enables us to reformulate our combinatorial model of \(\mathscr {B}(G/H)\) as follows.

Theorem 3

(Theorem 6.14) Let H be a spherical subgroup of G contained in B, then the map \((w,I) \mapsto \mathscr {S}_{w,I}\) defines a W-equivariant injective map \(\mathscr {B}(G/H) \longrightarrow \mathscr {S}(P)\). Moreover, \(\mathscr {S}_{w,I}\) is described as the intersection of P with a cone in \(\Lambda _\mathbb Q\) as follows
$$\begin{aligned} \mathscr {S}_{w,I} = P \cap \big (w\lambda +\mathbb Q_{\geqslant 0}(w(\varPhi ^-_I)) \big ). \end{aligned}$$

The map \((w,I) \mapsto \mathscr {S}_{w,I}\) is not surjective in general, but applying the last part of the previous theorem it is possible in any given example to describe explicitly the image of the map. Moreover, this map has the advantage of being compatible with the Bruhat order, i.e., the inclusion relation between B-orbit closures: if (vJ) and (wI) are reduced pairs such that \(\mathscr {S}_{v,J} \subset \mathscr {S}_{w,I}\), then \(\mathscr {O}_{v,J} \subset \overline{\mathscr {O}_{w,I}}\) (see Proposition 6.15). Unfortunately, the converse of the previous statement is false in general, and a complete description of the Bruhat order of \(\mathscr {B}(G/H)\) remains an open problem.

In case, \(H = TU'\) is easy to see that the image of the map \((w,I) \mapsto \mathscr {S}_{w,I}\) equals the set of faces of P. It was conjectured by Knop that \(G/TU'\) has the largest number of B-orbits among all the spherical homogeneous spaces for G. Using our formula for the cardinality of \(\mathscr {B}(G/H)\), we prove this in the solvable case.

Theorem 4

(Theorem 7.1) The spherical homogeneous space \(G/TU'\) has the largest number of B-orbits among the homogeneous spherical varieties G / H with H a solvable subgroup of G.

We explain now the structure of the paper. In Sect. 2, we explain our notations and collect some basic facts about spherical varieties and toric varieties. Then we restrict to the case of a spherical subgroup \(H \subset G\) contained in B, and in Sect. 3 we study the variety B / H: this is an affine toric T-variety whose set of T-stable prime divisors is naturally parametrized by \(\overline{\varPsi }\), and whose set of T-orbits is naturally parametrized by the subsets of \(\overline{\varPsi }\). In Sect. 4, we introduce the notion of weakly active root. These are the roots \(\alpha \in \varPhi ^+\) whose associated root subgroup \(U_\alpha \subset B\) acts non-trivially on B / H, and we use them to attach a root system \(\varPhi _I\) to every subset \(I \subset \overline{\varPsi }\) and thus to every T-orbit in B / H. In Sect. 5, we introduce the notion of reduced pair (and the analogous one of extended pair), and we prove Theorem 1. Finally, in Sect. 6 we study the action of W on \(\mathscr {B}(G/H)\) and we prove Theorems 2 and 3, and in Sect. 7 we prove Theorem 4.

2 Notations and preliminaries

2.1 Generalities

All varieties and algebraic groups that we will consider will be defined over the complex numbers. Let G be a connected reductive algebraic group. Given a subgroup \(K \subset G\), we denote by \(\mathcal X(K)\) the group of characters of K and by \(K^\mathrm u\) the unipotent radical of K. The Lie algebra of K will be denoted either by \({{\mathrm{Lie}}}K\) or by the corresponding fraktur letter (here \(\mathfrak k\)). Given \(g \in G\), we set \(K^g = g^{-1}Kg\) and \({}^gK = gKg^{-1}\). If K acts on an algebraic variety X, we denote by \({{\mathrm{Div}}}_K(X)\) the set of K-stable prime divisors of X. If S is a group acting on an algebraic variety X and if \(x \in X\), we denote by \({{\mathrm{Stab}}}_S(x)\) the stabilizer of x in S. If M is a lattice (that is, a free and finitely generated abelian group), the dual lattice is \(M^\vee = {{\mathrm{Hom}}}_{\mathbb Z}(M,\mathbb Z)\), and the corresponding \(\mathbb Q\)-vector space \(M \otimes _{\mathbb Z} \mathbb Q\) is denoted by \(M_\mathbb Q\).

Let B be a connected solvable group, and let \(T \subset B\) be a maximal torus. Suppose that Z is a B-homogeneous variety. The weight lattice of Z is the lattice
$$\begin{aligned} \mathcal X_B(Z) = \{B\text {-weights of nonzero rational } B\text {-eigenfunctions } f \in \mathbb C(Z) \}, \end{aligned}$$
and the rank of Z is by definition the rank of its weight lattice. When the acting group is clear from the context, we will drop the subscript and write simply \(\mathcal X(Z)\). By \(\mathbb C(Z)^{(B)}\), we will denote the set of nonzero rational B-eigenfunctions of Z. We say that \(z \in Z\) is a standard base point for Z (with respect to T) if \({{\mathrm{Stab}}}_T(z)\) is a maximal diagonalizable subgroup of \({{\mathrm{Stab}}}_B(z)\). Notice that standard base points always exist: indeed, since B is connected, every diagonalizable subgroup of B is contained in a maximal torus, and the maximal tori of B are all conjugated (see [18, Theorem 19.3 and Proposition 19.4]). Notice also that, if \(z \in Z\) is standard, then every \(z' \in Tz\) is standard as well.

Lemma 2.1

Let Z be a B-homogeneous variety and let \(z_0 \in Z\) be a standard base point, then \(T z_0\) is a closed T-orbit. If moreover \(H = {{\mathrm{Stab}}}_B(z_0)\), then \(H = (T \cap H) H^\mathrm u\) and the followings equalities hold:
$$\begin{aligned} \mathcal X_B(Z) = \mathcal X(B)^H =\mathcal X_T(Tz_0) \end{aligned}$$
(where \(\mathcal X(B)^{H}\) denotes the subgroup of \(\mathcal X(B)\) of characters that are trivial on H).

Proof

For all \(z \in Z\), the stabilizer \({{\mathrm{Stab}}}_T(z)\) is a diagonalizable subgroup of \({{\mathrm{Stab}}}_B(z)\). On the other hand, Z is B-homogeneous; therefore, all the stabilizers \({{\mathrm{Stab}}}_B(z)\) are isomorphic and the maximal dimension for \({{\mathrm{Stab}}}_T(z)\) is the dimension of a maximal torus of \({{\mathrm{Stab}}}_B(z)\). By definition, \({{\mathrm{Stab}}}_T(z_0)\) contains a maximal torus of \({{\mathrm{Stab}}}_B(z)\); therefore, \(T z_0\) has minimal dimension in Z; hence, it is closed. The last claim is immediate. \(\square \)

From now on, B will denote a Borel subgroup of G, \(T\subset B\) a maximal torus, and \(U = B^\mathrm u\) the unipotent radical of B. Let \(\varPhi \subset \mathcal X(T)\) be the root system of G associated with T, \(\varPhi ^+\) (resp. \(\varPhi ^-\)) the set of positive (resp. negative) roots determined by B and \(\Delta \subset \varPhi ^+\) the corresponding set of simple roots. When dealing with an explicit irreducible root system, we will enumerate the simple roots following Bourbaki’s notation [4]. We also set \(\Delta ^- = -\Delta \).

If \(\beta \in \varPhi ^+\) and \(\alpha \in \Delta \), we denote by \([\beta :\alpha ]\) the coefficient of \(\alpha \) in \(\beta \) as a sum of simple roots, and we define the support of \(\beta \) as
$$\begin{aligned} {{\mathrm{supp}}}(\beta ) = \{\alpha \in \Delta \, : \,[\beta :\alpha ] > 0\}. \end{aligned}$$
Let \(W = N_G(T)/T\) be the Weyl group of G with respect to T. If \(\alpha \in \varPhi \), we denote by \(s_\alpha \in W\) the corresponding reflection and by \(U_\alpha \subset G\) the unipotent root subgroup associated with \(\alpha \). If \(\alpha \in \Delta \), we denote by \(P_\alpha \) the minimal parabolic subgroup of G containing B associated with \(\alpha \). If \(w \in W\), we denote by \(\varPhi ^+(w)\) the corresponding inversion set, i.e.,
$$\begin{aligned} \varPhi ^+(w) = \{\alpha \in \varPhi ^+ \, : \,w(\alpha ) \in \varPhi ^-\}, \end{aligned}$$
and by l(w) the length of w that is the cardinality of \(\varPhi ^+(w)\). Denote by \(w_0\) be the longest element of W. If \(\mathscr {O}\) is a T-stable subset of a G-variety and \(n\in N_G(T)\), then \(n\mathscr {O}\) only depends on the class w of n in W; therefore, we will denote \(n\mathscr {O}\) simply by \(w\mathscr {O}\).

2.2 Spherical varieties and toric varieties

An irreducible normal G-variety X is called spherical if it contains an open B-orbit. See [19] as a general reference for spherical varieties. In particular, X contains an open G-orbit, which is a spherical homogeneous variety. Following [7] and [35], X is spherical if and only if B possesses finitely many orbits on it. Therefore, following the definition we gave in the introduction, a homogeneous variety G / H is spherical if and only if H is a spherical subgroup of G.

Let X be a spherical G-variety. Every B-stable prime divisor \(D \in {{\mathrm{Div}}}_B(X)\) induces a discrete valuation \(\nu _D\) on \(\mathbb C(X)\), which is trivial on the constant functions. On the other hand, since X contains an open B-orbit, every function \(f \in \mathbb C(X)^{(B)}\) is uniquely determined by its weight up to a scalar factor. By restricting valuations to \(\mathbb C(X)^{(B)}\), we get then a map
$$\begin{aligned} \rho : {{\mathrm{Div}}}_B(X) \longrightarrow \mathcal X(X)^\vee , \end{aligned}$$
defined by \(\langle \rho (D), \chi \rangle = \nu _D(f_\chi )\) where \(f_\chi \in \mathbb C(X)^{(B)}\) is any nonzero B-semiinvariant function of weight \(\chi \).

When \(G = B = T\) is a torus, we will also say that X is a toric variety. Notice that, differently form the standard literature on toric varieties, we will not assume that the action of T on X is effective. As a general reference on toric varieties, see [12].

Let X be an affine toric T-variety. The cone ofX is the rational polyhedral cone in \(\mathcal X(X)^\vee _\mathbb Q\) defined as
$$\begin{aligned} \sigma _X = {{\mathrm{cone}}}(\rho (D) \, : \,D \in {{\mathrm{Div}}}_T(X)) \end{aligned}$$
This cone encodes important information on the geometry of X. In particular, there is an order reversing one-to-one correspondence between the set of T-orbits on X (ordered with the inclusion of orbit closures) and the set of faces of \(\sigma _X\) (ordered with the inclusion). Given a face \(\tau \) of \(\sigma _X\), we denote by \(\mathscr {U}_\tau \) the corresponding T-orbit in X; in particular, \(\mathscr {U}_\sigma \) is the unique closed orbit of X and \(\mathscr {U}_0\) is the open orbit of X.

Definition 2.2

(see [23, Definition 2.3]) Let X be an affine toric T-variety. An element \(\alpha \in \mathcal X(X)\) is called a root of X if there exists \(\delta (\alpha ) \in {{\mathrm{Div}}}_T(X)\) such that \(\langle \rho (\delta (\alpha )), \alpha \rangle = -1\) and \(\langle \rho (D), \alpha \rangle \geqslant 0\) for all \(D \in {{\mathrm{Div}}}_T(X) {\smallsetminus }\{\delta (\alpha )\}\).

We will denote by \({{\mathrm{Root}}}(X)\) the set of roots of an affine toric variety X. Notice that by its definition \({{\mathrm{Root}}}(X)\) comes with a map
$$\begin{aligned} \delta : {{\mathrm{Root}}}(X) \longrightarrow {{\mathrm{Div}}}_T(X). \end{aligned}$$
(1)
For the notion of root of a toric variety, which goes back to Demazure, see also [1, Definition 2.1] and [29, Proposition 3.13]. Following [14] and [23], there is a one-to-one correspondence between the roots of X and the one-parameter unipotent subgroups of \({{\mathrm{Aut}}}(X)\) normalized by T, see [23, §2] and [1, §2]. More precisely, every \(\alpha \in {{\mathrm{Root}}}(X)\) defines a locally nilpotent derivation \(\partial _\alpha \) of the graded algebra \(\mathbb C[X]\), which acts on \(\mathbb C[X]\) by the rule
$$\begin{aligned} \partial _\alpha (f_\chi ) = \langle \rho (\delta (\alpha )), \chi \rangle f_{\chi + \alpha } \end{aligned}$$
(2)
where \(\chi \in \mathcal X(X)\) and where \(f_\chi \in \mathbb C(X)^{(T)}\) has weight \(\chi \). Notice that the line \(\mathbb C\partial _\alpha \) is fixed by the action of T, which acts on it via the weight \(\alpha \). By exponentiating \(\partial _\alpha \), we get then a one-parameter unipotent subgroup \(V_\alpha \subset {{\mathrm{Aut}}}(X)\) normalized by T, that is, \(V_\alpha = \lambda _\alpha (\mathbb C)\), where \(\lambda _\alpha \) denotes the one-parameter subgroup \(\xi \mapsto \exp (\xi \partial _\alpha )\).

We gather in the following proposition some properties that we will need later of the action of the group \(V_\alpha \) on the T-orbits of X.

Proposition 2.3

([1, Lemma 2.1, Proposition 2.1 and Lemma 2.2]) Let X be an affine toric T-variety, let \(\tau , \tau '\) be faces of \(\sigma _X\) and let \(\alpha \in {{\mathrm{Root}}}(X)\).
  1. (i)

    \(V_\alpha \mathscr {U}_\tau \) decomposes into the union of at most two T-orbits.

     
  2. (ii)

    \(\delta (\alpha )\) is the unique T-stable prime divisor of X which is not stable under the action of \(V_\alpha \).

     
  3. (iii)

    Suppose that \(\mathscr {U}_{\tau '} \subset \overline{\mathscr {U}_\tau }\), then \(\mathscr {U}_{\tau '} \subset V_\alpha \mathscr {U}_{\tau }\) if and only if \(\alpha _{|\tau '} \leqslant 0\) and \(\tau \) is the codimension one face of \(\tau '\) defined by the equation \(\tau = \tau ' \cap \ker \alpha \).

     

3 Strongly solvable spherical subgroups and associated toric varieties

From now on, if not differently stated, H will be a strongly solvable spherical subgroup of G. Up to conjugating H in G, we may assume that H is contained in B, in which case \(H^\mathrm u= U \cap H\). Up to conjugating H in B, we may and will also assume that \(T_H = T \cap H\) is a maximal diagonalizable subgroup of H. By Lemma 2.1, it follows then that \(TH/H \simeq T/T_H\) is a closed T-orbit in B / H and that \(\mathcal X_B(B/H) = \mathcal X_T(T/T_H)\). In particular, we get the equality \({{\mathrm{rk}}}B/H = {{\mathrm{rk}}}G - {{\mathrm{rk}}}H\).

In order to classify the strongly solvable spherical subgroups H of G, Avdeev introduced the active roots of H, defined as
$$\begin{aligned} \varPsi = \{ \alpha \in \varPhi ^+ \, : \,U_\alpha \not \subset H \}, \end{aligned}$$
and developed a combinatorial theory of such roots. We refer to [2, 3] for details on this construction.

3.1 The structure of B / H as a toric variety

Consider the projection \(G/H \rightarrow G/B\), denote by \(\mathscr {B}^*(G/H)\) the set of the B-orbits in G / H which project dominantly on G / B and by \({{\mathrm{Div}}}_B^*(G/H)\) the set of the B-stable prime divisors of G / H which project dominantly on G / B. Since G / H possesses finitely many B-orbits, \({{\mathrm{Div}}}_B^*(G/H)\) equals the set of closures of the codimension one B-orbits in \(\mathscr {B}^*(G/H)\).

Since it will be a fundamental object in what follows, we denote
$$\begin{aligned} \mathscr {D}= {{\mathrm{Div}}}_T(B/H). \end{aligned}$$

Proposition 3.1

The map \(\mathscr {O}\mapsto w_0 \mathscr {O}\cap B/H\) induces a bijection between \(\mathscr {B}^*(G/H)\) and the set of T-orbits in B / H, which preserves codimensions and inclusions of orbit closures. In particular, the map \(D \mapsto w_0 D \cap B/H\) induces a bijection between \({{\mathrm{Div}}}_B^*(G/H)\) and \(\mathscr {D}\).

Proof

Let \(\mathscr {O}\in \mathscr {B}^*(G/H)\). The image of \(\mathscr {O}\) in G / B is the dense B-orbit \(Bw_0B/B\); therefore, there exists \(u \in B\) such that \(\mathscr {O}= Bw_0 uH/H\). Then,
$$\begin{aligned} w_0 \mathscr {O}\cap B/H = (B \cap B^{w_0}) u H/H = T uH/H, \end{aligned}$$
that is, \(w_0 \mathscr {O}\cap B/H\) is a single T-orbit. Conversely, if \(\mathscr {U}\subset B/H\) is a T-orbit, then we get an element \(\mathscr {O}\in \mathscr {B}^*(G/H)\) by setting \(\mathscr {O}= Bw_0 \mathscr {U}\). The equalities \(\mathscr {O}= Bw_0(w_0 \mathscr {O}\cap B/H)\) and \(\mathscr {U}= w_0 (Bw_0\mathscr {U}) \cap B/H\) imply the first claim, and the rest is an obvious consequence. \(\square \)

Corollary 3.2

B / H is a smooth affine toric T-variety.

Proof

Since it is homogeneous under the action of a solvable group, B / H is smooth and affine (see, e.g., [34, Lemma 2.12]). Notice also that B / H is irreducible since B is connected. Therefore, the fact that B / H is toric under the action of T follows by the sphericity of H thanks to Proposition 3.1, since \(\mathscr {B}^*(G/H)\) is a finite set. \(\square \)

Proposition 3.3

The following hold.
  1. (i)

    As a T-variety, the weight lattice of B / H is \(w_0\mathcal X(G/H)\).

     
  2. (ii)

    For any \(D\in {{\mathrm{Div}}}_B^*(G/H)\), the T-invariant valuation of B / H defined by the T-stable prime divisor \(w_0 D\cap B/H \in \mathscr {D}\) coincides with \(w_0 \rho (D)\).

     

Proof

Up to twisting the T-action by \(w_0\), the T-varieties B / H and \(w_0B/H\) are isomorphic. Since the B-orbit of \(w_0B/B\) is dense in G / B and since \(B\cap B^{w_0}=T\), the T-stable prime divisors of \(w_0 B/H\) coincide with the intersections \(D \cap w_0 B/H\), where \(D \in {{\mathrm{Div}}}_B^*(G/H)\). To prove the proposition, we may then replace B / H with \(w_0 B/H\) and show instead the equality \(\mathcal X_T(w_0 B/H) = \mathcal X_B(G/H)\) and that the T-invariant valuation of \(w_0B/H\) defined by \(D\cap w_0B/H\) is \(\rho (D)\).

Consider the open subset \(Bw_0B/H \subset G/H\), and notice that we have an isomorphism \(Bw_0B/H \simeq B\times ^T w_0 B/H\). Then, restriction to \(w_0 B/H\) induces a bijection between B-semiinvariant rational functions on G / H and T-semiinvariant rational functions on \(w_0 B/H\), and i) follows.

Identifying \(\mathbb C(G/H)^{(B)}\) and \(\mathbb C(w_0B/H)^{(T)}\), we get as well an identification between the discrete valuation of \(\mathbb C(G/H)\) associated with a B-stable prime divisor \(D \subset G/H\) which intersects \(Bw_0B/H\) (hence \(w_0 B/H\)) with the discrete valuation of \(\mathbb C(w_0 B/H)\) associated with \(D\cap w_0 B/H\), and ii) follows. \(\square \)

The set of active roots of H is closely related to the structure of B / H as a toric variety. Consider indeed the Levi decompositions \(B =TU \simeq T \ltimes U\) and \(H = T_HH^\mathrm u\simeq T_H \ltimes H^\mathrm u\). These induce a projection \(B/H \rightarrow T/T_H\), and a T-equivariant isomorphism
$$\begin{aligned} B/H \simeq T\times ^{T_H} U/H^\mathrm u. \end{aligned}$$
(3)
In other words, B / H is a homogeneous vector bundle over \(T/T_H\), with fiber the \(T_H\)-module \(U/H^\mathrm u\). Therefore, the T-orbits in B / H correspond naturally to the \(T_H\)-orbits in \(U/H^\mathrm u\), and it follows that \(U/H^\mathrm u\) possesses an open \(T_H\)-orbit.
Denote \(\tau : \mathcal X(T) \rightarrow \mathcal X(T_H)\) the restriction map and set
$$\begin{aligned} \overline{\varPsi }= \{\tau (\alpha ) \, : \,\alpha \in \varPsi \}. \end{aligned}$$
Since \(T_H\) is a diagonalizable group, the \(T_H\)-module structure of \(U/H^\mathrm u\) is completely determined by its \(T_H\)-weights, namely by \(\overline{\varPsi }\). The fact that \(U/H^\mathrm u\) possesses an open \(T_H\)-orbit is then equivalent to the fact that it is a multiplicity-free\(T_H\)-module, that is \(\overline{\varPsi }\) is linearly independent. Given \(\pi \in \mathcal X(T_H)\), we denote by \(\mathbb C_\pi \) the one-dimensional \(T_H\)-module defined by \(\pi \). By [28, Lemma 1.4], the exponential map induces a \(T_H\)-equivariant isomorphism \(\mathfrak u/\mathfrak h^\mathrm u\rightarrow U/H^\mathrm u\); hence, we get isomorphisms of \(T_H\)-modules
$$\begin{aligned} U/H^\mathrm u\simeq \mathfrak u/\mathfrak h^\mathrm u\simeq \bigoplus _{\pi \in \overline{\varPsi }} \mathbb C_\pi . \end{aligned}$$
(4)
In particular, \(\overline{\varPsi }\) parametrizes the set \({{\mathrm{Div}}}_{T_H}(U/H^\mathrm u)\), and summarizing the previous discussion we get canonical bijections
$$\begin{aligned} {{\mathrm{Div}}}_B^*(G/H) \longleftrightarrow \mathscr {D}\longleftrightarrow {{\mathrm{Div}}}_{T_H}(U/H^\mathrm u) \longleftrightarrow \overline{\varPsi }. \end{aligned}$$
(5)

Definition 3.4

A root \(\alpha \in \varPhi ^+\) is called weakly active for H if \(U_\alpha \) acts non-trivially on B / H.

We denote by \(\varPsi ^\mathrm {\sharp }\subset \varPhi ^+\) the set of weakly active roots of H.

As an immediate consequence of the definitions, notice that every active root is weakly active. Moreover, if \(\alpha \in \varPsi ^\mathrm {\sharp }\), by definition we have a non-trivial homomorphism \(U_\alpha \rightarrow {{\mathrm{Aut}}}(B/H)\), which must be injective since \(U_\alpha \) has no non-trivial proper subgroups (see [18, Theorem 20.5]). On the other, hand \(U_\alpha \) is normalized by T, which by definition acts on the Lie algebra \(\mathfrak u_\alpha \) with the character \(\alpha \in \mathcal X(T)\). By the one-to-one correspondence between one-parameter unipotent subgroups of \({{\mathrm{Aut}}}(B/H)\) normalized by T and roots of B / H recalled in Sect. 2.2, it follows that every weakly active root for H is a root for B / H in the sense of Definition 2.2, and in the notation therein we have an isomorphism \(U_\alpha \simeq V_\alpha \subset {{\mathrm{Aut}}}(B/H)\). Therefore, we have \(\varPsi ^\mathrm {\sharp }\subset {{\mathrm{Root}}}(B/H)\), and in particular \(\varPsi ^\mathrm {\sharp }\subset \mathcal X_T(B/H)\).

Following Definition 2.2, the inclusion \(\varPsi ^\mathrm {\sharp }\subset {{\mathrm{Root}}}(B/H)\) yields by restriction a map
$$\begin{aligned} \delta _{|\varPsi ^\mathrm {\sharp }} : \varPsi ^\mathrm {\sharp }\longrightarrow \mathscr {D}\end{aligned}$$

Proposition 3.5

Let \(\alpha \in \varPsi ^\mathrm {\sharp }\). Then, \(\delta (\alpha )\) is uniquely determined by \(\tau (\alpha )\), and it is the unique T-stable prime divisor of B / H which is not \(U_\alpha \)-stable.

Proof

By (3), the T-stable prime divisors of B / H correspond to the \(T_H\)-stable prime divisors of \(U/H^\mathrm u\). If \(D \in \mathscr {D}\), it follows that the evaluation of \(\alpha \) along D coincides with the evaluation of its restriction \(\tau (\alpha ) \in \mathcal X_{T_H}(U/H^\mathrm u)\) along the intersection \(D \cap U/H^\mathrm u\in {{\mathrm{Div}}}_{T_H}(U/H^\mathrm u)\). In particular, \(\tau (\alpha ) \in {{\mathrm{Root}}}(U/H^\mathrm u)\), and \(\delta (\alpha )\) is uniquely determined by the restriction \(\tau (\alpha )\).

As we already noticed, the morphism of algebraic groups \(B \rightarrow {{\mathrm{Aut}}}(B/H)\) maps \(U_\alpha \) onto the one-parameter unipotent subgroup \(V_\alpha \subset {{\mathrm{Aut}}}(B/H)\) determined by \(\alpha \) as a root of B / H. Therefore, the second claim follows by Proposition 2.3. \(\square \)

We will need the following characterization of the active roots in terms of the corresponding evaluation.

Theorem 3.6

([3, Theorem 5.34 (b)]) Let \(\alpha \in \mathbb N\varPhi ^+\) and assume that \(\alpha \in \mathbb Z\varPsi \). Then, \(\alpha \in \varPsi \) if and only if there exists \(D_\alpha \in \mathscr {D}\) such that \(\langle \rho (D_\alpha ), \alpha \rangle = -1\) and \(\langle \rho (D), \alpha \rangle = 0\) for all \(D \in \mathscr {D}{\smallsetminus }\{D_\alpha \}\), in which case \(D_\alpha = \delta (\alpha )\).

For \(\alpha \in \varPsi ^\mathrm {\sharp }\), let \(f_{-\alpha } \in \mathbb C(B/H)^{(T)}\) be an eigenfunction of weight \(-\alpha \) (uniquely determined up to a scalar factor). Since \(\alpha \in {{\mathrm{Root}}}(B/H)\), it follows by Definition 2.2 that \(f_{-\alpha }\) vanishes with order 1 along \(\delta (\alpha )\), and \(\delta (\alpha )\) is the unique T-stable prime divisor of B / H where \(f_{-\alpha }\) vanishes. Notice that, when \(\alpha \in \varPsi \), the function \(f_{-\alpha }\) is nothing but the obvious lifting of the coordinate of \(\mathfrak u/\mathfrak h^\mathrm u\) corresponding to \(\tau (\alpha )\) via (3) and (4).

Corollary 3.7

Let \(\alpha \in \varPsi ^\mathrm {\sharp }\). Then, \(\alpha \in \varPsi \) if and only if \(\langle D, \alpha \rangle \leqslant 0\) for all \(D \in \mathscr {D}\). If moreover \(\alpha \in \varPsi \), then \(f_{-\alpha } \in \mathbb C[B/H]\) is a global equation for \(\delta (\alpha )\), and \(u \delta (\alpha ) \cap \delta (\alpha )= \varnothing \) for all non-trivial element \(u \in U_\alpha \).

Proof

The first implication follows by Theorem 3.6. Suppose that \(\alpha \in \varPsi ^\mathrm {\sharp }\) and suppose that \(\langle \rho (D), \alpha \rangle \leqslant 0\) for all \(D \in \mathscr {D}\). It follows that \(f_{-\alpha }\) has no pole on any T-stable divisor of the smooth variety B / H; hence, \(f_{-\alpha } \in \mathbb C[B/H]\). Moreover, \(f_{-\alpha }\) is a global equation for \(\delta (\alpha )\), because it vanishes with order 1 on \(\delta (\alpha )\), and it is nonzero on every other T-stable prime divisor of B / H.

We claim that \(u \delta (\alpha ) \cap \delta (\alpha ) = \varnothing \) for all non-trivial elements \(u \in U_\alpha \). Let indeed \(\partial _\alpha \) be the locally nilpotent derivation of \(\mathbb C[B/H]\) associated with \(\alpha \). By (2), it follows that \(\partial _\alpha (f_{-\alpha })\) is a nonzero function, which is constant on B / H. Exponentiating \(\partial _\alpha \), we obtain \(f_{-\alpha }(u_\alpha (\xi )x) = f_{-\alpha }(x)+\xi \) for all \(\xi \in \mathbb C\) (where \(\mathbb C\) is regarded as the T-module of weight \(\alpha \) and \(u_\alpha : \mathbb C\rightarrow U_\alpha \) is a T-equivariant parametrization). Therefore, \(u(\xi ) \delta (\alpha ) \cap \delta (\alpha ) = \varnothing \) for all \(\xi \in \mathbb C\) different from zero.

Since \(TH/H \subset B/H\) is the unique closed T-orbit, it holds \(TH/H \subset \delta (\alpha )\). Therefore, by the previous discussion we get \(U_\alpha T H/H \not \subset T H/H\), namely \(U_\alpha \not \subset H\). \(\square \)

Corollary 3.8

Let \(\alpha ,\beta \in \varPsi \), then \(\delta (\alpha ) = \delta (\beta )\) if and only if \(\tau (\alpha ) = \tau (\beta )\).

Proof

We already noticed in Proposition 3.5 that for all \(\alpha \in \varPsi ^\mathrm {\sharp }\) the divisor \(\delta (\alpha )\) is uniquely determined by the restriction \(\tau (\alpha )\). Conversely, if \(\alpha , \beta \in \varPsi \) and \(\delta (\alpha ) = \delta (\beta )\), then by Corollary 3.7 it follows that the restrictions \((f_{-\alpha })_{| U/H^\mathrm u}\) and \((f_{-\beta })_{| U/H^\mathrm u}\) both belong to \(\mathbb C[U/H^\mathrm u]^{(T_H)}\) and they are both global equation for the \(T_H\)-stable prime divisor \(\delta (\alpha ) \cap U/H^\mathrm u= \delta (\beta ) \cap U/H^\mathrm u\).

On the other hand, by (4) \(U/H^\mathrm u\) is \(T_H\)-equivariantly isomorphic to the toric module \(\bigoplus _{\pi \in \overline{\varPsi }} \mathbb C_\pi \), and the \(T_H\)-stable prime divisors of \(U/H^\mathrm u\) are precisely the coordinate hyperplanes of \(\bigoplus _{\pi \in \overline{\varPsi }} \mathbb C_\pi \). Therefore, under the isomorphism (4) the restrictions of \(f_{-\alpha }\) and \(f_{-\beta }\) both correspond to the unique coordinate corresponding to \(\delta (\alpha ) \cap U/H^\mathrm u= \delta (\beta ) \cap U/H^\mathrm u\). It follows that \((f_{-\alpha })_{| U/H^\mathrm u}\) and \((f_{-\beta })_{| U/H^\mathrm u}\) have the same \(T_H\)-weight, namely \(\tau (\alpha ) = \tau (\beta )\).

Corollary 3.9

Via the bijection \(\mathscr {D}\leftrightarrow \overline{\varPsi }\) of (5), the map \(\delta _{|\varPsi } : \varPsi \rightarrow \mathscr {D}\) is identified with \(\tau _{|\varPsi }: \varPsi \rightarrow \overline{\varPsi }\). In particular, \(\delta _{|\varPsi }\) is surjective.

3.2 Orbits of T on B / H via active roots

Since it is an affine toric variety, B / H possesses a unique closed T-orbit, namely TH / H, which is contained in every T-stable divisor of B / H. Moreover, since it is smooth, the cone associated with B / H is simplicial; therefore, the T-orbits in B / H are parametrized by the subsets of \(\mathscr {D}\). Given \(I \subset \mathscr {D}\), we denote by \(\mathscr {U}_I\) the corresponding T-orbit in B / H. To be more explicit, by Proposition 3.3\(\mathscr {U}_I\) is defined by the equality
$$\begin{aligned} I = \{D \in \mathscr {D}\, : \,\mathscr {U}_I \not \subset D\}. \end{aligned}$$
In particular, \(\mathscr {U}_\varnothing \) is the closed orbit of B / H, and \(\mathscr {U}_\mathscr {D}\) is the open orbit. The goal of this subsection is to give an explicit description of the T-orbits \(\mathscr {U}_I\), by giving canonical base points defined in terms of the active roots of H.

Proposition 3.10

Fix an enumeration \(\mathscr {D}= \{D_1, \ldots , D_m\}\), let \(1 \leqslant i_1< \ldots < i_p \leqslant m\), and set \(I = \{D_{i_1}, \ldots , D_{i_p}\}\). For every choice of elements \(\beta _1, \ldots , \beta _p \in \varPsi \) with \(\delta (\beta _1) = D_{i_1}, \ldots , \delta (\beta _p) = D_{i_p}\) and for every choice of non-trivial elements \(u_1 \in U_{\beta _1}, \ldots , u_p \in U_{\beta _p}\), it holds the equality
$$\begin{aligned} \mathscr {U}_I = T u_1 \cdots u_p H/H. \end{aligned}$$

Proof

Denote \(\mathscr {U}= T u_1 \cdots u_p H/H\). The point \(u_1 \cdots u_p H/H\) is obtained by successively applying the elements \(u_j\) on the base point \(H \in B/H\), which is contained in every T-stable prime divisor of B / H. By Proposition 3.5, the root group \(U_{\beta _j}\) stabilizes every \(D \in \mathscr {D}{\smallsetminus }\{\delta (\beta _j)\}\). Therefore, the definition of \(\beta _1, \ldots \beta _p\) implies that \(\mathscr {U}_I\) is contained in every D, for all \(D \in \mathscr {D}{\smallsetminus }\{ D_{i_1}, \ldots , D_{i_p} \}\).

It remains to prove that \(\mathscr {U}_I\) is not contained in \(\delta (\beta _j)\) for all \(j= 1, \ldots , p\). By Proposition 3.5, it follows that \(\delta (\beta _j)\) is stable under the action of \(U_{\beta _k}\) for all \(k \ne j\), whereas by Corollary 3.7 we have \(u_j \delta (\beta _j) \cap \delta (\beta _j) = \varnothing \). It follows that \(u_{j+1} \cdots u_p H/H \in \delta (\beta _j)\) and \(u_j \cdots u_p H/H \not \in \delta (\beta _j)\). Therefore, \(u_1 \cdots u_p H/H \not \in \delta (\beta _j)\) as well, and the claim follows. \(\square \)

We deduce immediately from Proposition 3.10 the following description of the T-stable prime divisors of B / H in terms of active roots.

Corollary 3.11

With the notation of Proposition 3.10, we have
$$\begin{aligned} \delta (\beta _i) = \overline{T U_{\beta _1} \cdots \hat{U}_{\beta _i} \cdots U_{\beta _m} H/H} \end{aligned}$$

4 Weakly active roots and root systems

In this section, we will extend the study of the active roots to the weakly active roots, and we will give two alternative combinatorial definitions of them. This will provide the technical tools needed for the description of \(\mathscr {B}(G/H)\) given in the next section. We keep the assumptions of the previous section.

Recall the dominance order on \(\mathcal X(T)\), defined by \(\lambda \leqslant \mu \) if and only if \(\mu - \lambda \in \mathbb N\Delta \). Given \(\alpha \in \varPsi \), denote \(F(\alpha )\) the family of active roots generated by\(\alpha \), defined as
$$\begin{aligned} F(\alpha ) = \{ \beta \in \varPsi \, : \,\beta \leqslant \alpha \}. \end{aligned}$$
We collect in the following proposition some properties of the families of active roots from [2] that will be very useful in what follows.

Proposition 4.1

([2, Lemma 5, Corollaries 2 and 3]) Let \(\alpha \in \varPsi \).
  1. (i)

    The family \(F(\alpha )\) is a linearly independent set of roots, and \(\delta (\beta ) \ne \delta (\beta ')\) for all \(\beta , \beta ' \in F(\alpha )\).

     
  2. (ii)

    If \(\beta \in \varPsi \), then \(\beta \in F(\alpha )\) if and only if \({{\mathrm{supp}}}(\beta ) \subset {{\mathrm{supp}}}(\alpha )\).

     

4.1 Weakly active roots

We will give two different combinatorial characterizations of the weakly active roots. To do this, we will need a couple of preliminary lemmas.

Lemma 4.2

Let \(\alpha \in \varPsi ^\mathrm {\sharp }\) and \(\beta \in \varPsi \) be such that \(\langle \rho (\delta (\beta )),\alpha \rangle > 0\). Then, \(\alpha + \beta \in \varPsi ^\mathrm {\sharp }\), and \(\delta (\alpha +\beta ) = \delta (\alpha )\).

Proof

Notice that \(\alpha +\beta \in {{\mathrm{Root}}}(B/H)\): indeed, \(\alpha + \beta \in \mathcal X(B/H)\), and by Theorem 3.6 it satisfies the condition of Definition 2.2. Moreover, as \(\langle \rho (\delta (\beta )), \alpha +\beta \rangle \geqslant 0\), it follows that \(\delta (\alpha +\beta ) = \delta (\alpha )\).

To conclude the proof, we need to show that \(\alpha +\beta \in \varPhi ^+\) and that \(U_{\alpha +\beta }\) acts non-trivially on B / H. As in Sect. 2.2, given \(\gamma \in {{\mathrm{Root}}}(B/H)\), we denote by \(V_\gamma \) the associated one-parameter unipotent subgroup of \({{\mathrm{Aut}}}(B/H)\), so that \({{\mathrm{Lie}}}(V_\gamma )\) acts as a derivation on \(\mathbb C(B/H)\) according to (2). In particular, given \(\gamma _1, \gamma _2 \in {{\mathrm{Root}}}(B/H)\), we get that \(\partial _{\gamma _1}\) and \(\partial _{\gamma _2}\) commute if and only if
$$\begin{aligned} \langle \rho (\delta (\gamma _1)), \gamma _2 \rangle = \langle \rho (\delta (\gamma _2)), \gamma _1 \rangle . \end{aligned}$$
Since \(\langle \rho (\delta (\alpha )), \beta \rangle = 0\) and \(\langle \rho (\delta (\beta )), \alpha \rangle > 0\), it follows by the previous discussion that \(\partial _\alpha \) and \(\partial _\beta \) do not commute; hence, \({{\mathrm{Lie}}}(V_{\alpha +\beta }) = [{{\mathrm{Lie}}}(V_\alpha ),{{\mathrm{Lie}}}(V_\beta )]\). Consider now the homomorphism of algebraic groups \(\psi : B \rightarrow {{\mathrm{Aut}}}(B/H)\) induced by the action of B on B / H. By definition, \(U_\alpha \) and \(U_\beta \) act non-trivially on B / H; hence, \(\psi (U_\alpha ) = V_\alpha \) and \(\psi (U_\beta ) = V_\beta \). Therefore,
$$\begin{aligned} {{\mathrm{Lie}}}(V_{\alpha +\beta }) = [{{\mathrm{Lie}}}(V_\alpha ),{{\mathrm{Lie}}}(V_\beta )] = d\psi [\mathfrak u_\alpha ,\mathfrak u_\beta ]. \end{aligned}$$
It follows that \([\mathfrak u_\alpha ,\mathfrak u_\beta ]\) is nonzero; hence, \(\alpha + \beta \in \varPhi ^+\) and \(\psi (U_{\alpha +\beta }) = V_{\alpha +\beta }\). \(\square \)
Given a root \(\alpha \in \varPhi ^+\), we define the following sets:
$$\begin{aligned} \varPhi ^+(\alpha ) = (\alpha + \mathbb N\varPhi ^+) \cap \varPhi ^+, \qquad \qquad \varPsi (\alpha ) = (\alpha + \mathbb N\varPsi ) \cap \varPsi . \end{aligned}$$

Lemma 4.3

Let \(\alpha \in \varPhi ^+\) and \(\beta \in \varPhi ^+(\alpha ) {\smallsetminus }\{\alpha \}\). There exist \(\beta _0, \ldots , \beta _n \in \varPhi ^+\) with \(\beta _0 = \alpha \), \(\beta _n = \beta \) and \(\beta _i - \beta _{i-1} \in \varPhi ^+\) for all \(i \leqslant n\).

Proof

Let \(\alpha _1, \ldots , \alpha _m \in \varPhi ^+\) with \(\beta = \alpha + \alpha _1 + \ldots + \alpha _m\). Assume that m is minimal with this property, and we prove the claim by induction on m. Set \(\gamma = \beta - \alpha \). If \(m=1\), the claim is true. Assume \(m>1\); then, by the minimality of m it follows that \(\gamma \not \in \varPhi ^+\). Since \(\gamma \in \mathbb N\varPhi ^+\), it follows then \(\gamma \not \in \varPhi \); hence, \((\beta ,\alpha ) \leqslant 0\), where \((-,-)\) denotes an ad-invariant scalar product on \(\mathfrak t^*\). Hence,
$$\begin{aligned} (\alpha ,\alpha ) + (\gamma ,\alpha ) = (\alpha + \gamma , \alpha ) = (\beta , \alpha ) \leqslant 0 \end{aligned}$$
an we get \((\gamma , \alpha ) \leqslant -(\alpha ,\alpha ) < 0\). Up to reordering the indices, we may assume that \((\alpha _1,\alpha ) < 0\); hence, \(\alpha ' = \alpha + \alpha _1 \in \varPhi ^+\). On the other hand, \(\beta \in \varPhi ^+(\alpha ')\) and \(\beta = \alpha ' + \alpha _2 + \cdots + \alpha _m\); therefore, the claim follows by induction. \(\square \)

Lemma 4.4

Given \(\alpha \in \varPhi ^+\), the ideal \((\mathfrak u_\alpha )\) generated by \(\mathfrak u_\alpha \) in \(\mathfrak u\) is the direct sum of the root spaces \(\mathfrak u_\beta \) with \(\beta \in \varPhi ^+(\alpha )\).

Proof

Denote \(\mathfrak r= \bigoplus _{\beta \in \varPhi ^+(\alpha )} \mathfrak u_\beta \). We show that \(\mathfrak r\subset \mathfrak u\) is an ideal and that \(\mathfrak r\subset (\mathfrak u_\alpha )\), whence the lemma. To show that \(\mathfrak r\) is an ideal, it is enough to notice that for all \(\beta \in \varPhi ^+(\alpha )\), and for all \(\gamma \in \varPhi ^+\) it holds either \([\mathfrak u_\beta , \mathfrak u_\gamma ] = 0\) or \([\mathfrak u_\beta , \mathfrak u_\gamma ] = \mathfrak u_{\beta + \gamma }\), in which case \(\beta + \gamma \in \varPhi ^+(\alpha )\).

Let \(\beta \in \varPhi ^+(\alpha )\). By Lemma 4.3, there exist \(\beta _0, \ldots , \beta _n \in \varPhi ^+\) with \(\beta _0 = \alpha \), \(\beta _n = \beta \) and \(\beta _i - \beta _{i-1} \in \varPhi ^+\) for all \(i \leqslant n\). For every \(i=1, \ldots , n-1\) define \(\alpha _i = \beta _i - \beta _{i-1}\); then, we have
$$\begin{aligned} \left[ \mathfrak u_{\beta _i},\mathfrak u_{\alpha _{i+1}}\right] = \mathfrak u_{\beta _{i+1}} \ne 0. \end{aligned}$$
It follows that \(\mathfrak u_\beta \subset (\mathfrak u_\alpha )\) for all \(\beta \in \varPhi ^+(\alpha )\); hence, \(\mathfrak r\subset (\mathfrak u_\alpha )\). \(\square \)

Theorem 4.5

Let \(\alpha \in \varPhi ^+\). The following statements are equivalent:
  1. (i)

    \(\alpha \in \varPsi ^\mathrm {\sharp }\);

     
  2. (ii)

    \(\varPsi (\alpha ) \ne \varnothing \);

     
  3. (iii)

    \(\varPhi ^+(\alpha ) \cap \varPsi \ne \varnothing \).

     
If moreover \(\alpha \in \varPsi ^\mathrm {\sharp }\), then we have \(\delta (\alpha ) = \delta (\beta )\) for all \(\beta \in \varPsi (\alpha )\).

Proof

  1. (i)
    \(\Rightarrow \) (ii) For all \(D \in \mathscr {D}\), fix a root \(\beta _D \in \varPsi \) such that \(\delta (\beta _D) = D\). This is possible thanks to Corollary 3.9. Let now \(\alpha \in \varPsi ^\mathrm {\sharp }\) and consider the element
    $$\begin{aligned} \beta = \alpha + \sum _{D\in \mathscr {D}{\smallsetminus }\{\delta (\alpha )\}} \langle \rho (D), \alpha \rangle \beta _D. \end{aligned}$$
    (6)
    By Lemma 4.2, it follows that \(\beta \in \varPsi ^\mathrm {\sharp }\). On the other hand, if \(D \in \mathscr {D}\), then Proposition 3.3 and Theorem 3.6 imply
    $$\begin{aligned} \langle \rho (D), \beta \rangle = \left\{ \begin{array}{ll} 0 &{} \text { if } D \ne \delta (\alpha ) \\ -1 &{} \text { if } D = \delta (\alpha ) \end{array}\right. \end{aligned}$$
    Therefore, \(\beta \in \varPsi \) by Corollary 3.7, and we get (ii).
     
  2. (ii)

    \(\Rightarrow \) (iii) Obvious.

     
  3. (iii)

    \(\Rightarrow \) (i) Suppose that \(U_\alpha \) acts trivially on B / H, we show that \(\varPhi ^+(\alpha ) \cap \varPsi = \varnothing \). Denote N the kernel of the action of U on B / H. Given \(u \in U\), we have \(u \in N\) if and only if \(ubH = bH\) for all \(b\in B\). On the other hand, for all \(b\in B\), the equality \(ubH = bH\) holds if and only if \(u\in bHb^{-1}\). Therefore, N is the intersection of U with the biggest normal subgroup of B contained in H. Equivalently, N is the biggest T-stable normal subgroup of U contained in H. It follows that N is stable under conjugation by T; hence, it is the product of the root subgroups \(U_\alpha \) which are contained in it.

     
By assumption, we have \(U_\alpha \subset N\); hence, \(\mathfrak u_\alpha \subset \mathfrak n\). By Lemma 4.4, the ideal \((\mathfrak u_\alpha )\) generated by \(\mathfrak u_\alpha \) in \(\mathfrak u\) is the direct sum of the root spaces \(\mathfrak u_\beta \) with \(\beta \in \varPhi ^+(\alpha )\). On the other hand, \(\mathfrak n\) is an ideal of \(\mathfrak u\), and by definition no root space \(\mathfrak u_\alpha \) with \(\alpha \in \varPsi \) is contained in \(\mathfrak h\). Therefore, \(\varPhi ^+(\alpha ) \cap \varPsi = \varnothing \).

For the last statement of the theorem, let \(\alpha \in \varPsi ^\mathrm {\sharp }\) and \(\beta \in \varPsi (\alpha )\). By Theorem 3.6, it follows that \(\langle \rho (\delta (\alpha )), \beta \rangle \leqslant -1\). On the other hand, \(\langle \rho (D), \beta \rangle = 0\) for all \(D \in \mathscr {D}{\smallsetminus }\{\delta (\beta )\}\); therefore, \(\delta (\beta ) = \delta (\alpha )\). \(\square \)

As a first consequence of the previous theorem, we generalize Lemma 4.2 to arbitrary pairs of weakly active roots.

Lemma 4.6

Let \(\alpha , \beta \in \varPsi ^\mathrm {\sharp }\) be such that \(\langle \rho (\delta (\beta )),\alpha \rangle > 0\). Then, \(\alpha + \beta \in \varPsi ^\mathrm {\sharp }\), and \(\delta (\alpha +\beta ) = \delta (\alpha )\).

Proof

Let \(\beta ' \in \varPsi (\beta )\), then \(\delta (\beta ') = \delta (\beta )\) and Lemma 4.2 implies that \(\alpha + \beta ' \in \varPsi ^\mathrm {\sharp }\), and \(\delta (\alpha +\beta ') = \delta (\alpha )\). Let \(\gamma \in \varPsi (\alpha +\beta ')\), then \(\gamma = \alpha + \beta + \gamma _1+ \ldots +\gamma _n\) for some \(\gamma _1, \ldots , \gamma _n \in \varPsi \) and Theorem 4.5 implies \(\delta (\gamma ) = \delta (\alpha )\). Therefore, by Theorem 3.6, \(\langle \rho (D), \gamma \rangle = 0\) for all \(D \in \mathscr {D}{\smallsetminus }\{\delta (\alpha )\}\), and \(\langle \rho (\delta (\alpha )), \gamma \rangle = -1\). As \(\{\gamma _1, \ldots , \gamma _n\} \subset F(\gamma )\), by Proposition 4.1 i) it follows \(\delta (\gamma _i) \ne \delta (\gamma )\) for all i; hence, \(\langle \rho (\delta (\alpha )), \gamma _i \rangle = 0\) for all i by Theorem 3.6, and it follows
$$\begin{aligned} \langle \rho (\delta (\alpha )), \alpha +\beta \rangle = \langle \rho (\delta (\alpha )), \gamma \rangle = -1. \end{aligned}$$
On the other hand, the hypothesis of the lemma implies \(\langle \rho (\delta (\beta )), \alpha + \beta \rangle \geqslant 0\); hence, \(\langle \rho (D), \alpha + \beta \rangle \geqslant 0\) for all \(D \in \mathscr {D}{\smallsetminus }\{\delta (\alpha )\}\), and following Definition 2.2 we get \(\alpha + \beta \in {{\mathrm{Root}}}(B/H)\) and \(\delta (\alpha +\beta ) = \delta (\alpha )\).

To conclude the proof, we need to show that \(\alpha +\beta \in \varPhi ^+\) and that \(U_{\alpha +\beta }\) acts non-trivially on B / H. This is shown with the same argument used in Lemma 4.2, which applies without any change in this more general case. \(\square \)

Given \(I\subset \mathscr {D}\), we denote
$$\begin{aligned} \varPsi _I = \{\alpha \in \varPsi \, : \,\delta (\alpha ) \in I\} \end{aligned}$$
In case \(I = \{D\}\) is a single element, then we will also denote \(\varPsi _I\) simply by \(\varPsi _D\).

Definition 4.7

Let \(I \subset \mathscr {D}\) and \(\alpha \in \varPsi ^\mathrm {\sharp }\), then we say that \(\alpha \) is activated byI if there is \(\beta \in \mathbb N\varPsi _I\) with \(\alpha + \beta \in \varPsi \), and we say that \(\alpha \)stabilizesI if there is \(\beta \in \mathbb N\varPsi _I\) with \(\alpha + \beta \in \varPsi _I\).

We denote by
$$\begin{aligned} \varTheta _I = \{\alpha \in \varPsi ^\mathrm {\sharp }\, \, : \,\, \exists \beta \in \mathbb N\varPsi _I \text { with } \alpha + \beta \in \varPsi \} \end{aligned}$$
the set of the weakly active roots activated by I and by \(\varPsi ^\mathrm {\sharp }_I\) the set of the weakly active roots which stabilize I. By Theorem 4.5, we have
$$\begin{aligned} \varPsi ^\mathrm {\sharp }_I = \{\alpha \in \varTheta _I \, \, : \,\, \delta (\alpha ) \in I \}. \end{aligned}$$
Notice that, for all \(I \subset \mathscr {D}\), we have \(\varPsi \subset \varTheta _I\) and \(\varPsi _I \subset \varPsi ^\mathrm {\sharp }_I\).

In the language just introduced, we have the following consequence of Theorem 3.6.

Proposition 4.8

Let \(\alpha \in \varPsi ^\mathrm {\sharp }\) and \(I \subset \mathscr {D}\). Then,
  1. (i)

    \(\alpha \in \varTheta _I\) if and only if \(\langle \rho (D), \alpha \rangle \leqslant 0\) for all \(D \in \mathscr {D}{\smallsetminus }I\),

     
  2. (ii)

    \(\alpha \in \varPsi ^\mathrm {\sharp }_I\) if and only if \(\langle \rho (D), \alpha \rangle = 0\) for all \(D \in \mathscr {D}{\smallsetminus }I\).

     
In particular, we have the equality \(\varPsi ^\mathrm {\sharp }_I = \mathbb N\varPsi ^\mathrm {\sharp }_I \cap \varPsi ^\mathrm {\sharp }\).

Proof

Let \(\alpha \in \varTheta _I\), then there exists \(\beta \in \varPsi \) such that \(\beta -\alpha \in \mathbb N\varPsi _I\). Applying Theorem 3.6 to \(\beta \), it follows that \(\langle \rho (D), \alpha \rangle \leqslant 0\) for all \(D \in \mathscr {D}{\smallsetminus }I\). Conversely, if such inequalities are all satisfied, then we may consider the element \(\beta \in \varPsi (\alpha )\) defined as in (6). Then, by Theorem 4.5 we have \(\delta (\beta ) = \delta (\alpha )\), and since \(\langle \rho (D), \alpha \rangle \geqslant 0\) for all \(D \in \mathscr {D}{\smallsetminus }\{\delta (\alpha )\}\) it follows that \(\beta - \alpha \in \mathbb N\varPsi _I\). This shows (i). Claim (ii) follows by noticing that if \(\alpha \in \varTheta _I\), then by definition \(\alpha \in \varPsi ^\mathrm {\sharp }_I\) if and only if \(\delta (\alpha ) \in I\), if and only if \(\langle \rho (D), \beta \rangle \geqslant 0\) for all \(D \in \mathscr {D}{\smallsetminus }I\). \(\square \)

Given \(I \subset \mathscr {D}\), let \(\sigma _I \subset \mathcal X(B/H)^\vee _\mathbb Q\) be the cone of the affine open subset of B / H with closed T-orbit \(\mathscr {U}_I\). By definition \(\sigma _I\) is generated by the one-dimensional rays in \(\mathcal X(B/H)^\vee _\mathbb Q\) which correspond to the T-stable prime divisors of B / H which contain \(\mathscr {U}_I\), namely the elements of \(\mathscr {D}{\smallsetminus }I\). Hence
$$\begin{aligned} \sigma _I = {{\mathrm{cone}}}(\rho (D) \, : \,D \in \mathscr {D}{\smallsetminus }I), \end{aligned}$$
and we may rephrase Proposition 4.8 as follows.

Proposition 4.9

Let \(\alpha \in \varPsi ^\mathrm {\sharp }\) and \(I \subset \mathscr {D}\). Then, \(\alpha \in \varTheta _I\) if and only if \(\alpha _{|\sigma _I} \leqslant 0\), and \(\alpha \in \varPsi ^\mathrm {\sharp }_I\) if and only if \(\alpha _{|\sigma _I} = 0\).

Given \(\alpha \in \varPsi ^\mathrm {\sharp }\), we are now ready to describe combinatorially the action of the root subgroup \(U_\alpha \) on the set of T-orbits on B / H in terms of weakly active roots.

Proposition 4.10

Let \(\alpha \in \varPsi ^\mathrm {\sharp }\) and \(I \subset \mathscr {D}\). Then, \(\mathscr {U}_I\) is not stable under the action of \(U_\alpha \) if and only if \(\alpha \in \varTheta _I\). If this is the case, then we have
$$\begin{aligned} U_\alpha \mathscr {U}_I = \left\{ \begin{array}{ll} \mathscr {U}_I \cup \mathscr {U}_{I \cup \{\delta (\alpha )\}} &{} \text { if }\alpha \in \varTheta _I {\smallsetminus }\varPsi ^\mathrm {\sharp }_I \\ \mathscr {U}_I \cup \mathscr {U}_{I {\smallsetminus }\{\delta (\alpha )\}} &{} \text { if } \alpha \in \varPsi ^\mathrm {\sharp }_I \end{array} \right. \end{aligned}$$
In particular, \(U_\alpha \mathscr {U}_I \subset \overline{\mathscr {U}}_I\) if and only if either \(\alpha \in \varPsi ^\mathrm {\sharp }_I\) or \(\alpha \not \in \varTheta _I\).

Proof

Consider the homomorphism of algebraic groups \(B \rightarrow {{\mathrm{Aut}}}(B/H)\); then, in the notation of Sect. 2.2 the root group \(U_\alpha \) acts on B / H as the one-parameter unipotent subgroup \(V_\alpha \subset {{\mathrm{Aut}}}(B/H)\). Suppose that \(\mathscr {U}_I\) is not stable under the action of \(U_\alpha \). Then, by Proposition 2.3 (i) the set \(U_\alpha \mathscr {U}_I\) decomposes in the union of two T-orbits \(\mathscr {U}_I \cup \mathscr {U}_J\) for some J.

Suppose \(|I| < |J|\), then \(\mathscr {U}_I \subset \overline{\mathscr {U}_J}\) and Proposition 2.3 (iii) implies \(\alpha _{|\sigma _I} \leqslant 0\) and \(\sigma _J = \sigma _I \cap \ker \alpha \) is a facet of \(\sigma _I\). By Proposition 4.9, we get then \(\alpha \in \varTheta _I {\smallsetminus }\varPsi ^\mathrm {\sharp }_I\). If instead \(|I| > |J|\), then \(\mathscr {U}_J \subset \overline{\mathscr {U}_I}\) and the same proposition implies \(\alpha _{|\sigma _J} \leqslant 0\) and \(\sigma _I = \sigma _J \cap \ker \alpha \); hence, \(\alpha \in \varPsi ^\mathrm {\sharp }_I\) by Proposition 4.9.

Conversely, suppose that \(\alpha \in \varTheta _I\). Then, Proposition 4.9 implies \(\alpha _{|\sigma _I} \leqslant 0\), and by the description of \(\sigma _I\) it follows \(\alpha _{|\sigma _I} = 0\) if and only if \(\delta (\alpha ) \in I\). Then the claim follows by Proposition 2.3 (iii) applied to the pair \(\mathscr {U}_{I {\smallsetminus }\delta (\alpha )} \subset \overline{\mathscr {U}_I}\) or to the pair \(\mathscr {U}_I \subset \overline{\mathscr {U}_{I \cup \delta (\alpha )}}\), depending on whether \(\delta (\alpha ) \in I\) or \(\delta (\alpha ) \not \in I\). \(\square \)

Remark 4.11

In particular, given \(\alpha \in \varPsi ^\mathrm {\sharp }\) and \(I \subset \mathscr {D}\), we have the following properties:
  1. (i)

    If \(U_\alpha \mathscr {U}_I \ne \mathscr {U}_I\), then the closure \(\overline{\mathscr {U}_I}\) is \(U_\alpha \) stable if and only if \(\alpha \in \varPsi ^\mathrm {\sharp }_I\).

     
  2. (ii)

    If \(\alpha \in \varPsi ^\mathrm {\sharp }_I\), then \(\alpha \in \varTheta _{I {\smallsetminus }\{\delta (\alpha )\}}\).

     

4.2 Combinatorics related to the weakly active roots

Recall the following fundamental property of the active roots.

Lemma 4.12

[2, Lemma 7] For all \(\beta \in F(\alpha ){\smallsetminus }\{\alpha \}\), it holds \(\alpha - \beta \in \varPhi ^+\).

We will need a generalization of the previous property, which holds for any combination of roots in \(F(\alpha )\).

Lemma 4.13

Let \(\alpha \in \varPsi \) and let \(\beta \in \mathbb N\varPsi \) be such that \(\beta < \alpha \), then \(\alpha - \beta \in \varPhi ^+\).

Proof

Write \(\alpha -\beta = \alpha _1 + \cdots + \alpha _n\) with \(\alpha _i \in \varPhi ^+\), and we show the claim by induction on n. We have \(\alpha \in \varPhi ^+(\alpha _i)\cap \varPsi \) for all \(i\in \{1,\ldots ,n\}\), so Theorem 4.5 yields \(\alpha _i\in \varPsi ^\mathrm {\sharp }\).

Notice that by Theorem 3.6 we have \(\langle \rho (D),\beta \rangle \leqslant 0\) for all \(D \in \mathscr {D}\); hence, \(\langle \rho (D),\alpha -\beta \rangle < 0\) at most for one \(D \in \mathscr {D}\), and if such D exists, then it is \(\delta (\alpha )\). On the other hand, write \(\beta = \beta _1 + \cdots + \beta _m\) for \(\beta _1,\ldots ,\beta _m \in \varPsi \): then \(\beta _j\leqslant \alpha \) for all j and from Proposition 4.1 (i) we deduce \(\delta (\beta _j) \ne \delta (\alpha )\). Therefore, \(\langle \rho (\delta (\alpha )),\alpha -\beta \rangle =-1\), and following Definition 2.2 we get \(\alpha _1+\ldots +\alpha _n \in {{\mathrm{Root}}}(B/H)\).

Suppose that \(n > 1\), then there is at least one index i with \(\delta (\alpha _i) \ne \delta (\alpha )\), say \(i=1\): this implies \(\langle \rho (\delta (\alpha _1)), \alpha \rangle = 0\) by Theorem 3.6. Since \(\langle \rho (\delta (\alpha _1)), \alpha _1 \rangle = -1\) and \(\langle \rho (\delta (\alpha _1)), \beta \rangle \leqslant 0\), it follows that there is at least one index i such that \(\langle \rho (\delta (\alpha _1)), \alpha _i \rangle > 0\), say \(i=2\). It follows by Lemma 4.6 that \(\alpha _1+\alpha _2 \in \varPsi ^\mathrm {\sharp }\); therefore, we can apply the inductive hypothesis and we get \(\alpha \in \varPhi ^+\). \(\square \)

Thanks to Theorem 4.5, we deduce the following descriptions.

Corollary 4.14

Let \(I \subset \mathscr {D}\), then we have the equalities
$$\begin{aligned} \varTheta _I= & {} \{\alpha - \beta \, : \,\alpha \in \varPsi , \; \beta \in \mathbb N\varPsi _I, \; \beta< \alpha \},\\ \varPsi ^\mathrm {\sharp }_I= & {} \{\alpha - \beta \, : \,\alpha \in \varPsi _I, \; \beta \in \mathbb N\varPsi _I, \; \beta < \alpha \}. \end{aligned}$$

Let \(\alpha \in \varPsi ^\mathrm {\sharp }\) and let \(\beta \in \varPsi \) be such that \(\alpha + \beta \in \varPsi \), then [2, Proposition 1] shows that \(\alpha + \beta ' \in \varPsi \) for all \(\beta ' \in \varPsi \) with \(\delta (\beta ') = \delta (\beta )\). We will need the following generalization of this property.

Proposition 4.15

Let \(\alpha \in \mathbb N\varPhi ^+\) and let \(\beta _1, \ldots , \beta _n \in \varPsi \) be such that \(\alpha + \sum _i a_i \beta _i \in \varPsi \) for some \(a_1, \ldots , a_n \in \mathbb N\). Then, \(\alpha \in \varPsi ^\mathrm {\sharp }\), and \(\alpha + \sum _i a_i \beta _i' \in \varPsi \) for all \(\beta '_1, \ldots , \beta '_n \in \varPsi \) such that \(\delta (\beta _1) = \delta (\beta _1'), \ldots , \delta (\beta _n) = \delta (\beta _n')\). Moreover, we have \(\delta (\alpha + \sum _i a_i \beta _i) = \delta (\alpha + \sum _i a_i \beta '_i)\).

Proof

Denote \(\gamma = \alpha + \sum _i a_i \beta _i\). We show the claim by induction on the sum \(a = \sum _i a_i\). Suppose that \(a = 1\), then \(n=1\) and Lemma 4.12 shows \(\alpha = \gamma - \beta _1 \in \varPhi ^+\). Therefore, the claim follows by [2, Proposition 1].

Suppose now that \(n >1\), then \(\beta _n \in F(\gamma )\) and Lemma 4.12 implies
$$\begin{aligned} \alpha + a_1 \beta _1 + \cdots + a_{n-1} \beta _{n-1} + (a_n-1)\beta _n \in \varPhi ^+. \end{aligned}$$
Setting \(\alpha ' = \alpha + \beta _n'\), by [2, Proposition 1] it follows
$$\begin{aligned} \alpha ' + a_1 \beta _1 + \cdots + a_{n-1} \beta _{n-1} + (a_n-1)\beta _n \in \varPsi , \end{aligned}$$
and we conclude that \(\alpha + \sum _i a_i \beta _i' \in \varPsi \) by the inductive hypothesis.

For the last claim, notice that \(\alpha \in \varPsi ^\mathrm {\sharp }\) by Lemma 4.13. Denote \(\gamma ' = \alpha + \sum a_i \beta '_i\), and then, we have \(\gamma , \gamma ' \in \varPsi (\alpha )\); therefore, \(\delta (\gamma ') = \delta (\gamma ) = \delta (\alpha )\) by Theorem 4.5. \(\square \)

We now show that the semigroup generated by the set of weakly active roots \(\varPsi ^\mathrm {\sharp }_I\) which stabilize a given subset \(I \subset \mathscr {D}\) is saturated in the root lattice \(\mathbb Z\Delta \), that is,
$$\begin{aligned} \mathbb N\varPsi ^\mathrm {\sharp }_I = {{\mathrm{cone}}}(\varPsi ^\mathrm {\sharp }_I) \cap \mathbb Z\Delta . \end{aligned}$$
More precisely, we will show that \(\mathbb N\varPsi ^\mathrm {\sharp }_I\) is the intersection of the semigroup generated by the positive roots with the rational vector space generated by \(\varPsi _I\). In order to do this, we will need a couple of preliminary lemmas.

Definition 4.16

We denote by \(\Sigma \) the union of the sets \({{\mathrm{supp}}}(\beta )\) for \(\beta \in \varPsi \).

Proposition 4.17

We have the equality \(\Sigma = \varPsi ^\mathrm {\sharp }\cap \Delta \). In particular, \(\mathbb Z\varPsi = \mathbb Z\varPsi ^\mathrm {\sharp }= \mathbb Z\Sigma \) and \(\mathbb Z\varPsi \cap \mathbb N\Delta = \mathbb N\varPsi ^\mathrm {\sharp }= \mathbb N\Sigma \).

Proof

Let \(\alpha \in \varPsi ^\mathrm {\sharp }\cap \Delta \), then by Theorem 4.5 it follows \(\varPsi (\alpha ) \ne \varnothing \); hence, \(\alpha \in \Sigma \). Suppose conversely that \(\alpha \in \Sigma \), then by definition \(\varPhi ^+(\alpha ) \cap \varPsi \ne \varnothing \); therefore, \(\alpha \in \varPsi ^\mathrm {\sharp }\) by Theorem 4.5 again. The last equalities follow immediately by the inclusions \(\mathbb Z\Sigma \subset \mathbb Z\varPsi ^\mathrm {\sharp }\subset \mathbb Z\varPsi \subset \mathbb Z\Sigma \) and \(\mathbb N\Sigma \subset \mathbb N\varPsi ^\mathrm {\sharp }\subset \mathbb Z\varPsi \cap \mathbb N\Delta \subset \mathbb N\Sigma \). \(\square \)

Remark 4.18

Up to a twist, the set \(\Sigma \) of Definition 4.16 is the set of the spherical roots attached to G / H following the theory of spherical varieties (see, e.g., [21, Theorem 1.3] for the general case, and [3, Theorem 5.28] for the case where H is a strongly solvable spherical subgroup). More precisely, if \(\Sigma _{G/H}\) is the set of the spherical roots of G / H, then we have \(\Sigma _{G/H} = -w_0(\Sigma )\): the twist appearing here is due to the equality of weight lattices \(\mathcal X(G/H) = -w_0\mathcal X(B/H)\) proved in Proposition 3.3 (for this reason, the subgroup H in [3] is assumed to be contained in the opposite Borel subgroup \(B^{w_0}\)).

Proposition 4.19

Given \(I \subset \mathscr {D}\), we have
$$\begin{aligned} \mathbb N\varPsi ^\mathrm {\sharp }_I = \mathbb Z\varPsi _I \cap \mathbb N\Delta = \mathbb Q\varPsi _I \cap \mathbb N\Delta . \end{aligned}$$

Proof

By the definition of \(\Sigma \) together with Proposition 4.17, we have
$$\begin{aligned} \mathbb Q\varPsi \cap \mathbb N\Delta = \mathbb Q\varPsi \cap \mathbb N\Sigma = \mathbb N\Sigma = \mathbb N\varPsi ^\mathrm {\sharp }. \end{aligned}$$
Therefore, the claim is equivalent to the equalities
$$\begin{aligned} \mathbb N\varPsi ^\mathrm {\sharp }_I = \mathbb Z\varPsi _I \cap \mathbb N\varPsi ^\mathrm {\sharp }= \mathbb Q\varPsi _I \cap \mathbb N\varPsi ^\mathrm {\sharp }. \end{aligned}$$
By the definition of \(\varPsi ^\mathrm {\sharp }_I\), we have the inclusions \(\mathbb N\varPsi ^\mathrm {\sharp }_I \subset \mathbb Z\varPsi _I \cap \mathbb N\varPsi ^\mathrm {\sharp }\subset \mathbb Q\varPsi _I \cap \mathbb N\varPsi ^\mathrm {\sharp }\). Let \(\alpha \in \mathbb Q\varPsi _I \cap \mathbb N\varPsi ^\mathrm {\sharp }\) and write \(\alpha = \beta _1 + \cdots + \beta _n\) with \(\beta _i \in \varPsi ^\mathrm {\sharp }\). By Proposition 4.8, we have \(\langle \rho (D), \alpha \rangle = 0\) for every \(D \in \mathscr {D}{\smallsetminus }I\). Proceeding by induction on n, we show that we may choose \(\beta _i\) in \(\varPsi ^\mathrm {\sharp }_I\) for all i. If \(n=1\), then we have \(\alpha \in \varPsi ^\mathrm {\sharp }\); hence, \(\alpha \in \varPsi ^\mathrm {\sharp }_I\) by Proposition 4.8. Suppose \(n >1\) and assume that \(\beta _i \not \in \varPsi ^\mathrm {\sharp }_I\) for some i. Then, the same corollary implies \(\langle \rho (D), \beta _i \rangle \ne 0\) for some \(D \in \mathscr {D}{\smallsetminus }I\).

Suppose that \(\langle \rho (D), \beta _i \rangle > 0\). As \(\langle \rho (D), \alpha \rangle = 0\), it follows \(\langle \rho (D), \beta _j \rangle < 0\) for some \(j \ne i\); hence, \(D = \delta (\beta _j)\) and \(\beta _i + \beta _j \in \varPsi ^\mathrm {\sharp }\) by Lemma 4.6. Similarly, if \(\langle \rho (D), \beta _i \rangle < 0\), then \(D = \delta (\beta _i)\) and there is some \(j \ne i\) with \(\langle \rho (\delta (\beta _i)), \beta _j \rangle > 0\); hence, \(\beta _i + \beta _j \in \varPsi ^\mathrm {\sharp }\) by Lemma 4.6 again. Therefore, \(\alpha \) can be written as a sum of \(n-1\) weakly active roots, and we conclude by the inductive assumption. \(\square \)

4.3 The root system associated with a T-orbit in B / H.

Recall that a root subsystem \(\varPhi ' \subset \varPhi \) is closed if, for all \(\alpha , \beta \in \varPhi '\) such that \(\alpha + \beta \in \varPhi \), it holds \(\alpha + \beta \in \varPhi '\) as well. Given \(I \subset \mathscr {D}\), we denote
$$\begin{aligned} \varPhi _I = \mathbb Z\varPsi _I \cap \varPhi , \qquad \qquad \varPhi _I^{\pm } = \mathbb Z\varPsi _I \cap \varPhi ^\pm . \end{aligned}$$
It follows easily by its definition that \(\varPhi _I\) is a closed root subsystem of \(\varPhi \), and that \(\varPhi _I^+ \subset \varPhi _I\) is a system of positive roots. We denote by \(\Delta _I \subset \varPhi ^+_I\) the corresponding basis and by \(W_I\) the Weyl group of \(\varPhi _I\) (notice that in general \(\Delta _I \not \subset \Delta \)).

Proposition 4.19 readily implies the following property of the root system \(\varPhi _I\).

Proposition 4.20

Given \(I \subset \mathscr {D}\), we have the following equalities:
  1. (i)

    \(\varPhi ^+_I = \mathbb Q\varPsi _I \cap \varPhi ^+ = \mathbb N\varPsi ^\mathrm {\sharp }_I \cap \varPhi ^+\);

     
  2. (ii)

    \(\varPsi ^\mathrm {\sharp }_I = \varPhi ^+_I \cap \varPsi ^\mathrm {\sharp }\) and \(\varPsi _I = \varPhi ^+_I \cap \varPsi \).

     
In particular, \(\Delta _I \subset \varPsi ^\mathrm {\sharp }_I\), and I is recovered by \(\varPhi _I\).

Proof

By Proposition 4.19, we get the equalities \(\varPhi ^+_I = \mathbb Q\varPsi _I \cap \varPhi ^+ = \mathbb N\varPsi ^\mathrm {\sharp }_I \cap \varPhi ^+\), and the inclusion \(\Delta _I \subset \varPsi ^\mathrm {\sharp }_I\) follows as well. Combining with Proposition 4.8, we get then \(\varPsi ^\mathrm {\sharp }_I = \varPhi ^+_I \cap \varPsi ^\mathrm {\sharp }\), and since \(\varPsi _I = \varPsi ^\mathrm {\sharp }_I \cap \varPsi \) the last equality follows as well. The last claim follows by noticing that \(I = \delta (\varPsi ^\mathrm {\sharp }_I) = \delta (\varPhi ^+_I \cap \varPsi ^\mathrm {\sharp })\). \(\square \)

We say that a root subsystem \(\varPhi ' \subset \varPhi \) is parabolic if there exists \(w \in W\) such that \(w(\varPhi ')\) is generated by a subset of simple roots of \(\varPhi \). This is equivalent to the property that \(\varPhi ' = \mathbb Q\varPhi ' \cap \varPhi \) (see [4, Ch. VI, § 1, Proposition 24]); therefore, we get the following corollary.

Corollary 4.21

Let \(I \subset \mathscr {D}\), then \(\varPhi _I\) is a parabolic root subsystem of \(\varPhi \).

Since \(\varPhi _I \subset \varPhi \) is a closed root subsystem, to every \(I \subset \mathscr {D}\) we may also attach a reductive subgroup \(G_I\) of G, namely the subgroup generated by T together with the root subgroups \(U_\alpha \) with \(\alpha \in \varPhi _I\). We denote by \(B_I\) the Borel subgroup of \(G_I\) associated with \(\varPhi ^+_I\), that is, \(B_I = G_I \cap B\), and by \(U_I\) its unipotent radical. The following proposition provides a first link between the root system \(\varPhi _I\) and the geometry of the corresponding T-orbit \(\mathscr {U}_I\).

Proposition 4.22

Let \(I \subset \mathscr {D}\), then \(\overline{\mathscr {U}_I} = \overline{B_I H/H}\) and \(\mathcal X(\mathscr {U}_I) = \mathbb Z\varPhi _I + \mathcal X(\mathscr {U}_\varnothing )\).

Proof

By Proposition 4.10, it follows that \(\overline{\mathscr {U}_I}\) is \(U_\alpha \)-stable for all \(\alpha \in \varPsi ^\mathrm {\sharp }_I\). On the other hand by Proposition 4.20, every \(\alpha \in \varPhi ^+_I {\smallsetminus }\varPsi ^\mathrm {\sharp }_I\) acts trivially on B / H; therefore, \(\overline{\mathscr {U}}_I\) is \(B_I\)-stable. Moreover, we have \(\mathscr {U}_I \subset B_I H/H\) by Proposition 3.10 and \(B_I H/H\subseteq \overline{\mathscr {U}_I}\) by Propositions 4.10 and 4.20, and the first claim follows.

To show the second claim, it follows by the previous discussion that \(\mathcal X(\mathscr {U}_I) = \mathcal X(B_I H/H)\). Reasoning as in (3), the projection \(B_I/ B_I \cap H \rightarrow T/T_H\) induces a T-equivariant isomorphism
$$\begin{aligned} B_I/ B_I \cap H \simeq T \times ^{T_H} U_I/U_I \cap H. \end{aligned}$$
Being a smooth and affine toric variety, \(B_I/ B_I \cap H\) has trivial class group; therefore, we get a surjective homomorphism \(\phi : \mathcal X(B_I H/H) \rightarrow \mathrm {Div}^T(B_I/ B_I \cap H)\) defined by sending the character \(\chi \) to the divisor of a rational T-eigenfunction of weight \(\chi \) (see, e.g., [12, Theorem 4.1.3]). Since the T-stable prime divisors of \(B_I/ B_I \cap H\) correspond to the \(T_H\)-stable hyperplanes of the fiber \(U_I/U_I \cap H\) (which is a affine space), it follows that the kernel of \(\phi \) equals \(\mathcal X(T/T_H) = \mathcal X(\mathscr {U}_\varnothing )\). On the other hand by Corollary 3.7, every T-stable prime divisor on \(B_I H/H\) has a global equation of weight \(-\alpha \) with \(\alpha \in \varPsi _I\); therefore, \(\mathcal X(\mathscr {U}_I) = \mathcal X(B_I H/H)\) is generated by \(\mathcal X(\mathscr {U}_\varnothing )\) together with \(\mathbb Z\varPsi _I = \mathbb Z\varPhi _I\). \(\square \)

Corollary 4.23

Let \(I \subset \mathscr {D}\), then the following inequalities hold
$$\begin{aligned} |I| \leqslant {{\mathrm{rk}}}\varPhi _I \leqslant \dim (T/T_H) + |I| \end{aligned}$$

Proof

This follows from the equality \({{\mathrm{rk}}}\mathcal X(\mathscr {U}_I) = \dim (\mathscr {U}_I) = \dim (T/T_H) + |I|\) together with the inclusion \(\varPsi _I \subset \varPhi _I\).

Remark 4.24

By the definition of \(\varPhi _I\), we have \({{\mathrm{rk}}}(\varPhi _I) = \dim (\mathbb Q\varPsi _I)\). In some cases, this rank is easily computable:
  • \(0 \leqslant {{\mathrm{rk}}}(\varPhi _I) \leqslant |\Sigma |\).

  • \({{\mathrm{rk}}}(\varPhi _I) = 0\) if and only if \(I = \varnothing \).

  • \({{\mathrm{rk}}}(\varPhi _I) = |\Sigma |\) if and only if \(I = \mathscr {D}\).

  • If \(I = \{D\}\), then \({{\mathrm{rk}}}(\varPhi _I) = |\delta ^{-1}(D)|\) (this follows by [2, Corollary 1]).

  • If \(T \subset H\), then \({{\mathrm{rk}}}(\varPhi _I) = |I|\) for all \(I \subset \mathscr {D}\) (this follows by [2, Theorem 1]).

4.4 Explicit description of the basis \(\Delta _I\)

By making use of the classification of the active roots, we will give in this subsection some more explicit description of the root systems \(\varPhi _I\) and of their bases \(\Delta _I\). We will not use these results until the last section of the paper, where we will prove a bound for the number of B-orbits in G / H.

Suppose that \(H \subset B\) is a spherical subgroup of G (possibly not containing T). In the following theorem, we recall Avdeev’s classification of the active roots, as well as some of their properties (see [2], Proposition 3, Corollary 6, Theorem 3, and Lemma 10) and deduce some corollaries.

Theorem 4.25

[2] Let \(\beta \in \varPsi \), then the following properties hold.
  1. (i)

    There exists a unique simple root \(\pi (\beta ) \in {{\mathrm{supp}}}(\beta )\) with the following property: if \(\beta = \beta _1 + \beta _2\) for some \(\beta _1, \beta _2 \in \varPhi ^+\), then \(\beta _1 \in \varPsi \) if and only if \(\pi (\beta ) \not \in {{\mathrm{supp}}}(\beta _1)\).

     
  2. (ii)

    The map \(\beta ' \mapsto \pi (\beta ')\) induces a bijection between \(F(\beta )\) and \({{\mathrm{supp}}}(\beta )\).

     
  3. (iii)

    If \(\beta _1, \beta _2 \in \varPsi \) and \(\pi (\beta _1) = \pi (\beta _2)\), then \(\delta (\beta _1) = \delta (\beta _2)\).

     
  4. (iv)

    The active root \(\beta \) appears in Table 11, and \([\beta : \pi (\beta )] = 1\).

     
Table 1

Active roots

Type

\(\hbox {Supp}(\beta )\)

\(\beta \)

1.

Any of rank n

\(\alpha _1+\cdots +\alpha _n\)

2.

\(\mathsf {B}_n\)

\(\alpha _1+\cdots +\alpha _{n-1} + 2\alpha _n\)

3.

\(\mathsf {C}_n\)

\(2\alpha _1+\cdots +2\alpha _{n-1} + \alpha _n\)

4.

\(\mathsf {F}_4\)

\(\alpha _1 + \alpha _2 + 2\alpha _3 + 2\alpha _4\)

5.

\(\mathsf {G}_2\)

\(2\alpha _1+\alpha _2\)

6.

\(\mathsf {G}_2\)

\(3\alpha _1+\alpha _2\)

We will call the integer appearing in the first column in Table 1 the type of an active root.

Let \(\beta \in \varPsi \), we say that a subset \(A \subset {{\mathrm{supp}}}(\beta )\) is connected (resp. co-connected) if A (resp. \({{\mathrm{supp}}}(\beta ) {\smallsetminus }A\)) is connected as a set of vertices in the Dynkin diagram of \(\varPhi \). If \(\beta ' \in F(\beta )\), it follows by a direct inspection in Table 1 that \({{\mathrm{supp}}}(\beta ') \subset {{\mathrm{supp}}}(\beta )\) is co-connected, whereas Theorem 4.25 (i) shows that if \(\beta ' \ne \beta \), then \(\pi (\beta ) \not \in {{\mathrm{supp}}}(\beta ')\).

Notice that Theorem 4.25 allows to construct the whole family \(F(\beta )\) from \(\beta \) and \(\pi (\beta )\). In particular, we have the following property.

Corollary 4.26

Let \(\beta \in \varPsi \) and let \(A \subset {{\mathrm{supp}}}(\beta )\). Then, \(A = {{\mathrm{supp}}}(\beta ')\) for some \(\beta ' \in F(\beta ) {\smallsetminus }\{\beta \}\) if and only if A is connected and co-connected, and \(\pi (\beta ) \not \in A\).

Proof

As already noticed, every active root \(\beta ' \in F(\beta ) {\smallsetminus }\{\beta \}\) satisfies the properties of the claim. The corollary follows then by Theorem 4.25 (ii) by noticing that the number of subsets \(A\subset {{\mathrm{supp}}}(\beta )\) which are connected and co-connected such that \(\pi (\beta ) \not \in A\) is the number of edges of the Dynkin diagram of \({{\mathrm{supp}}}(\beta )\). Indeed, associating with A the unique edge connecting a simple root of A to a simple root of \({{\mathrm{supp}}}(\beta ){\smallsetminus }A\) is a bijection. On the other hand, the number of such edges is precisely \(|{{\mathrm{supp}}}(\beta )|-1\). \(\square \)

In particular, we get the following characterization of the pairs \((\alpha , \beta ) \in \varPsi ^\mathrm {\sharp }\times \varPsi \) with \(\beta \in \varPsi (\alpha )\).

Corollary 4.27

Let \(\alpha \in \varPhi ^+\) and \(\beta \in \varPsi \), and suppose that \(\alpha < \beta \). Then, \(\beta \in \varPsi (\alpha )\) if and only if \(\pi (\beta ) \in {{\mathrm{supp}}}(\alpha )\).

Proof

Since \(\beta \in \varPhi ^+(\alpha )\), by Theorem 4.5 it follows that \(\alpha \in \varPsi ^\mathrm {\sharp }\). If \({{\mathrm{supp}}}(\alpha ) \ne {{\mathrm{supp}}}(\beta )\), let \(A \subset {{\mathrm{supp}}}(\beta ) {\smallsetminus }{{\mathrm{supp}}}(\alpha )\) be a connected component. Notice that A is both connected and co-connected in \({{\mathrm{supp}}}(\beta )\); therefore, by Corollary 4.26 there is \(\beta ' \in \varPsi \) with \({{\mathrm{supp}}}(\beta ') = A\). On the other hand, \(\langle \beta ', \alpha ^\vee \rangle < 0\); therefore, \(\alpha + \beta ' \in \varPhi ^+\), and since \(\alpha + \beta ' \leqslant \beta \) by Theorem 4.5, we still have \(\alpha + \beta ' \in \varPsi ^\mathrm {\sharp }\). On the other hand, \(\beta -\beta ' \in \varPhi ^+\) by Lemma 4.12; hence, \(\pi (\beta ) \not \in {{\mathrm{supp}}}(\beta ')\) by Theorem 4.25. Therefore, \(\pi (\beta ) \in {{\mathrm{supp}}}(\alpha )\) if and only if \(\pi (\beta ) \in {{\mathrm{supp}}}(\alpha +\beta ')\).

It follows that the claim holds for \(\alpha \) if and only if it holds for \(\alpha +\beta '\). Therefore, we may assume that \({{\mathrm{supp}}}(\alpha ) ={{\mathrm{supp}}}(\beta )\), in which case the claim can be easily checked by making use of Theorem 4.25 and Table 1:
  1. (i)

    If \(\beta \) is of type 1, then \(\alpha = \beta \), which is absurd.

     
  2. (ii)

    If \(\beta \) is of type 2, then \(\alpha = \beta - \alpha _n\), and \(\alpha _n \in \varPsi \).

     
  3. (iii)

    If \(\beta \) is of type 3, then \(\alpha = \beta - (\alpha _1 + \ldots + \alpha _i)\) for some \(i < n\), and \(\alpha _1 + \cdots + \alpha _i \in \varPsi \) for all \(i < n\).

     
  4. (iv)

    If \(\beta \) is of type 4, then either \(\alpha = \beta - (\alpha _3 + \alpha _4)\) or \(\alpha = \beta - \alpha _4\), and both \(\alpha _3 + \alpha _4\) and \(\alpha _4\) are active roots.

     
  5. (v)

    If \(\beta \) is of type 5, then \(\alpha = \beta - \alpha _1\), and \(\alpha _1 \in \varPsi \).

     
  6. (vi)

    If \(\beta \) is of type 6, then either \(\alpha = \beta - \alpha _1\) or \(\alpha = \beta - 2\alpha _1\), and \(\alpha _1 \in \varPsi \). \(\square \)

     

In particular, the previous corollary shows that \(\beta \in \varPsi (\pi (\beta ))\) for all \(\beta \in \varPsi \). In this case, we can be even more precise.

Corollary 4.28

Let \(\beta \in \varPsi \) and \(\beta ' \in F(\beta )\), then \(\beta '\) is maximal in \(F(\beta ) {\smallsetminus }\{\beta \}\) if and only if \({{\mathrm{supp}}}(\beta ')\) is a connected component of \({{\mathrm{supp}}}(\beta ) {\smallsetminus }\{\pi (\beta )\}\) if and only if \(\pi (\beta )\) and \(\pi (\beta ')\) are non-orthogonal. Moreover,
$$\begin{aligned} \pi (\beta ) = \beta - \sum _{\gamma \in \max (F(\beta ) {\smallsetminus }\{\beta \}) } a_{\beta ,\gamma } \gamma \end{aligned}$$
where \(\max (F(\beta ) {\smallsetminus }\{\beta \})\) denotes the set of the maximal elements in \(F(\beta ) {\smallsetminus }\{\beta \}\) and where \(a_{\beta ,\gamma } \in \mathbb N\) is the maximum such that \(a_{\beta ,\gamma } \gamma < \beta \).

Proof

Notice that every connected component of \({{\mathrm{supp}}}(\beta ) {\smallsetminus }\{\pi (\beta )\}\) is co-connected. The first claim follows then by Corollary 4.26, together with Proposition 4.1 (ii).

Let \(\alpha ' \in {{\mathrm{supp}}}(\beta ')\) non-orthogonal to \(\pi (\beta )\), notice that such root is unique since \({{\mathrm{supp}}}(\beta )\) contains no loops. Then \({{\mathrm{supp}}}(\beta ') {\smallsetminus }\{\alpha '\}\) is connected and co-connected in \({{\mathrm{supp}}}(\beta )\), and since it does not contain \(\pi (\beta )\) by Corollary 4.26, there is \(\beta '' \in F(\beta )\) such that \({{\mathrm{supp}}}(\beta '') = {{\mathrm{supp}}}(\beta ') {\smallsetminus }\{\alpha '\}\). Since \({{\mathrm{supp}}}(\beta '') \subset {{\mathrm{supp}}}(\beta ')\), Proposition 4.1 implies that \(\beta '' \in F(\beta ')\) and \(\beta '-\beta '' \in \varPhi ^+\). On the other hand, by Theorem 4.25 (i) \(\pi (\beta ') \in {{\mathrm{supp}}}(\beta ') {\smallsetminus }{{\mathrm{supp}}}(\beta '')\); therefore, it must be \(\pi (\beta ') = \alpha '\).

The last claim can be easily deduced by the previous discussion together with a direct inspection based on Theorem 4.25 and Table 1. \(\square \)

Another technical property that we will need and that we can deduce from Theorem 4.25 is the following.

Lemma 4.29

Let \(\beta \in \varPsi \) and let \(\beta _1, \beta _2 \in F(\beta )\). If \({{\mathrm{supp}}}(\beta _1) \cup {{\mathrm{supp}}}(\beta _2)\) is connected, then \(\beta _1\) and \(\beta _2\) are comparable.

Proof

By Corollary 4.26, the complementary
$$\begin{aligned} {{\mathrm{supp}}}(\beta ) {\smallsetminus }({{\mathrm{supp}}}(\beta _1)\cup {{\mathrm{supp}}}(\beta _2)) = ({{\mathrm{supp}}}(\beta ){\smallsetminus }{{\mathrm{supp}}}(\beta _1))\cap ({{\mathrm{supp}}}(\beta ){\smallsetminus }{{\mathrm{supp}}}(\beta _2)) \end{aligned}$$
is the intersection of two connected subsets of \({{\mathrm{supp}}}(\beta )\), so it is connected since the Dynkin diagram of \({{\mathrm{supp}}}(\beta )\) has no loops. Therefore, \({{\mathrm{supp}}}(\beta _1)\cup {{\mathrm{supp}}}(\beta _2)\) is connected and co-connected, and by Corollary 4.26 there exists \(\gamma \in F(\beta )\) such that \({{\mathrm{supp}}}(\gamma )={{\mathrm{supp}}}(\beta _1)\cup {{\mathrm{supp}}}(\beta _2)\).

Suppose that \(\gamma \) is different both from \(\beta _1\) and \(\beta _2\). For \(i = 1,2\), we have then \(\beta _i\in F(\gamma )\) by Proposition 4.1 (ii), and \(\gamma -\beta _i\in \varPhi ^+\) by Lemma 4.12. Therefore, by Theorem 4.25 (i) \(\pi (\gamma )\notin {{\mathrm{supp}}}(\beta _i)\), which is absurd by the definition of \(\gamma \). Therefore, \(\gamma \) is either \(\beta _1\) or \(\beta _2\); hence, \(\beta _1\) and \(\beta _2\) are comparable by Proposition 4.1 (ii). \(\square \)

Let \(I\subset \mathscr {D}\) and \(\beta \in \varPsi _I\). Denote \(F_I(\beta ) = (F(\beta ) \cap \varPsi _I) {\smallsetminus }\{\beta \}\) and define
$$\begin{aligned} \beta ^\sharp _I = \beta - \sum _{\gamma \in \max (F_I(\beta )) } a_{\beta ,\gamma } \gamma \end{aligned}$$
where \(\max (F_I(\beta ))\) is the set of the maximal elements in \(F_I(\beta )\) and where \(a_{\beta ,\gamma } \in \mathbb N\) is the maximum such that \(a_{\beta ,\gamma } \gamma < \beta \). By Theorem 4.5, it holds \(\beta _I^\sharp \in \varPsi ^\mathrm {\sharp }_I\), and \(\delta (\beta _I^\sharp ) = \delta (\beta )\): indeed, being pairwise incomparable, by Lemma 4.29 the elements in \(\max (F_I(\beta ))\) have pairwise disjoint support.

Theorem 4.30

Let \(I \subset \mathscr {D}\), then \(\Delta _I = \{\beta ^\sharp _I \, : \,\beta \in \varPsi _I\}\).

Proof

Suppose that \(\alpha \in \Delta _I\), then by Corollary 4.14 we may write \(\alpha = \beta - (\beta _1 + \cdots + \beta _p)\) for some \(\beta \in \varPsi _I\) and some \(\beta _1, \ldots , \beta _p \in F_I(\beta )\). Suppose that \(p > 0\), otherwise there is nothing to show. We claim that
$$\begin{aligned} \beta _1 + \cdots + \beta _p \leqslant \sum _{\gamma \in \max (F_I(\beta )) } a_{\beta ,\gamma }\gamma . \end{aligned}$$
(7)
The inequality is clear if the supports of \(\beta _1, \ldots , \beta _p\) are pairwise disconnected, e.g., if the active root \(\beta \) is of type 1. Suppose that this is not the case and that (up to reordering the indices) \({{\mathrm{supp}}}(\beta _1) \cap {{\mathrm{supp}}}(\beta _2) \ne \varnothing \). Then, \(\beta _1\) and \(\beta _2\) are comparable by Lemma 4.29, and we may assume that \(\beta _1 \leqslant \beta _2\). Since \(\beta _1 + \beta _2 < \beta \), arguing with Theorem 4.25 and Table 1 we have the following possibilities:
  1. (i)

    \(\beta \) is of type 2, and \(\beta _1 = \beta _2 = \alpha _n\);

     
  2. (ii)

    \(\beta \) is of type 3, and there are indices ij with \(1 \leqslant i \leqslant j < n\) such that \(\beta _1 = \alpha _1 + \cdots + \alpha _i\) and \(\beta _2 = \alpha _1+ \cdots + \alpha _j\);

     
  3. (iii)

    \(\beta \) is of type 4, and there are indices ij with \(3 \leqslant i \leqslant j \leqslant 4\) such that \(\beta _1 = \alpha _j + \cdots + \alpha _4\) and \(\beta _2 = \alpha _i+ \cdots + \alpha _4\);

     
  4. (iv)

    \(\beta \) is of type 5 or 6, and \(\beta _1 = \beta _2 = \alpha _1\).

     
If \(\gamma \in \max (F_I(\beta ))\) is such that \(\beta _2 \leqslant \gamma \), in all these cases we get that \(\beta _1 + \beta _2 \leqslant a_{\beta ,\gamma } \gamma \).

By making use of Theorem 4.25, it is easy to see that for all \(\beta ' \in F(\beta ) {\smallsetminus }\{\beta \}\) the family \(F(\beta ')\) is totally ordered by the dominance order. On the other hand, we see in Table 1 that the coefficient of the active root \(\beta \) along a simple root can be at most 2, unless \(\beta \) is of Type 6. Provided that \(\beta \) is not of type 6, it follows that no root \(\beta _i\) with \(i > 2\) is in \(F(\gamma )\), since otherwise \(\beta _1 + \beta _2 + \beta _i \not \leqslant \beta \), and inequality (7) follows by summing up all the roots with intersecting supports. Finally, the inequality is immediately checked if \(\beta = 3\alpha _1 + \alpha _2\) is of type 6, in which case \(\alpha = \alpha _2\) and \(\beta _1 = \beta _2 = \beta _3 = \alpha _1\).

As a consequence of (7), we get the inequality \(\beta ^\sharp _I \leqslant \alpha \). Since by assumption \(\alpha \) is minimal in \(\varPhi ^+_I\) and since \(\beta ^\sharp _I \in \varPhi ^+_I\), it follows that \(\alpha = \beta ^\sharp _I\). Therefore, we have proved the inclusion \(\Delta _I \subset \{\beta ^\sharp _I \, : \,\beta \in I\}\). We now show that for all \(\beta \in \varPsi \) the element \(\beta _I^\sharp \) is indecomposable in \(\varPhi _I^+\).

Suppose that \(\beta \in \varPsi _I\) and let \(\gamma _1, \ldots , \gamma _n \in \varPsi _I\) be such that
$$\begin{aligned} \beta _I^\sharp = (\gamma _1)_I^\sharp + \cdots + (\gamma _n)_I^\sharp . \end{aligned}$$
By Definition 2.2, for all i the evaluation of \((\gamma _i)_I^\sharp \) takes exactly one negative value on \(\mathscr {D}\). As \(\delta (\beta _I^\sharp ) = \delta (\beta )\), there exists \(\gamma \in \{\gamma _1, \ldots , \gamma _n\}\) with \(\delta (\beta ) = \delta (\gamma )\). Therefore, \(\gamma _I^\sharp \leqslant \beta _I^\sharp < \beta \), and \(\pi (\gamma ) = \pi (\gamma _I^\sharp ) \in {{\mathrm{supp}}}(\beta )\). By [2, Corollary 11], we get then \(\pi (\beta ) = \pi (\gamma _I^\sharp )\), and Corollary 4.27 shows that \(\beta \in \varPsi (\gamma _I^\sharp )\). Therefore, there are \(\beta _1, \ldots , \beta _m \in F(\beta )\) such that \(\beta = \gamma _I^\sharp + \beta _1+ \cdots + \beta _m\). Then \(\langle \rho (\delta (\beta _i)),\gamma _I^\sharp \rangle > 0\) for all \(i=1, \ldots , m\), and by Proposition 4.8 it follows \(\delta (\beta _i) \in I\), that is, \(\beta _i \in \varPsi _I\). Therefore, the definition of \(\beta _I^\sharp \) implies \(\beta _I^\sharp \leqslant \gamma _I^\sharp \), that is, \(n=1\) and \(\beta _I^\sharp = \gamma _I^\sharp \).

We assume for the rest of the section that \(H \subset G\) is a spherical subgroup such that \(T \subset H \subset B\). Then, \(\delta _{|\varPsi } : \varPsi \rightarrow \mathscr {D}\) is bijective by Corollary 3.9, and \(\pi : \varPsi \rightarrow \Delta \) is injective by Theorem 4.25 (iii). We will identify the set of divisors \(\mathscr {D}\) with the set of active roots \(\varPsi \).

Define a graph \(\mathcal G(\varPsi )\) with set of vertices \(\varPsi \) as follows: two active roots \(\beta , \beta '\) are connected by an edge if and only if they are not strongly orthogonal (namely, neither their sum nor their difference is in \(\varPhi \)). Given \(I \subset \varPsi \), we denote by \(\mathcal G(I)\) the associated subgraph of \(\mathcal G(\varPsi )\). Notice that, if \(\beta , \beta '\) belong to the same connected component of \(\mathcal G(I)\), then they belong to the same irreducible component of \(\varPhi _I\): if indeed \(\{\beta , \beta '\} \subset I\) is an edge of \(\mathcal G(I)\), then either \(\beta + \beta ' \in \varPhi _I\) or \(\beta - \beta ' \in \varPhi _I\).

If \(I \subset \varPsi \), we denote by \(\varPhi '_I\) the root system generated by the simple roots \(\pi (\beta )\) with \(\beta \in I\) and by \(W'_I\) the Weyl group of \(\varPhi '_I\).

Proposition 4.31

Suppose that \(T \subset H\), let \(I \subset \varPsi \) and suppose that \(\pi (I)\) is connected. Then, the followings hold:
  1. (i)

    \(\varPhi _I\) is an irreducible parabolic subsystems of \(\varPhi \) of rank |I|.

     
  2. (ii)

    If \(\varPhi \) is simply laced, then \(\varPhi _I\) and \(\varPhi '_I\) are isomorphic root systems.

     
  3. (iii)

    If \(\varPhi \) is not simply laced and \(\pi (I)\) contains both long and short roots, then \(\varPhi _I\) and \(\varPhi '_I\) are isomorphic root systems.

     

Proof

  1. (i)

    Let \(\beta , \beta ' \in I\), denote \(\alpha =\pi (\beta )\) and \(\alpha ' =\pi (\beta ')\) and suppose \(\langle \alpha ', \alpha ^\vee \rangle < 0\). Suppose that there is some active root whose family contains both \(\beta \) and \(\beta '\): then \(\beta \) and \(\beta '\) are comparable by Lemma  4.29; hence, \(\{\beta ,\beta '\}\) is an edge of \(\mathcal G(I)\). Therefore, \(\beta , \beta '\) belong to the same component of \(\varPhi _I\).

    Suppose that there is no active root whose family contains both \(\beta \) and \(\beta '\). We claim that in this case \({{\mathrm{supp}}}(\beta ) \cap {{\mathrm{supp}}}(\beta ') = \varnothing \). Indeed, if this intersection is not empty then \(\alpha \in {{\mathrm{supp}}}(\beta ')\) or \(\alpha '\in {{\mathrm{supp}}}(\beta )\) (because the Dynkin diagram of G has no loops). Assume that \(\alpha '\in {{\mathrm{supp}}}(\beta )\), and then, by Theorem 4.25 (ii) there is \(\beta ''\in F(\beta )\) with \(\pi (\beta '')=\alpha '\), which implies \(\beta ''=\beta '\) by the injectivity of \(\pi \). Therefore, \(\beta '\in F(\beta )\), contradicting the assumption.

    Therefore, \({{\mathrm{supp}}}(\beta ) \cap {{\mathrm{supp}}}(\beta ') = \varnothing \), and since \(\alpha \in {{\mathrm{supp}}}(\beta )\) and \(\alpha ' \in {{\mathrm{supp}}}(\beta ')\) are non-orthogonal, Theorem 4.25 (iv) implies that \(\langle \beta ', \beta ^\vee \rangle = \langle \alpha ', \alpha ^\vee \rangle < 0\). Therefore, \(\beta + \beta ' \in \varPhi ^+_I\); hence, \(\beta , \beta '\) belong to the same component of \(\varPhi _I\). It follows that \(\varPhi _I\) is an irreducible subsystem of \(\varPhi \) (whose rank is |I| by Remark 4.24), and it is parabolic by Corollary 4.21.

     
  2. (ii)

    By Theorem 4.30, \(\Delta _I = \{\beta ^\sharp _I \, : \,\beta \in I\}\) is a basis of \(\varPhi _I\). Let \(\beta \in I\). By Corollary 4.28, if \(\alpha '\in \pi (I)\) adjacent to \(\pi (\beta )\), then the root \(\beta '\in I\) such that \(\pi (\beta ')=\alpha \) is maximal in \(F(\beta ){\smallsetminus }\{\beta \}\); hence, \(\beta '\in \max (F_I(\beta ))\), which implies that \(\beta ^\sharp _I\) does not have \(\alpha '\) in its support. In other words, \({{\mathrm{supp}}}(\beta _I^\sharp ) \cap \pi (I)\) contains \(\pi (\beta )\) but none of the roots of \(\pi (I)\) adjacent to it: thanks to the fact that \({{\mathrm{supp}}}(\beta _I^\sharp ) \cap \pi (I)\) is connected, we deduce that \({{\mathrm{supp}}}(\beta _I^\sharp ) \cap \pi (I) = \{\pi (\beta )\}\). Notice also that all elements of \({{\mathrm{supp}}}(\beta _I^\sharp ){\smallsetminus }\{\pi (\beta )\}\) are orthogonal to the elements of \(\pi (I){\smallsetminus }\{\pi (\beta )\}\); otherwise, the Dynkin diagram of \(\varPhi \) would have a loop. Since all the roots in \(\varPhi \) have the same length, it follows that the root systems generated by \(\Delta _I\) and by \(\pi (I)\) are isomorphic.

     
  3. (iii)

    Let \(\Delta _0 \subset \Delta \) be the connected component containing \(\pi (I)\). Denote \(\varPhi _0\) the corresponding irreducible subsystem of \(\varPhi \) and enumerate \(\Delta _0 = \{\alpha _1, \ldots , \alpha _n\}\) as in [4]. We also enumerate I by setting \(\beta _i = \pi ^{-1}(\alpha _i)\) for all \(\alpha _i \in \pi (I)\). If \(\pi (I) = \Delta _0\), then the claim is clear; since (i) implies then \(\varPhi _I = \varPhi '_I = \varPhi _0\). In particular, we may assume that \(\varPhi '_I\) is not of type \(\mathsf F_4\) nor of type \(\mathsf G_2\).

     
Suppose first that \(\varPhi _0\) is of type \(\mathsf B_n\) or \(\mathsf C_n\). Then, \(\pi (I) = \{\alpha _p, \ldots , \alpha _n\}\) for some p with \(1< p < n\). Notice that the claim follows once we prove that \(\varPhi _I\) contains roots of different length: indeed, by (i) \(\varPhi _I\) and \(\varPhi '_I\) are both irreducible parabolic subsystems of \(\varPhi _0\) of rank |I|. By Corollary 4.28, it follows that I contains all the maximal elements in \(F(\beta _n) {\smallsetminus }\{\beta _n\}\); therefore, \((\beta _n)_I^\sharp = \pi (\beta _n) = \alpha _n\) and we get \(\alpha _n \in \varPhi ^+_I\).

Suppose that \(\varPhi _0\) is of type \(\mathsf B_n\), then \(\alpha _n\) is short. On the other hand, a positive root \(\gamma \) with \([\gamma : \alpha _{n-1}] = 1\) is short if and only if \(\gamma = \alpha _i + \cdots + \alpha _n\) for some \(i \leqslant n\); therefore, either \(\beta _{n-1}\) is long or \(\beta _{n-1}-\alpha _n\) is long, and the claim follows since \(\varPhi _I\) is a parabolic subsystem of \(\varPhi _0\).

Suppose that \(\varPhi _0\) is of type \(\mathsf C_n\), then \(\alpha _n\) is long. On the other hand, a positive root \(\gamma \) with \([\gamma :\alpha _{n-1}] = 1\) is necessarily short, and the claim follows.

Suppose now that \(\varPhi _0\) is of type \(\mathsf F_4\), then by the assumption \(\pi (I) \supset \{\alpha _2, \alpha _3\}\). If \(\bigcup _{\beta \in I} {{\mathrm{supp}}}(\beta )\) is of type \(\mathsf B\) or \(\mathsf C\) the same arguments as before apply, suppose that this is not the case. A direct inspection based on Theorem 4.25 shows that \(\beta _3\) is necessarily a short root, and that either \(\beta _2\) is a long root or \(\beta _2 - \beta _3\) is a long root. Therefore, \(\varPhi _I\) contains roots of different lengths; hence, it is of type \(\mathsf B_{|I|}\) or \(\mathsf C_{|I|}\). If \(|I| = 2\), then the claim follows.

Suppose that \(|I| = 3\). If \(\beta _1 \in I\), then it follows by Corollary 4.28 that I contains all the maximal elements in \(F(\beta _1) {\smallsetminus }\{\beta _1\}\) and in \(F(\beta _2) {\smallsetminus }\{\beta _2\}\). Therefore, \((\beta _i)^\sharp _I = \pi (\beta _i) = \alpha _i\) for \(i=1,2\), and it follows that \(\Delta _I\) contains two long roots; hence, \(\varPhi _I\) is of type \(\mathsf B_3\). Similarly, if \(\beta _4 \in I\) we conclude that \(\Delta _I\) contains two short roots; hence, \(\varPhi _I\) is of type \(\mathsf C_3\). \(\square \)

5 Combinatorial parameters for the orbits of B on G / H

We are now ready to give a combinatorial parametrization of the B-orbits in G / H, where H is a spherical subgroup of G contained in B, in terms of the root systems introduced in the previous sections.

5.1 Reduced and extended pairs

Consider the projection \(G/H \rightarrow G/B\). By the Bruhat decomposition, every B-orbit \(\mathscr {O}\) in G / H uniquely determines a Weyl group element, namely the element \(w\in W\) such that the image of \(\mathscr {O}\) in G / B is the Schubert cell BwB / B. Given \(w \in W\) and \(I \subset \mathscr {D}\), notice that \(w\mathscr {U}_I\) is a well-defined T-orbit in G / H, mapping on the T-fixed point \(wB/B \in G/B\). Therefore, to every pair (wI) with \(w \in W\) and \(I \subset \mathscr {D}\), we may associate a B-orbit in G / H by setting \(\mathscr {O}_{w,I} = Bw \mathscr {U}_I\).

Notice that every B-orbit in G / H is of the shape \(\mathscr {O}_{w,I}\) for some \(w \in W\) and some \(I \subset \mathscr {D}\). Suppose indeed that \(\mathscr {O}\) is a B-orbit in G / H which projects on the Schubert cell BwB / B. Then, reasoning as in Proposition 3.1, the intersection \(w^{-1} \mathscr {O}\cap B/H\) is a \((B \cap B^w)\)-orbit in B / H; hence, it is T-stable. If \(\mathscr {U}_I\) is any T-orbit in \(w^{-1} \mathscr {O}\cap B/H\), we have then the equality \(\mathscr {O}= \mathscr {O}_{w,I}\). While w is uniquely determined by \(\mathscr {O}\), in general there are several choices for \(I \subset \mathscr {D}\).

We will say that \(I \subset \mathscr {D}\) is a representative for a B-orbit \(\mathscr {O}\) if \(\mathscr {O}= \mathscr {O}_{w,I}\) for some \(w \in W\). The goal of the following theorem is to show that there are canonical minimal and maximal representatives for the B-orbits in G / H and to give combinatorial characterizations of such pairs.

Theorem 5.1

Let \(\mathscr {O}\) be a B-orbit in G / H. There exist a unique minimal representative \(\mathrm m\) and a unique maximal representative \(\mathrm M\) for \(\mathscr {O}\), and I is a representative for \(\mathscr {O}\) if and only if \(\mathrm m\subset I \subset \mathrm M\). Suppose moreover that \(\mathscr {O}= \mathscr {O}_{w,I}\), and denote
$$\begin{aligned} I(w) = \delta \big (\varTheta _I {\smallsetminus }\varPhi ^+(w) \big ). \end{aligned}$$
  1. (i)

    We have the equalities \(\mathrm m= I {\smallsetminus }I(w)\) and \(\mathrm M= I \cup I(w)\).

     
  2. (ii)
    The following formulae hold:
    $$\begin{aligned} \dim (\mathscr {O}) = \dim (B/H) + l(w) - |\mathscr {D}{\smallsetminus }\mathrm M|,\\ {{\mathrm{rk}}}(\mathscr {O}) = {{\mathrm{rk}}}(B/H) + |\mathrm m|. \end{aligned}$$
    More precisely, \(\mathcal X_B(\mathscr {O}) = w \mathcal X_T(\mathscr {U}_\mathrm m)\), and in particular, \(w(\varPhi _\mathrm m) \subset \mathcal X_B(\mathscr {O})\).
     

Proof

Let \(w \in W\) and \(I \subset \mathscr {D}\) be such that \(\mathscr {O}= \mathscr {O}_{w,I}\) and denote \(Z = w^{-1}\mathscr {O}\cap B/H\). By the discussion at the beginning of this section, the first claim is equivalent to the fact that Z contains a unique minimal T-orbit and a unique maximal T-orbit (ordered via inclusion of closures). Since \(Z = (B \cap B^w) \mathscr {U}_I\), it follows that Z is homogeneous under the action of a connected solvable group, and reasoning as in Corollary 3.2 it follows that Z is an irreducible T-stable affine subvariety of B / H.

In particular, Z is itself a toric variety under the action of T. Since it is irreducible, there exists a unique maximal T-orbit \(\mathscr {U}_\mathrm M\subset Z\), and since it is affine there exists a unique closed T-orbit \(\mathscr {U}_\mathrm m\subset Z\). To show that \(\mathscr {U}_J \subset Z\) for every J with \(\mathrm m\subset J \subset \mathrm M\), notice that Z is open in its closure. Therefore, the boundary \(\overline{Z} {\smallsetminus }Z\) is closed and T-stable, and if \(\mathscr {U}_J \subset \overline{Z} {\smallsetminus }Z\), then it must be \(\mathrm m\not \subset J\) because \(\mathscr {U}_\mathrm m\subset Z\). Therefore, Z is the union of all \(\mathscr {U}_J \subset B/H\) such that \(\mathscr {U}_\mathrm m\subset \overline{\mathscr {U}}_J \subset \overline{\mathscr {U}}_\mathrm M\).
  1. (i)

    First of all, notice that \(I(w) \subset \mathrm M\). Indeed, \(B\cap B^w\) is the product of the root subgroups \(U_\alpha \) with \(\alpha \in \varPhi ^+ {\smallsetminus }\varPhi ^+(w)\). If \(\alpha \in \varTheta _I {\smallsetminus }\varPhi ^+(w)\), by Proposition 4.10 it follows then \(\mathscr {U}_{I\cup \delta (\alpha )} \subset U_\alpha \mathscr {U}_I \subset (B \cap B^w) \mathscr {U}_I\); hence, \(I \cup \{\delta (\alpha )\} \subset \mathrm M\) by the maximality of \(\mathrm M\).

    Similarly, notice that \(\mathrm m\subset I {\smallsetminus }I(w)\): if indeed \(\alpha \in \varTheta _I {\smallsetminus }\varPhi ^+(w)\) and \(\delta (\alpha ) \in I\), then \(\mathscr {U}_{I{\smallsetminus }\delta (\alpha )} \subset U_\alpha \mathscr {U}_I \subset (B \cap B^w) \mathscr {U}_I\); hence, \(\mathrm m\subset I {\smallsetminus }\{\delta (\alpha )\}\) by the minimality of \(\mathrm m\). At this point, to conclude it is enough to show the equality \(\mathrm M= \mathrm m\cup I(w)\). We have already proved the inclusion \(\mathrm m\cup I(w)\subset \mathrm M\). Since \(\mathrm m(w) \subset I(w)\), the reverse inclusion \(\mathrm M\subset \mathrm m\cup I(w)\) follows if we prove that \(\mathrm M{\smallsetminus }\mathrm m\subset \mathrm m(w)\).

    We claim that
    $$\begin{aligned} \forall D \in \mathrm M{\smallsetminus }\mathrm m\qquad \exists \beta \in \varPsi _D \quad \exists \gamma \in \mathbb N\varPsi _\mathrm m\quad \, : \,\quad \beta - \gamma \in \mathbb N(\varPsi ^\mathrm {\sharp }{\smallsetminus }\varPhi ^+(w)). \end{aligned}$$
    (8)
    Indeed, given \(\alpha \in \varPhi ^+\), we have \(U_\alpha \subset B \cap B^w\) if and only if \(\alpha \not \in \varPhi ^+(w)\). Since \(\mathscr {U}_\mathrm M\subset (B \cap B^w) \mathscr {U}_\mathrm m\), it follows then by Proposition 4.10 that there is a sequence \(\alpha _1, \ldots , \alpha _n \in \varPsi ^\mathrm {\sharp }{\smallsetminus }\varPhi ^+(w)\) such that \(\mathrm M{\smallsetminus }\mathrm m= \{\delta (\alpha _1), \ldots , \delta (\alpha _n)\}\) and \(U_{\alpha _n} \cdots U_{\alpha _1} \mathscr {U}_\mathrm m\supset \mathscr {U}_M\). Moreover, every root \(\alpha _i\) has to be activated by \(\mathrm m\cup \{\delta (\alpha _1), \ldots , \delta (\alpha _{i-1})\}\), that is, for all \(i = 1, \ldots , n\) there are \(\gamma _i \in \mathbb N\varPsi _\mathrm m\), \(\beta _{i,1} \in \varPsi _{\delta (\alpha _1)}\), \(\ldots \), \(\beta _{i,i-1} \in \varPsi _{\delta (\alpha _{i-1})}\) and nonnegative integers \(a_{i,1}\), \(\ldots \), \(a_{i,i-1}\) such that
    $$\begin{aligned} \beta _1&:= \alpha _1 + \gamma _1 \in \varPsi _{\delta (\alpha _1)} \\ \beta _2&:= \alpha _2 + \gamma _2 + a_{1,1} \beta _{1,1} \in \varPsi _{\delta (\alpha _2)} \\ \beta _3&:= \alpha _3 + \gamma _3 + a_{2,1} \beta _{2,1} + a_{2,2} \beta _{2,2} \in \varPsi _{\delta (\alpha _3)} \\ \qquad \qquad&\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \qquad \qquad \\ \beta _n&:= \alpha _n + \gamma _n + a_{n,1} \beta _{n,1} + \cdots + a_{n,n-1} \beta _{n,n-1} \in \varPsi _{\delta (\alpha _n)} \end{aligned}$$
    By Proposition 4.15, it follows then
    $$\begin{aligned} \beta _1'&:= \alpha _1 + \gamma _1 \in \varPsi _{\delta (\alpha _1)} \\ \beta _2'&:= \alpha _2 + \gamma _2 + a_{1,1} \beta _1' \in \varPsi _{\delta (\alpha _2)} \\ \beta _3'&:= \alpha _3 + \gamma _3 + a_{2,1} \beta _1' + a_{2,2} \beta _2' \in \varPsi _{\delta (\alpha _3)} \\ \qquad \qquad&\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \qquad \qquad \\ \beta _n'&:= \alpha _n + \gamma _n + a_{n,1} \beta _1' + \cdots + a_{n,n-1} \beta _{n-1}' \in \varPsi _{\delta (\alpha _n)} \end{aligned}$$
    In particular, for every \(i = 1, \ldots , n\), it follows that \(\beta '_i = \alpha '_i + \gamma '_i\) with \(\gamma '_i \in \mathbb N\varPsi _\mathrm m\) and \(\alpha '_i \in \mathbb N\alpha _1 + \cdots + \mathbb N\alpha _i\), and (8) follows.

    Let now \(D \in \mathrm M{\smallsetminus }\mathrm m\), \(\beta \in \varPsi _D\) and \(\gamma \in \mathbb N\varPsi _\mathrm m\) as in (8), and set \(\alpha = \beta - \gamma \). Then, by Lemma 4.13 we have \(\alpha \in \varPhi ^+\); hence, \(\alpha \in \varTheta _{\mathrm m}\). On the other hand, we have \(w(\alpha ) \in \mathbb N\varPhi ^+\); hence, \(\alpha \in \varPhi ^+ {\smallsetminus }\varPhi ^+(w)\). Therefore, \(\alpha \in \varTheta _{\mathrm m} {\smallsetminus }\varPhi ^+(w)\), and by Theorem 4.5 we get \(D = \delta (\alpha )\), which shows that \(D \in \mathrm m(w)\).

     
  2. (ii)
    For the dimension formula, notice that the action of G on G / H induces an isomorphism of varieties
    $$\begin{aligned} U \cap (wU^-w^{-1}) \times wB/H \longrightarrow BwB/H \end{aligned}$$
     
If \(\mathscr {O}\subset BwB/H\), it follows that \(\dim (\mathscr {O}) = l(w) + \dim (w^{-1}\mathscr {O}\cap B/H)\). On the other hand, since B / H is a smooth affine toric variety, if \(I \subset \mathscr {D}\) we have
$$\begin{aligned} \dim (\mathscr {U}_I) = \dim (B/H) - |\mathscr {D}{\smallsetminus }I| = \dim (\mathscr {U}_\varnothing ) + |I|; \end{aligned}$$
therefore, the formula follows by the definition of \(\mathrm M\).

For the rank formula, fix a point \(x_\mathrm m\in \mathscr {U}_\mathrm m\). Notice that \(x_\mathrm m\) is a standard base point in \((B \cap B^w)\mathscr {U}_\mathrm m\), as a homogeneous space under \(B \cap B^w\). Indeed, if \(x \in (B \cap B^w)\mathscr {U}_\mathrm m\) is a standard base point, then Tx is a closed T-orbit by Lemma 2.1. On the other hand, \(\mathscr {U}_\mathrm m\subset (B \cap B^w)\mathscr {U}_\mathrm m\) is the unique closed T-orbit; therefore, \(Tx=Tx_\mathrm m\) and \(x_m\) is a standard base point too.

It follows that \(wx_\mathrm m\in \mathscr {O}\) is a standard base point as well. Indeed, since \(x_\mathrm m\in B/H\) and \(H\subset B\), we have \({{\mathrm{Stab}}}_G(x_\mathrm m) = bHb^{-1} = {{\mathrm{Stab}}}_B(x_\mathrm m)\); hence,
$$\begin{aligned} {{\mathrm{Stab}}}_B(wx_\mathrm m) = B \cap w {{\mathrm{Stab}}}_G(x_\mathrm m) w^{-1} = B \cap w {{\mathrm{Stab}}}_B(x_\mathrm m) w^{-1}. \end{aligned}$$
On the other hand, the latter equals \(w {{\mathrm{Stab}}}_{B \cap B^w} (x_\mathrm m)w^{-1}\), and since \({{\mathrm{Stab}}}_T(wx_\mathrm m) = w{{\mathrm{Stab}}}_T(x_\mathrm m) w^{-1}\), it follows that \(wx_\mathrm m\in \mathscr {O}\) is a standard base point because \(x_\mathrm m\in (B \cap B^w)\mathscr {U}_\mathrm m\) is so.
By Lemma 2.1, we have then
$$\begin{aligned} \mathcal X_B(\mathscr {O}) = \mathcal X_T(w\mathscr {U}_\mathrm m) = w\mathcal X_T(\mathscr {U}_\mathrm m); \end{aligned}$$
therefore, \({{\mathrm{rk}}}(\mathscr {O}) = \dim (\mathscr {U}_\mathrm m) = \dim (\mathscr {U}_\varnothing ) + |\mathrm m|\). On the other hand, by Lemma 2.1 again, we have \({{\mathrm{rk}}}(B/H) = \dim (\mathscr {U}_\varnothing )\), and the rank formula follows. Finally, the inclusion \(w(\varPhi _\mathrm m) \subset \mathcal X_B(\mathscr {O})\) follows by Proposition 4.22. \(\square \)

Given \(w \in W\) and \(I\subset \mathscr {D}\), we will denote by \(\mathrm m_{w,I}\) the minimal representative of \(\mathscr {O}_{w,I}\) and by \(\mathrm M_{w,I}\) the maximal representative of \(\mathscr {O}_{w,I}\).

Definition 5.2

Let \(w \in W\) and \(I\subset \mathscr {D}\). We say that (wI) is a reduced pair if \(I = \mathrm m_{w,I}\) (i.e., if \(I \cap I(w) = \varnothing \)), and we say that it is an extended pair if \(I = \mathrm M_{w,I}\) (i.e., if \(I(w) \subset I\)). Given \(w \in W\) and \(I \subset \mathscr {D}\), we will call \((w,\mathrm m_{w,I})\) and \((w,\mathrm M_{w,I})\), respectively, the reduction and the extension of (wI).

Remark 5.3

Given \(I \subset \mathscr {D}\), we have by definition \(\varPsi ^\mathrm {\sharp }_I = \varTheta _I \cap \delta ^{-1}(I)\), whereas by Theorem 5.1 we have the inclusion \(\varTheta _I {\smallsetminus }\varPsi ^\mathrm {\sharp }_{\mathrm M_{w,I}} \subset \varPhi ^+(w)\). The following characterizations of reduced pairs and of extended pairs follow immediately:
  1. (i)

    The pair (wI) is reduced if and only if \(\varPsi ^\mathrm {\sharp }_I \subset \varPhi ^+(w)\), if and only if \(\varPhi ^+_I \subset \varPhi ^+(w)\).

     
  2. (ii)

    The pair (wI) is extended if and only if \(\varTheta _I {\smallsetminus }\varPsi ^\mathrm {\sharp }_I \subset \varPhi ^+(w)\).

     

To summarize,

Corollary 5.4

The map \((w,I) \mapsto \mathscr {O}_{w,I}\) gives a parametrization by reduced (resp. extended) pairs
$$\begin{aligned} \{(w,I) \, : \,\varPhi ^+_I \subset \varPhi ^+(w) \} \longleftrightarrow \mathscr {B}(G/H) \longleftrightarrow \{(w,I) \, : \,\varTheta _I {\smallsetminus }\varPsi ^\mathrm {\sharp }_I \subset \varPhi ^+(w) \} . \end{aligned}$$
Moreover,
  1. (i)

    If (wI) is reduced, then \({{\mathrm{rk}}}(\mathscr {O}_{w,I}) = {{\mathrm{rk}}}(B/H) + |I|\).

     
  2. (ii)

    If (wI) is extended, then \(\dim (\mathscr {O}_{w,I}) = \dim (B/H) + l(w) - |\mathscr {D}{\smallsetminus }I|\).

     

Remark 5.5

The following properties easily follow from Remark 5.3:
  1. (i)

    All the pairs of the shape \((w,\varnothing )\) are reduced, and all the pairs of the shape \((w,\mathscr {D})\) are extended. All the pairs of the shape \((w_0, I)\) are both reduced and extended.

     
  2. (ii)

    If (wI) is reduced and if \(J \subset I\), then (wJ) is also reduced.

     
  3. (iii)

    If (wI) is reduced (resp. extended) and if \(w \preceq v\) (i.e., w is a right subexpression of v), then (vI) is also reduced (resp. extended).

     

Corollary 5.6

Suppose that (wI) is reduced. Then, \(\mathscr {O}_{w,I}\) is closed in G / H if and only if all of the following conditions hold:
  1. (i)

    \(I = \varnothing \);

     
  2. (ii)

    \(\varPhi ^+(w) \subset \varPsi \);

     
  3. (iii)

    \(\varPsi _{\delta (\varPhi ^+(w))} = \varPhi ^+(w)\);

     
  4. (iv)

    \(\delta _{|\varPhi ^+(w)} : \varPhi ^+(w) \rightarrow \mathscr {D}\) is injective.

     

Proof

As a consequence of the main theorem in [30], every closed B-orbit has minimal rank. Therefore, every closed B-orbit in G / H is of the shape \(\mathscr {O}_{w,\varnothing }\) for some \(w \in W\). On the other hand, B / H is a closed B-orbit, and by [9, Proposition 2.2] all closed B-orbits have the same dimension; therefore, \(\mathscr {O}_{w,\varnothing }\) is closed if and only if \(\dim (\mathscr {O}_{w,\varnothing }) = \dim (B/H)\).

By Theorem 5.1, the maximal representative of \(\mathscr {O}_{w,\varnothing }\) is \(\mathrm M= \delta (\varPsi {\smallsetminus }\varPhi ^+(w))\); therefore, \(\dim (\mathscr {O}_{w,\varnothing }) = \dim (B/H)\) if and only if \(|\mathscr {D}{\smallsetminus }\mathrm M| = l(w)\). On the other hand,
$$\begin{aligned} \mathscr {D}{\smallsetminus }\delta (\varPsi {\smallsetminus }\varPhi ^+(w)) \subset \delta (\varPsi \cap \varPhi ^+(w)) \end{aligned}$$
and the latter has cardinality at most l(w); therefore, \(|\mathscr {D}{\smallsetminus }\mathrm M| = l(w)\) if and only if \(\varPhi ^+(w) \subset \varPsi \) and \(\mathscr {D}{\smallsetminus }\delta (\varPsi {\smallsetminus }\varPhi ^+(w)) = \delta (\varPhi ^+(w))\) has cardinality l(w). Assuming (ii), the claim follows by noticing that the equality \(\mathscr {D}{\smallsetminus }\delta (\varPsi {\smallsetminus }\varPhi ^+(w)) = \delta (\varPhi ^+(w))\) is equivalent to (iii), whereas the equality \(|\delta (\varPhi ^+(w))| = l(w)\) is equivalent to (iv). \(\square \)

Example 5.7

In this example, we study the special case where H contains a maximal torus of G, and we show how the theory developed in this section simplifies under this assumption. Assume that \(T \subset H\).
  1. (i)

    Notice that \(\dim (B/H) = |\varPsi |\) and \({{\mathrm{rk}}}(B/H) = 0\). The first formula is clear, whereas the second one follows by Lemma 2.1 because B / H contains a T-fixed point.

     
  2. (ii)

    The restriction \(\delta _{|\varPsi } : \varPsi \rightarrow \mathscr {D}\) is bijective by Corollary 3.9; therefore, we may identify \(\varPsi \) and \(\mathscr {D}\) (as we will in the following points) and regard \(\delta _{|\varPsi ^\mathrm {\sharp }}\) as a map \(\varPsi ^\mathrm {\sharp }\rightarrow \varPsi \) which extends the identity.

     
  3. (iii)
    Let (wI) be a reduced pair, then by Corollary 5.6 we have
    $$\begin{aligned} \mathscr {O}_{w,I} \text { is closed} \iff I = \varnothing \text { and } \varPhi ^+(w) \subset \varPsi \end{aligned}$$
     
  4. (iv)
    Let (wI) be an extended pair and denote \(\varPhi ^+(w,I) = \varPhi ^+(w) {\smallsetminus }(\varPsi {\smallsetminus }I)\), then we have
    $$\begin{aligned} \dim \mathscr {O}_{w,I} = |\varPsi | + |\varPhi ^+(w,I)|. \end{aligned}$$
    Indeed, \(\varPsi {\smallsetminus }I = \varPsi \cap (\varTheta _I {\smallsetminus }\varPsi ^\mathrm {\sharp }_I)\); therefore, \(\varPsi {\smallsetminus }I \subset \varPhi ^+(w)\) by Remark 5.3 and the formula follows by Corollary 5.4.
     
  5. (v)
    Let (wI) be a reduced pair, then \({{\mathrm{rk}}}(\mathscr {O}_{w,I}) = |I|\) and
    $$\begin{aligned} \dim \mathscr {O}_{w,I} \geqslant |\varPsi | + |\varPhi ^+_I|. \end{aligned}$$
    The rank formula follows by Theorem 5.1 thanks to (i). The dimension formula follows by (iii), thanks to the inclusion \(\varPhi ^+_I \subset \varPhi ^+(w,\mathrm M)\) (where \((w,\mathrm M)\) denotes the extension of (wI)): indeed, \(\varPhi ^+_I \subset \varPhi ^+(w)\) by Remark 5.3, and if we regard \(\delta _{|\varPsi ^\mathrm {\sharp }}\) as a map \(\varPsi ^\mathrm {\sharp }\rightarrow \varPsi \), then by Proposition 4.20 we have \(\varPsi \cap \varPhi ^+_I = I \subset \mathrm M\).
     

Example 5.8

(see [34]) Let \(U' = \prod _{\alpha \in \varPhi ^+ {\smallsetminus }\Delta } U_\alpha \) be the derived subgroup of U, then \(H = T U'\) is a spherical subgroup of G which is contained in B. In particular, \(T \subset H\); therefore, the discussion of Example 5.7 applies. Notice that \(\varPsi = \Delta \) is the set of the simple roots; therefore, we can identify \(\mathscr {D}\) with \(\Delta \) via the map \(\delta \). Notice that \(\varPsi ^\mathrm {\sharp }= \varPsi = \Delta \): indeed by definition \(\varPsi ^\mathrm {\sharp }\subset \varPhi ^+\), and by Theorem 4.5 for all \(\alpha \in \varPsi ^\mathrm {\sharp }\) there exists \(\beta \in \varPsi \) such that \(\alpha \leqslant \beta \).

Given \(w \in W\) and \(I \subset \Delta \), we have by definition \(I(w) = I {\smallsetminus }\varPhi ^+(w)\). Therefore, by Remark 5.3 the pair (wI) is reduced if and only if \(I \subset \varPhi ^+(w)\), whereas it is extended if and only if \(\Delta {\smallsetminus }\varPhi ^+(w) \subset I\). In particular, if (wI) is reduced, then we have \({{\mathrm{rk}}}(\mathscr {O}_{w,I}) = |I|\) and
$$\begin{aligned} \dim (\mathscr {O}_{w,I}) = l(w) + |I| + |\Delta {\smallsetminus }\varPhi ^+(w)|. \end{aligned}$$

5.2 Comparison with the wonderful case

An important class of spherical subgroups of a reductive group G is that of wonderful subgroups (see, e.g., [5]). This class plays a prominent role in the theory of spherical varieties, both in their classification and in the study of their geometry. Wonderful subgroups can be characterized in terms of their spherical roots: by definition, the spherical roots of a spherical subgroup \(H \subset G\) are a distinguished set of elements in the weight lattice \(\mathcal X(G/H)\) (see the references in Remark 4.18 for the definition), and H is wonderful if and only if the corresponding spherical roots form a basis of \(\mathcal X(G/H)\).

To any spherical subgroup H, one may canonically associate a wonderful subgroup \(\overline{H}\) containing H, called the spherical closure of H. Given a spherical subgroup H contained in B, the aim of this subsection is to compare the set of B-orbits in G / H with that of \(G/\overline{H}\), and to show that these two sets are canonically identified. We keep the notations introduced in Sect. 2.2.

Restricting to the strongly solvable case, we say that a spherical subgroup \(H \subset G\) contained in B is wonderful if \(w_0 \mathcal X(G/H) = \mathbb Z\Sigma \), where \(\Sigma \) is the set introduced in Definition 4.16. The fact that this definition agrees with the general one is a consequence of Proposition 3.3 (see Remark 4.18).

If H is a spherical subgroup of G, then \(N_G(H)\) acts by conjugation on \(\mathcal X(H)\). The spherical closure \(\overline{H}\) of H is by definition the kernel of this action, and we say that H is spherically closed if \(H = \overline{H}\). Notice that we have inclusions \(H \subset \overline{H}\) and \(Z(G) \subset \overline{H} \subset N_G(H)\), where Z(G) denotes the center of G. By [3, Corollary 5.25], it follows that if H is contained in B, then \(\overline{H}\) is also contained in B.

By a general theorem of Knop [21, Theorem 7.5 and Corollary 7.6], the spherical closure of a spherical subgroup of G is wonderful. In the case of a strongly solvable spherical subgroup \(H \subset G\), the converse also is true: H is wonderful if and only if it is spherically closed (see [3, Corollary 3.42]).

Let \(H \subset G\) be a wonderful spherical subgroup contained in B, and denote \(\Sigma _{G/H} = -w_0(\Sigma )\). Then, by definition \(\Sigma _{G/H}\) is a basis for \(\mathcal X(G/H)\), whose elements are simple roots of G.

Lemma 5.9

Suppose that G is adjoint and let \(H \subset G\) be a strongly solvable wonderful subgroup. Then, H is connected.

Proof

Let \(H^\circ \subset H\) be the identity component. Since \(H^\circ \) has finite index in H and since H has an open orbit on G / B, it follows that \(H^\circ \) has an open orbit on G / B as well; hence, it is spherical. The pullback of rational functions along the projection \(G/H^\circ \rightarrow G/H\) identifies the weight lattice \(\mathcal X(G/H)\) with a sublattice of \(\mathcal X(G/H^\circ )\), which has finite index because \(H/H^\circ \simeq \mathcal X(G/H^\circ )/\mathcal X(G/H)\) (see, e.g., [16, Lemma 2.4]).

Since G is adjoint, we have \(\mathcal X(G/H^\circ ) \subset \mathcal X(T)= \mathbb Z\Delta \). On the other hand, \(\mathcal X(G/H) = \mathbb Z\Sigma _{G/H}\) is generated by a subset of \(\Delta \). Since \(\mathcal X(G/H) \subset \mathcal X(G/H^\circ ) \subset \mathbb Z\Delta \) and since the first inclusion has finite index, it follows the equality \(\mathcal X(G/H) = \mathcal X(G/H^\circ )\). The equality \(H = H^\circ \) follows then by applying the isomorphism \(H/H^\circ \simeq \mathcal X(G/H^\circ )/\mathcal X(G/H)\) once more. \(\square \)

Corollary 5.10

Suppose that \(H \subset G\) is spherical and strongly solvable. Then, the projection \(G/H\rightarrow G/\overline{H}\) induces a bijection between \(\mathscr {B}(G/H)\) and \(\mathscr {B}(G/\overline{H})\).

Proof

Denote \(H' = H Z(G)\). Since, as we already recalled, \(\overline{H}\) contains both H and Z(G), we have \(H' \subset \overline{H}\). The projection \(G/H\rightarrow G/H'\) induces a bijection between \(\mathscr {B}(G/H)\) and \(\mathscr {B}(G/H')\). Therefore, we may replace H with \(H'\), and since \(Z(G) \subset H'\), we may also assume that G is adjoint. On the other hand, by Lemma 5.9 it follows that the quotient \(\overline{H}/H\) is the image of a connected variety; hence, it is connected as well and the claim follows by [10, Lemma 3]. \(\square \)

6 The Weyl group action on \(\mathscr {B}(G/H)\)

6.1 Preliminaries

We denote by M(W) the Richardson–Springer monoid, namely the monoid generated by elements \(m(s_\alpha )\) (\(\alpha \in \Delta \)) with defining relations \(m(s_\alpha )^2 = m(s_\alpha )\) for all \(\alpha \in \Delta \) and the braid relations
$$\begin{aligned} m(s_\alpha ) m(s_\beta ) m(s_\alpha ) \ldots = m(s_\beta ) m(s_\alpha ) m(s_\beta ) \ldots \end{aligned}$$
for all \(\alpha , \beta \in \Delta \), the number of factors on both sides being the order of \(s_\alpha s_\beta \) in W (see [31, 3.10]).
As a set, M(W) is identified with W, and given \(w \in W\) we will denote by \(m(w) \in M(W)\) the corresponding element. Hence, we may consider the Richardson–Springer monoid as the Weyl group with a different multiplication, defined by the following rule: if \(w \in W\) and \(\alpha \in \Delta \), then
$$\begin{aligned} m(s_\alpha ) m(w) = \left\{ \begin{array}{ll} m(s_\alpha w) &{} \text { if } l(s_\alpha w) > l(w)\\ m(w) &{} \text { if } l(s_\alpha w) < l(w) \end{array}\right. \end{aligned}$$
From a geometrical point of view, the multiplication on M(W) coincides with the one defined on the Weyl group by the multiplication of Bruhat cells, namely we have \(\overline{B wB w' B} = \overline{Bw''B}\), where \(w'' \in W\) is the element defined by the equality \(m(w'') = m(w) m(w')\). Sometime, we will identify M(W) and W as sets; in that case, we will denote the product in M(W) of two elements \(w, w' \in W\) by \(w * w'\).
Given a spherical homogeneous variety G / H, both W and M(W) act on the set of B-orbits \(\mathscr {B}(G/H)\). The action of M(W) was defined by Richardson and Springer in the case of a symmetric homogeneous variety (see [31, 4.7]), and the definition carries over without modifications to the spherical case. To define the action of M(W) in the case of a simple reflection, given \(\alpha \in \Delta \) and \(\mathscr {O}\in \mathscr {B}(G/H)\), consider \(P_\alpha \mathscr {O}\). This is an irreducible B-variety, and since G / H is spherical, it decomposes into finitely many B-orbits. The element \(m(s_\alpha ) \cdot \mathscr {O}\) is defined then as the open B-orbit in \(P_\alpha \mathscr {O}\). This definition extends to an action of the monoid M(W) on \(\mathscr {B}(G/H)\), and it allows to define a partial order \(\preceq \) on \(\mathscr {B}(G/H)\) (called the weak order) as follows:
$$\begin{aligned} \mathscr {O}\preceq \mathscr {O}' \text { if and only if } \quad \exists w \in W \; : \; \mathscr {O}' = m(w) \cdot \mathscr {O}\end{aligned}$$
The action of W on \(\mathscr {B}(G/H)\) is much more subtle than that of M(W) and was defined by Knop [20]. We recall the definition of this action in the case of a spherical subgroup \(H \subset G\) contained in B, where the involved considerations turn out to be easier. By a case-by-case consideration (see [20, Lemma 3.2] and [10, Lemma 5 (iv)]), the B-stable variety \(P_\alpha \mathscr {O}\) decomposes in the union of two B-orbits or in the union of three B-orbits. More precisely, we have the following possibilities:
(U)

Suppose that \(P_\alpha \mathscr {O}= \mathscr {O}\cup \mathscr {O}'\) decomposes in the union of two orbits, and assume for simplicity that \(\mathscr {O}\) is the open one. Then, \(\dim \mathscr {O}' = \dim \mathscr {O}-1\) and \(\mathcal X(\mathscr {O}') = s_\alpha \mathcal X(\mathscr {O})\), and we define \(s_\alpha \cdot \mathscr {O}= \mathscr {O}'\).

(T)

Suppose that \(P_\alpha \mathscr {O}= \mathscr {O}\cup \mathscr {O}' \cup \mathscr {O}''\) decomposes in the union of three orbits, and assume for simplicity that \(\mathscr {O}\) is the open one. Then, \(\dim \mathscr {O}' = \dim \mathscr {O}'' = \dim \mathscr {O}-1\) and \(s_\alpha \mathcal X(\mathscr {O}') = \mathcal X(\mathscr {O}'') \subset \mathcal X(\mathscr {O}) = s_\alpha \mathcal X(\mathscr {O})\), where \(\mathcal X(\mathscr {O})/\mathcal X(\mathscr {O}') \simeq \mathbb Z\), and we define \(s_\alpha \cdot \mathscr {O}= \mathscr {O}\) and \(s_\alpha \cdot \mathscr {O}' = \mathscr {O}''\).

By [20, Theorem 5.9], the \(s_\alpha \)-actions defined above glue together into an action of W on \(\mathscr {B}(G/H)\). As a consequence of the previous analysis, notice that the rank of a B-orbit is invariant for this action, which agrees with the action of W on the weight lattices \(\mathcal X(\mathscr {O})\), where \(\mathscr {O}\in \mathscr {B}(G/H)\).

Example 6.1

Consider the case \(H = B\). Then, \(\mathscr {B}(G/B) = \{BwB/B \, : \,w \in W \}\) is the set of the Schubert cells. Notice that all B-orbits of G / B have rank 0: indeed, any such orbit is homogeneous under the action of a suitable unipotent subgroup of B; hence, every B-semiinvariant function on such orbit is constant. If \(\alpha \in \Delta \) and \(w \in W\), it follows that \(P_\alpha w B/B = BwB/B \cup Bs_\alpha w B/B\) is of type (U); hence, \(s_\alpha \cdot BwB/B = Bs_\alpha w B/B\). Therefore, the W action on \(\mathscr {B}(G/B)\) is induced by the action of W on itself by right multiplication.

In some special cases, the rank uniquely identifies the W-orbit; in particular, this happens when the rank of a B-orbit is maximal and minimal. This is summarized in the following theorem, which holds for any spherical subgroup H: the case of maximal rank being due to Knop (see [20, Theorem 6.2]) and that of minimal rank to Ressayre (see [30, Corollary 3.1 and Theorem 4.2]).

Theorem 6.2

Let \(\mathscr {O}, \mathscr {O}' \in \mathscr {B}(G/H)\) and suppose that \({{\mathrm{rk}}}\mathscr {O}= {{\mathrm{rk}}}\mathscr {O}' = {{\mathrm{rk}}}G/H\) (resp. \({{\mathrm{rk}}}\mathscr {O}= {{\mathrm{rk}}}\mathscr {O}' = {{\mathrm{rk}}}G - {{\mathrm{rk}}}H\)). Then, there exists \(w \in W\) such that \(\mathscr {O}' = w \cdot \mathscr {O}\).

6.2 Stabilizers for the W-action on \(\mathscr {B}(G/H)\)

We now describe the actions of W and of M(W) on \(\mathscr {B}(G/H)\) in the case of a strongly solvable spherical subgroup \(H \subset G\) contained in B, in terms of the combinatorial parametrization of Corollary 5.4.

In order to study the actions of W and of M(W) in terms of reduced and extended pairs, we take a closer look at the possible cases arising in the decomposition of the B-stable subsets \(P_\alpha \mathscr {O}_{w,I}\), where \(\mathscr {O}_{w,I} \in \mathscr {B}(G/H)\) and \(\alpha \in \Delta \).

Lemma 6.3

Let \(w \in W\) and \(\alpha \in \Delta \). Let \(\beta \) be the unique positive root in the set \(\{w^{-1}(\alpha ), -w^{-1}(\alpha )\}\), then the following hold.
  1. (i)
    Suppose that (wI) is a reduced pair, then
    $$\begin{aligned} s_\alpha \cdot \mathscr {O}_{w,I} = \left\{ \begin{array}{ll} \mathscr {O}_{w,I} &{} \text { if } \beta \in \varPsi ^\mathrm {\sharp }_I \\ \mathscr {O}_{s_\alpha w,I} &{} \text { otherwise } \end{array}\right. \end{aligned}$$
    (where all the orbits above are expressed in terms of reduced pairs).
     
  2. (ii)
    Suppose that (wI) is an extended pair, then
    $$\begin{aligned} m(s_\alpha ) \cdot \mathscr {O}_{w,I} = \left\{ \begin{array}{ll} \mathscr {O}_{s_\alpha * w, I\cup \{\delta (\beta )\}} &{} \text { if } \beta \in \varTheta _I \\ \mathscr {O}_{s_\alpha * w,I} &{} \text { if } \beta \not \in \varTheta _I \end{array}\right. \end{aligned}$$
    (where all the orbits above are expressed in terms of extended pairs). In particular, \(m(s_\alpha ) \cdot \mathscr {O}_{w,I} = \mathscr {O}_{w,I}\) if and only if \(\beta \in \varPhi ^+(w) {\smallsetminus }(\varTheta _I {\smallsetminus }\varPsi ^\mathrm {\sharp }_I)\).
     

Proof

Let \(w \in W\) and \(I \subset \mathscr {D}\). By the results recalled in Sect. 6.1, the B-stable subset \(P_\alpha \mathscr {O}_{w,I} = P_\alpha w\mathscr {U}_I\) decomposes either in the union of two B-orbits which are permuted by the action of \(s_\alpha \) or in the union of three B-orbits, an open one fixed by \(s_\alpha \) and two of codimension one which are permuted by \(s_\alpha \).

Notice that if \(\mathscr {O}_{v,J} \subset P_\alpha \mathscr {O}_{w,I}\) is the open orbit, then it must be \(v = s_\alpha *w\), namely v is the longest element between w and \(s_\alpha w\). Indeed, \(P_\alpha \mathscr {O}_{w,I} \cap B(s_\alpha *w)B/H\) is a B-stable dense open subset; hence, it must be \(\mathscr {O}_{v,J} \subset B(s_\alpha *w)B/H\) because \(P_\alpha \mathscr {O}_{w,I}\) is irreducible and the intersection of two open non-empty subsets therein is always non-empty. It follows that \(\mathscr {O}_{v,J} \subset P_\alpha \mathscr {O}_{w,I}\) is the open orbit if and only if v has maximal length and J has maximal cardinality among all the reduced pairs \((v',J')\) with \(\mathscr {O}_{v',J'} \subset P_\alpha \mathscr {O}_{w,I}\).

Recall that \(w < s_\alpha w\) in W if and only if \(w^{-1}(\alpha ) \in \varPhi ^+\), in which case \(\varPhi ^+(s_\alpha w) = \varPhi ^+(w) \cup \{w^{-1}(\alpha )\}\). We will distinguish two different cases, depending on \(s_\alpha w < w\) or \(s_\alpha w > w\).

Case 1. Suppose that \(\beta = -w^{-1}(\alpha )\) and denote \(v = s_\alpha w\), so that \(l(v) = l(w)-1\). The decomposition
$$\begin{aligned} P_\alpha = Bs_\alpha \cup Bs_\alpha B s_\alpha = B s_\alpha \cup B U_{-\alpha } \end{aligned}$$
implies that \(P_\alpha \mathscr {O}_{w,I} = Bv \mathscr {U}_I \cup Bw U_{\beta } \mathscr {U}_I\). By Proposition 4.10, we get
$$\begin{aligned} U_\beta \mathscr {U}_I = \left\{ \begin{array}{ll} \mathscr {U}_I &{} \text { if }\beta \in \varPhi ^+ {\smallsetminus }\varTheta _I\\ \mathscr {U}_I \cup \mathscr {U}_{I \cup \{\delta (\beta )\}} &{} \text { if }\beta \in \varTheta _I {\smallsetminus }\varPsi ^\mathrm {\sharp }_I\\ \mathscr {U}_I \cup \mathscr {U}_{I {\smallsetminus }\{\delta (\beta )\}} &{} \text { if }\beta \in \varPsi ^\mathrm {\sharp }_I \end{array} \right. \end{aligned}$$
By the discussion in Sect. 6.1, we have the following three possibilities, that we denote by (U), (T1) and (T2) and that we describe in detail here below. Denote \(\mathrm m= \mathrm m_{w,I}\) and \(\mathrm M= \mathrm M_{w,I}\):
(U)

Suppose that \(\beta \in \varPhi ^+ {\smallsetminus }\varTheta _{\mathrm m}\) or that \(\beta \in \varTheta _{\mathrm m} \cap \varPsi ^\mathrm {\sharp }_{\mathrm M} {\smallsetminus }\varPsi ^\mathrm {\sharp }_\mathrm m\).

By Theorem 5.1, we have in both these cases that \(U_\beta \mathscr {U}_I \subset (B \cap B^w)\mathscr {U}_I\): indeed, in the first case \(U_\beta \mathscr {U}_\mathrm m= \mathscr {U}_\mathrm m\), whereas in the second case \(U_\beta \mathscr {U}_\mathrm m= \mathscr {U}_\mathrm m\cup \mathscr {U}_{\mathrm m\cup \{\delta (\beta )\}}\), so that all subsets of \(\mathscr {D}\) that arise by applying \(U_\beta \) to \(\mathscr {U}_\mathrm m\) are still representatives for \(\mathscr {O}_{w,I}\). Therefore, \(BwU_\beta \mathscr {U}_I = \mathscr {O}_{w,I}\), and \(P_\alpha \mathscr {O}_{w,I} = \mathscr {O}_{v,I} \cup \mathscr {O}_{w,I}\) decomposes in the union of two orbits, and since \(v < w\), the open orbit is \(\mathscr {O}_{w,I}\). Therefore, we have:
  • \(\dim \mathscr {O}_{v,I} = \dim \mathscr {O}_{w,I} -1\);

  • \({{\mathrm{rk}}}\mathscr {O}_{v,I} = {{\mathrm{rk}}}\mathscr {O}_{w,I}\);

  • \(\mathrm M_{v,I} = \mathrm M\);

  • \(\mathrm m_{v,I} =\mathrm m\).

(T1)

Suppose that \(\beta \in \varPsi ^\mathrm {\sharp }_\mathrm m\).

We have in this case \(U_\beta \mathscr {U}_I = \mathscr {U}_I \cup \mathscr {U}_{I'}\), where \(I' = I {\smallsetminus }\{\delta (\beta )\}\), and since \(\mathrm m\not \subset I'\), Theorem 5.1 shows that \(\mathscr {O}_{w,I'} \ne \mathscr {O}_{w,I}\). Therefore, \(P_\alpha \mathscr {O}_{w,I} = \mathscr {O}_{v,I} \cup \mathscr {O}_{w,I'} \cup \mathscr {O}_{w,I}\) decomposes in the union of three orbits. Since \(v < w\) and \(\mathrm m_{w,I'} = \mathrm m{\smallsetminus }\{\delta (\beta )\}\), it follows that \(\mathscr {O}_{w,I}\) is the open orbit. On the other hand, by Proposition 4.10 we have \(\beta \in \varTheta _{I'}\), and since \(\beta \not \in \varPhi ^+(v)\) by assumption, by Theorem 5.1 we get \(\delta (\beta ) \in I(v)\), namely \(\mathscr {O}_{v,I} = \mathscr {O}_{v,I'}\). Therefore, the following hold:
  • \(\dim \mathscr {O}_{v,I} = \dim \mathscr {O}_{w,I'} = \dim \mathscr {O}_{w,I} - 1\);

  • \({{\mathrm{rk}}}\mathscr {O}_{v,I} = {{\mathrm{rk}}}\mathscr {O}_{w,I'} = {{\mathrm{rk}}}\mathscr {O}_{w,I} - 1\);

  • \(\mathrm M_{v,I} = \mathrm M\), \(\mathrm M_{w,I'} = \mathrm M{\smallsetminus }\{\delta (\beta )\}\);

  • \(\mathrm m_{v,I} = \mathrm m_{w,I'} = \mathrm m{\smallsetminus }\{\delta (\beta )\}\).

(T2)

Suppose that \(\beta \in \varTheta _{\mathrm m} {\smallsetminus }\varPsi ^\mathrm {\sharp }_\mathrm M\).

Then, \(U_\beta \mathscr {U}_I = \mathscr {U}_I \cup \mathscr {U}_{I'}\), where we set \(I' = I \cup \{\delta (\beta )\}\), and by Theorem 5.1 we have \(\mathscr {O}_{w,I'} \ne \mathscr {O}_{w,I}\). Therefore, \(P_\alpha \mathscr {O}_{w,I} = \mathscr {O}_{v,I} \cup \mathscr {O}_{w,I} \cup \mathscr {O}_{w,I'}\) decomposes in the union of three orbits as represented by the following diagram, and we have:
  • \(\dim \mathscr {O}_{v,I} = \dim \mathscr {O}_{w,I} = \dim \mathscr {O}_{w,I'} - 1\);

  • \({{\mathrm{rk}}}\mathscr {O}_{v,I} = {{\mathrm{rk}}}\mathscr {O}_{w,I} = {{\mathrm{rk}}}\mathscr {O}_{w,I'} - 1\);

  • \(\mathrm M_{v,I} = \mathrm M_{w,I'} = \mathrm M\cup \{\delta (\beta )\}\);

  • \(\mathrm m_{v,I} = \mathrm m\), \(\mathrm m_{w,I'} = \mathrm m\cup \{\delta (\beta )\}\).

Case 2. Suppose now that \(\beta = w^{-1}(\alpha )\) and denote \(v=s_\alpha w\), so that \(l(v) = l(w)+1\). The decomposition
$$\begin{aligned} P_\alpha = B \cup B s_\alpha B = B \cup B s_\alpha U_\alpha . \end{aligned}$$
implies that \(P_\alpha \mathscr {O}_{w,I} = Bw\mathscr {U}_I \cup B v U_\beta \mathscr {U}_I\). Since (wI) is reduced, we have \(\beta \not \in \varPsi ^\mathrm {\sharp }_I\) and \(\beta \not \in \varTheta _I {\smallsetminus }\varPsi ^\mathrm {\sharp }_M\). By the discussion in Sect. 6.1, we have the following two possibilities that we denote by (U) and (T) and that we describe in detail here below. Denote \(\mathrm m= \mathrm m_{w,I}\), \(\mathrm M= \mathrm M_{w,I}\), \(\mathrm M' = \mathrm M_{v,I}\):
(U)

Suppose that \(\beta \in \varPhi ^+ {\smallsetminus }\varTheta _{\mathrm m}\) or \(\beta \in \varTheta _{\mathrm m} \cap \varPsi ^\mathrm {\sharp }_{\mathrm M'} {\smallsetminus }\varPsi ^\mathrm {\sharp }_\mathrm m\).

Then, \(U_\beta \mathscr {U}_I \subset (B \cap B^v)\mathscr {U}_I\), hence \(BvU_\beta \mathscr {U}_I = \mathscr {O}_{v,I}\). Therefore, \(P_\alpha \mathscr {O}_{w,I} = \mathscr {O}_{w,I} \cup \mathscr {O}_{v,I}\) decomposes in the union of two orbits as represented in the following diagram, where we have:
  • \(\dim \mathscr {O}_{w,I} = \dim \mathscr {O}_{v,I} -1\);

  • \({{\mathrm{rk}}}\mathscr {O}_{w,I} = {{\mathrm{rk}}}\mathscr {O}_{v,I}\);

  • \(\mathrm M_{v,I} = \mathrm M\);

  • \(\mathrm m_{v,I} = \mathrm m\).

(T)

Suppose that \(\beta \in \varTheta _{\mathrm m} {\smallsetminus }\varPsi ^\mathrm {\sharp }_{\mathrm M'}\).

Then, \(U_\beta \mathscr {U}_I = \mathscr {U}_I \cup \mathscr {U}_{I'}\), where we set \(I' = I \cup \{\delta (\beta )\}\), and by Theorem 5.1 it follows \(\mathscr {O}_{v,I'} \ne \mathscr {O}_{v,I}\). Therefore, \(P_\alpha \mathscr {O}_{w,I} = \mathscr {O}_{w,I} \cup \mathscr {O}_{v,I} \cup \mathscr {O}_{v,I'}\) decomposes in the union of three orbits as represented by the following diagram, where we have:
  • \(\dim \mathscr {O}_{v,I} = \dim \mathscr {O}_{w,I} = \dim \mathscr {O}_{v,I'} - 1\);

  • \({{\mathrm{rk}}}\mathscr {O}_{v,I} = {{\mathrm{rk}}}\mathscr {O}_{w,I} = {{\mathrm{rk}}}\mathscr {O}_{v,I'} - 1\);

  • \(\mathrm M_{v,I} = \mathrm M{\smallsetminus }\{\delta (\beta )\}\), \(\mathrm M_{v,I'} = \mathrm M\);

  • \(\mathrm m_{v,I} = \mathrm m\), \(\mathrm m_{v,I'} = \mathrm m\cup \{\delta (\beta )\}\).

The claims follow now by applying the definitions of the actions of \(s_\alpha \) and of \(m(s_\alpha )\) in all the possibilities presented above. \(\square \)

Remark 6.4

In what follows, reduced pairs will play a main role in the understanding of the Weyl group action on \(\mathscr {B}(G/H)\). In particular, we will be interested in the stabilizer of \(\mathscr {O}_{w,I}\) under the action of W. If we restrict to the simple reflections which stabilize \(\mathscr {O}_{w,I}\), we see by Lemma 6.3 that \(s_\alpha \cdot \mathscr {O}_{w,I} =\mathscr {O}_{w,I}\) if and only if \(-w^{-1}(\alpha )\) is a weakly active root which stabilizes I in the sense of Definition 4.7. Such simple reflections are parametrized by the set of weakly active roots \(-w^{-1}(\alpha ) \in w^{-1}(\Delta ^-) \cap \varPsi ^\mathrm {\sharp }_I\). Notice that
$$\begin{aligned} w^{-1}(\Delta ^-) \cap \varPhi ^+_I = w^{-1}(\Delta ^-) \cap \varPsi ^\mathrm {\sharp }_I = w^{-1}(\Delta ^-) \cap \Delta _I. \end{aligned}$$
Indeed, we have the inclusions \(\Delta _I \subset \varPsi ^\mathrm {\sharp }_I \subset \varPhi ^+_I \subset \varPhi ^+(w)\), and by definition \(\Delta _I\) is a basis for \(\varPhi _I\). If \(\alpha \in \Delta \) and \(-w^{-1}(\alpha ) \in \varPhi ^+_I\), it follows that there exist \(\beta _1,\ldots , \beta _n \in \Delta _I\) such that \(-w^{-1}(\alpha ) = \beta _1+ \cdots + \beta _n\). Therefore, \(-\alpha = w(\beta _1) + \cdots + w(\beta _n)\), and since \(\alpha \in \Delta \) we get \(n=1\).

Corollary 6.5

Let (wI) be an extended pair. The orbit \(\mathscr {O}_{w,I}\) is minimal w.r.t. the weak order on \(\mathscr {B}(G/H)\) if and only if
$$\begin{aligned} w^{-1}(\Delta ^-) \cap \varPhi ^+ \subset \varTheta _I {\smallsetminus }\varPsi ^\mathrm {\sharp }_I. \end{aligned}$$

Proof

Let \(\alpha \in \Delta \). Then, \(P_\alpha \mathscr {O}_{w,I} \cap B s_\alpha wH/H \ne \varnothing \); therefore, \(P_\alpha \mathscr {O}_{w,I}\) contains at least two B-orbits. In particular, it follows that \(\mathscr {O}_{w,I}\) is minimal w.r.t. the weak order if and only if \(m(s_\alpha ) \cdot \mathscr {O}_{w,I} \ne \mathscr {O}_{w,I}\) for all \(\alpha \in \Delta \). On the other hand, by the analysis in the proof of Proposition 6.3 we have that \(m(s_\alpha ) \cdot \mathscr {O}_{w,I} \ne \mathscr {O}_{w,I}\) if and only if either \(w^{-1}(\alpha ) \in \varPhi ^+\) or \(-w^{-1}(\alpha ) \in \varTheta _I {\smallsetminus }\varPsi ^\mathrm {\sharp }_I\). Therefore, \(\mathscr {O}_{w,I}\) is minimal if and only if \(w^{-1}(\Delta ^-) \cap \varPhi ^+ \subset \varTheta _I {\smallsetminus }\varPsi ^\mathrm {\sharp }_I\). \(\square \)

We now focus on the action of W on \(\mathscr {B}(G/H)\). First, we show that the minimal representative of an orbit is a complete invariant for the action of W; namely, it is invariant and it distinguishes the W-orbits. Then, we will describe the stabilizers of the action in terms of reduced pairs.

Theorem 6.6

Let (wI), (vJ) be reduced pairs. Then, \(\mathscr {O}_{w,I}\) and \(\mathscr {O}_{v,J}\) are in the same W-orbit if and only if \(I = J\), in which case we have \(\mathscr {O}_{v,I} = vw^{-1} \cdot \mathscr {O}_{w,I}\).

Proof

By Lemma 6.3 (i), it follows that the minimal representative I is an invariant for the action of W. To show that it uniquely determines the W-orbit, we show that if (wI) is reduced, then \(\mathscr {O}_{w_0,I} = w_0 w^{-1} \cdot \mathscr {O}_{w,I}\). In particular, this will imply that every W-orbit in \(\mathscr {B}(G/H)\) contains a unique element which projects dominantly on G / B.

Suppose that (wI) is reduced and let \(w_0 w^{-1} = s_{\alpha _1} \cdots s_{\alpha _n}\) be a reduced expression. Then, \(w_{i+1}^{-1}(\alpha _i) \in \varPhi ^+\) for every \(i=1, \ldots n\), where we denote
$$\begin{aligned} w_j = \left\{ \begin{array}{ll} w &{} \text { if } j = n+1 \\ s_{\alpha _j} \cdots s_{\alpha _n} w &{} \text { if } 0 < j \leqslant n \end{array}\right. \end{aligned}$$
Therefore, for every \(i \leqslant n\), we have \(\varPhi ^+(w_i) = \varPhi ^+(w_{i+1}) \cup \{w_{i+1}^{-1}(\alpha _i)\}\). Since (wI) is reduced, it follows that \(\varPhi ^+_I \subset \varPhi ^+(w)\); thus, \(\varPhi ^+_I \subset \varPhi ^+(w_i)\) for all \(i \leqslant n\). It follows that \((w_i,I)\) is reduced as well for all \(i \leqslant n\), and by Lemma 6.3 we get \(\mathscr {O}_{w_{i-1},I} = s_{i-1} \cdot \mathscr {O}_{w_i,I}\). Combining all the steps we get \(\mathscr {O}_{w_0,I} = w_0 w^{-1} \cdot \mathscr {O}_{w,I}\), and the last claim also follows. \(\square \)

Corollary 6.7

The Weyl group W has \(2^{|\mathscr {D}|}\) orbits in \(\mathscr {B}(G/H)\), and a complete set of representatives is given by the orbits which project dominantly on G / B, namely by the subsets of \(\mathscr {D}\).

It follows by Theorem 6.6 that every W-orbit contains a distinguished element.

Corollary 6.8

Let (wI) be a reduced pair, then the element \(w^{-1} \cdot \mathscr {O}_{w,I}\) depends only on I and not on w.

Given a reduced pair (wI), we set \(\mathscr {O}_I^\sharp = w^{-1} \cdot \mathscr {O}_{w,I}\). We now turn to the description of the stabilizers for the action of W on \(\mathscr {B}(G/H)\), and we prove the following theorem.

Theorem 6.9

Let (wI) be a reduced pair. Then, \({{\mathrm{Stab}}}_W(\mathscr {O}_{w,I}) = wW_Iw^{-1}\).

Together with Theorem 6.6, the previous theorem gives a formula to compute the number of B-orbits in G / H in terms of the root systems \(\varPhi _I\).

Corollary 6.10

The number of B-orbits in G / H is given by the formula
$$\begin{aligned} \sum _{I \subset \mathscr {D}} |W/W_I|. \end{aligned}$$

To prove Theorem 6.9, we proceed by steps. First, we prove one of the two inclusions.

Lemma 6.11

Let (wI) be a reduced pair. Then, \({{\mathrm{Stab}}}_W(\mathscr {O}_{w,I}) \subset wW_Iw^{-1}\).

Proof

Let \(v \in {{\mathrm{Stab}}}_W(\mathscr {O}_{w,I})\), we proceed by induction on the length of v. If \(l(v) = 1\), then by Lemma 6.3 we have \(v = s_\alpha \) for some \(\alpha \in w(\varPsi ^\mathrm {\sharp }_I)\); hence, we assume \(l(v) > 1\). Let \(v = s_{\alpha _m} \cdots s_{\alpha _1}\) be a reduced expression, for \(i \leqslant m\) we set \(v_i = s_{\alpha _i} \cdots s_{\alpha _1}\) and \(\mathscr {O}_i = v_i \cdot \mathscr {O}_{w,I}\). Suppose \(\mathscr {O}_i \ne \mathscr {O}_{i-1}\) for all \(i \leqslant m\), then Lemma 6.3 implies \(v \cdot \mathscr {O}_{w,I} = \mathscr {O}_{vw,I}\); hence, \(vw = w\) and \(v=e\). Otherwise, let \(n \leqslant m\) be such that \(\mathscr {O}_n = \mathscr {O}_{n-1}\), and assume that n is minimal with this property.

If \(n = 1\), the claim follows by the inductive hypothesis, suppose \(n > 1\). Denote \(\alpha = v_{n-1}^{-1} (\alpha _n)\); then, \(v = v' s_\alpha \), where \(v' = s_{\alpha _m} \cdots \hat{s}_{\alpha _n} \cdots s_{\alpha _1}\). By construction
$$\begin{aligned} \mathscr {O}_{n-1} = v_{n-1} \cdot \mathscr {O}_{w,I} = \mathscr {O}_{v_{n-1} w,I} \end{aligned}$$
and \(s_{\alpha _n} \in {{\mathrm{Stab}}}_W(\mathscr {O}_{n-1})\), and it follows \(s_\alpha \in {{\mathrm{Stab}}}_W(\mathscr {O}_{w,I})\). On the other hand, we have \(w^{-1} v_{n-1}^{-1} (\alpha _n) \in \varPsi ^\mathrm {\sharp }_I\) by Lemma 6.3; hence, \(\alpha \in w(\varPsi ^\mathrm {\sharp }_I)\) is of the desired shape and \(s_\alpha \in w W_I w^{-1}\). Finally, \(v' \in {{\mathrm{Stab}}}_W(\mathscr {O}_{w,I})\) and \(l(v') < l(v)\); therefore, we may apply the inductive hypothesis and it follows \(v \in w W_I w^{-1}\).

Corollary 6.12

Let \(w_I \in W_I\) be the longest element, then we have \(\mathscr {O}_I^\sharp = \mathscr {O}_{w_I,I}\).

Proof

Let \(w \in W\) be such that \(\mathscr {O}^\sharp _I = \mathscr {O}_{w,I}\). By the definition of \(\mathscr {O}^\sharp _I\), we have the equality \(\mathscr {O}_{w,I} = w^{-1} \cdot \mathscr {O}_{w,I}\); hence, \(w \in {{\mathrm{Stab}}}_W(\mathscr {O}_{w,I})\), and by Lemma 6.11 we get \(w \in W_I\). On the other hand, \(W_I\) contains a unique element such that \(\varPhi ^+_I \subset \varPhi ^+(w)\), namely \(w_I\). \(\square \)

Proof of Theorem 6.9

Since \(\mathscr {O}_{w,I} = w \cdot \mathscr {O}^\sharp _I\), by the previous corollary it is enough to show the equality \({{\mathrm{Stab}}}_W(\mathscr {O}^\sharp _I) = W_I\). The inclusion \({{\mathrm{Stab}}}_W(\mathscr {O}^\sharp _I) \subset W_I\) follows from Lemma 6.11, and we show the other inclusion.

Let \(v \in W_I\), we show \(v \in {{\mathrm{Stab}}}_W(\mathscr {O}^\sharp _I)\) proceeding by induction on l(v), where we regard v as an element of W. If \(l(v) = 1\), then we have \(v = s_\alpha \) for some \(\alpha \in \Delta \cap \varPhi ^+_I \subset \Delta _I\); hence, \(\alpha = -w_I(\beta )\) for some other root \(\beta \in \Delta _I\). On the other hand, \(\Delta _I \subset \varPsi ^\mathrm {\sharp }_I\), and Lemma 6.3 implies \(s_\alpha \in {{\mathrm{Stab}}}_W(\mathscr {O}_I^\sharp )\).

Suppose now that \(l(v) > 1\), and let \(v = s_{\alpha _n} \cdots s_{\alpha _1}\) be a reduced expression of v as an element of W. If \(i \leqslant n\), we denote \(v_i = s_{\alpha _i} \cdots s_{\alpha _1}\) and \(\mathscr {O}_i = v_i \cdot \mathscr {O}^\sharp _I\). Suppose \(\mathscr {O}_i \ne \mathscr {O}_{i-1}\) for all \(i \leqslant n\), then Lemma 6.3 implies that \(v \cdot \mathscr {O}_{w_I,I} = \mathscr {O}_{vw_I,I}\) and that \((vw_I,I)\) is a reduced pair. On the other hand, \(vw_I \in W_I\), and \(w_I\) is the unique element in \(W_I\) such that \(\varPhi ^+_I \subset \varPhi ^+(w_I)\). Therefore, \(vw_I = w_I\), which is absurd since \(v \ne e\). Therefore, \(s_{\alpha _k} \in {{\mathrm{Stab}}}_W(\mathscr {O}_{k-1})\) for some \(k \leqslant n\). Assume that k is minimal with this property, then Lemma 6.3 implies
$$\begin{aligned} \mathscr {O}_{k-1} = v_{k-1} \cdot \mathscr {O}_I^\sharp = \mathscr {O}_{v_{k-1} w_I,I}. \end{aligned}$$
By construction \(s_{\alpha _k} \in {{\mathrm{Stab}}}_W(\mathscr {O}_{k-1})\); hence, \(-w_I v^{-1}_{k-1} (\alpha _k) \in \varPsi ^\mathrm {\sharp }_I\) by Lemma 6.3. Denote \(\alpha = v^{-1}_{k-1} (\alpha _k)\), then \(s_\alpha \in {{\mathrm{Stab}}}_W(\mathscr {O}_I^\sharp )\) and \(\alpha \in \varPhi ^+_I\); hence, \(s_\alpha \in W_I\). On the other hand, \(v = s_{\alpha _n} \cdots \hat{s}_{\alpha _k} \cdots s_{\alpha _1} s_\alpha \); therefore, \(s_{\alpha _n} \cdots \hat{s}_{\alpha _k} \cdots s_{\alpha _1} \in W_I\), and applying the inductive hypothesis it follows \(v \in {{\mathrm{Stab}}}_W(\mathscr {O}_I^\sharp )\). \(\square \)

6.3 Reduced pairs and weight polytopes

Building upon Theorem 6.9, we now produce a combinatorial model for the action of W on \(\mathscr {B}(G/H)\) in terms of weight polytopes.

Consider the Weyl group \(W_I\). The system of positive roots \(\varPhi ^+_I\) induces a Bruhat order \(\leqslant _I\) on \(W_I\), which is compatible with the restriction of the Bruhat order \(\leqslant \) on W in the following sense.

Lemma 6.13

Let (wI) be a reduced pair. Let \(v_1, v_2 \in W_I\) be such that \(v_1 \leqslant _I v_2\), then \(wv_2 \leqslant wv_1\). In particular, the left coset \(wW_I\) possesses a unique minimal element and a unique maximal element with respect to \(\leqslant \), namely \(ww_I\) and w.

Proof

Recall that \(G_I \subset G\) is the reductive subgroup generated by T together with the root spaces \(U_\alpha \) with \(\alpha \in \varPhi _I\), and \(B_I \subset G_I\) is the Borel subgroup \(B \cap G_I\). As \(v_1 \leqslant _I v_2\), it follows \(w_I v_2 \leqslant _I w_I v_1\); hence, \(w_I v_2 \in \overline{B_I w_I v_1 B_I}\).

Since \(\varPhi _I^+ \subset \varPhi ^+(w) \cap \varPhi ^+(w_I)\), it follows \(\varPhi ^+(ww_I) \cap \varPhi ^+_I = \varnothing \); hence, we get the equality \(Bww_IT = Bww_I B_I\). Therefore,
$$\begin{aligned} w v_2 \in Bw w_I B_I w_I v_2 B_I \subset \overline{Bw w_I B_I w_I v_1 B_I} \subset \overline{Bwv_1 B_I} \subset \overline{Bwv_1 B} \end{aligned}$$
and the claim follows.
Denote by \(\Lambda \) the weight lattice of T and let \(\lambda \in \Lambda \) be a regular dominant weight. Denote \(P = {{\mathrm{conv}}}(W\lambda )\) the weight polytope of \(\lambda \) in \(\Lambda _\mathbb Q\); then, the vertices of P correspond bijectively to the elements of W. By a subpolytope of P, we mean the convex hull of a subset of vertices of P. Denote \(\mathscr {S}(P)\) the set of subpolytopes of P; then, the Weyl group acts naturally on \(\mathscr {S}(P)\). Given \(I \subset \mathscr {D}\), we set \(\mathscr {S}_I = {{\mathrm{conv}}}(W_I \lambda )\), and for a reduced pair (wI) we set
$$\begin{aligned} \mathscr {S}_{w,I} = w \mathscr {S}_I = {{\mathrm{conv}}}(wW_I \lambda ) = {{\mathrm{conv}}}\big (vw (\lambda ) \, : \,v \in {{\mathrm{Stab}}}_W(\mathscr {O}_{w,I})\big ) \end{aligned}$$
(where the last equality follows from Theorem 6.9). Denote moreover
$$\begin{aligned} \mathscr {C}_{w,I} = {{\mathrm{cone}}}(-w(\varPsi ^\mathrm {\sharp }_I)) = {{\mathrm{cone}}}(-w(\varPhi ^+_I)) = -w\big (\mathbb Q\varPsi _I \cap \mathbb Q_{\geqslant 0} \Delta \big ) \end{aligned}$$

Theorem 6.14

The map \(\mathscr {O}_{w,I} \mapsto \mathscr {S}_{w,I}\) is a W-equivariant embedding of \(\mathscr {B}(G/H)\) into \(\mathscr {S}(P)\). Moreover, we have the equality
$$\begin{aligned} \mathscr {S}_{w,I} = P \cap (\mathscr {C}_{w,I}+w\lambda ), \end{aligned}$$
and in particular \(\dim (\mathscr {S}_{w,I}) = {{\mathrm{rk}}}(\varPhi _I)\).

Proof

Let \(I \subset \mathscr {D}\) and denote by \(\mathscr {B}_I(G/H)\) the corresponding W-orbit in \(\mathscr {B}(G/H)\). We claim that the map \(\mathscr {O}_{w,I} \mapsto \mathscr {S}_{w,I}\), regarded as a map \(\mathscr {B}_I(G/H) \rightarrow \mathscr {S}(P)\), is injective and W-equivariant. Indeed, since (wI) is reduced, by construction we have
$$\begin{aligned} {{\mathrm{Stab}}}_W(\mathscr {O}_{w,I}) = {{\mathrm{Stab}}}_W(\mathscr {S}_{w,I}). \end{aligned}$$
Since \(\mathscr {B}_I(G/H)\) is a single W-orbit, this shows that the map is injective, and since by Lemma 6.3 the actions of the simple reflections on \(\mathscr {O}_{w,I}\) and on \(\mathscr {S}_{w,I}\) coincide, it shows that it is W-equivariant as well.

In particular, the map \(\mathscr {B}(G/H) \rightarrow \mathscr {S}(P)\) is W-equivariant and we need only to show the injectivity, namely that I is determined by \(\mathscr {S}_{w,I}\). First of all, notice that w is determined by \(\mathscr {S}_{w,I}\). Indeed, by Lemma 6.13, w is maximal in \(wW_I\) w.r.t. the Bruhat order and is uniquely determined by this property. It follows that \(w\lambda \) is the unique minimal vertex of \(\mathscr {S}_{w,I}\) w.r.t. the dominance order; hence, w is uniquely determined by \(\mathscr {S}_{w,I}\). Then, the equality \(\mathscr {S}_{w,I} = {{\mathrm{conv}}}(wW_I \lambda )\) implies that the set of differences \(\{v\lambda - w\lambda \, : \,v\lambda \in \mathscr {S}_{w,I} \}\) generates the semigroup \(w(\mathbb N\varPhi ^+_I)\); hence, we recover I from w and \(\mathscr {S}_{w,I}\) thanks to Proposition 4.20. The last claim also follows. \(\square \)

We conclude this subsection by showing that the parametrization of orbits via subpolytopes of P of Theorem 6.14 is compatible with the Bruhat order on \(\mathscr {B}(G/H)\) in the following sense.

Proposition 6.15

Let (wI) and (vJ) be reduced pairs and suppose that \(\mathscr {S}_{v,J} \subset \mathscr {S}_{w,I}\). Then, we have \(\mathscr {O}_{v,J} \subset \overline{\mathscr {O}_{w,I}}\).

The proposition is an easy consequence of the following lemma.

Lemma 6.16

Let (wI) be a reduced pair, and let \(J \subset \mathrm M_{w,I}\). If \(v \in wW_I\), then \(\mathscr {O}_{v,J} \subset \overline{\mathscr {O}_{w,I}}\).

Proof

We will make use of the following fact, which follows by the description of the T-stable curves in G / B (see [11, Proposition 3.9]): if \(x \in W\) and \(\alpha \in \varPhi ^+(x)\), then \(x s_\alpha \in \overline{B x U_\alpha }\).

Let \(\alpha _1, \ldots , \alpha _m \in \Delta _I\) be such that \(w^{-1} v = s_{\alpha _1} \cdots s_{\alpha _m}\) is a reduced expression and, for \(i \leqslant m\), denote \(w_i = w s_{\alpha _1} \cdots s_{\alpha _i}\). Then, by Lemma 6.13 we have the following chain in the Bruhat order
$$\begin{aligned} v = w_n< \ldots< w_i< \ldots< w_1 < w. \end{aligned}$$
Denote \(\mathrm M= \mathrm M_{w,I}\). As \(J \subset \mathrm M\), it follows \(\mathscr {O}_{v,J} \subset \overline{\mathscr {O}_{v,\mathrm M}}\); therefore, to prove the theorem it is enough to consider the case \(J = \mathrm M\). In particular, we have the inclusion \(I \subset J\); hence, \(\Delta _I \subset \varPsi ^\mathrm {\sharp }_I \subset \varPsi ^\mathrm {\sharp }_J\). It follows then by Proposition 4.22 that \(\overline{\mathscr {U}_J} = \overline{B_J \mathscr {U}_J}\) is \(U_{\alpha _i}\)-stable for all i. In particular, we have that \(U_{\alpha _1} U_{\alpha _2} \cdots U_{\alpha _n} \mathscr {U}_J \subset \overline{\mathscr {U}_J}\), and by the remark at the beginning of the proof we get the inclusions
$$\begin{aligned} \overline{\mathscr {O}_{w,I}}= & {} \overline{Bw \mathscr {U}_J} = \overline{Bw U_{\alpha _1} \cdots U_{\alpha _n} \mathscr {U}_J} \supset \overline{Bw_1 U_{\alpha _2} \cdots U_{\alpha _n} \mathscr {U}_J} \supset \\&\cdots \supset \overline{Bw_n \mathscr {U}_J} \supset Bv \mathscr {U}_J. \end{aligned}$$
\(\square \)

Proof of Proposition 6.15

Notice that \(vW_J \subset wW_I\); hence, \(w^{-1}v \in W_I\) implies \(W_J \subset W_I\). In particular, it follows \(\varPhi ^+_J \subset \varPhi ^+_I\); therefore,
$$\begin{aligned} J = \delta (\varPhi ^+_J \cap \varPsi ^\mathrm {\sharp }) \subset \delta (\varPhi ^+_I \cap \varPsi ^\mathrm {\sharp }) = I, \end{aligned}$$
and the claim follows by Lemma 6.16. \(\square \)

7 A bound for the number of B-orbits in G / H.

A conjecture of Knop states that the homogeneous variety \(G/TU'\) of Example 5.8 has the largest number of B-orbits among all the homogeneous spherical G-varieties. In this section, we prove such bound in the setting of solvable spherical subgroups. That is, we prove the following theorem.

Theorem 7.1

The spherical variety \(G/TU'\) has the largest number of B-orbits among the homogeneous spherical varieties G / H with H a solvable subgroup of G.

The set \(\mathscr {B}(G/TU')\) is nicely described in terms of faces \(\mathscr {F}(P)\) of the weight polytope P of a regular dominant weight. Indeed, in this case the embedding of Theorem 6.14 induces a W-equivariant bijection between \(\mathscr {B}(G/TU')\) and \(\mathscr {F}(P)\) (see also [34, Proposition 3.5] and [36, Section 3] for a description of \(\mathscr {F}(P)\) and of \(\mathscr {B}(G/TU')\) in terms of the W-action).

We first prove Theorem 7.1 in the basic case of a maximal rank strongly solvable spherical subgroup of G.

Lemma 7.2

The spherical variety \(G/TU'\) has the largest number of B-orbits among the homogeneous spherical varieties G / H with H a strongly solvable subgroup of G of maximal rank.

Proof

Suppose that \(\pi (I) \subset \varPsi \) is connected. By Proposition 4.31, \(\varPhi _I\) and \(\varPhi _I'\) are both irreducible of rank |I|. In particular, if \(\varPhi _I'\) is of type \(\mathsf A\), then it follows the inequality \(|W_I'| \leqslant |W_I|\). Otherwise, if \(\varPhi _I'\) is not of type \(\mathsf A\), then Proposition 4.31 shows that \(W'_I = W_I\).

Consider now an arbitrary subset \(I \subset \varPsi \) and let \(I = I_1 \cup \ldots \cup I_m\) correspond to the decomposition of \(\pi (I)\) into connected components. Then, \(\varPhi '_I = \varPhi '_{I_1} \times \ldots \times \varPhi '_{I_m}\), and for all \(j=1, \ldots , m\) it holds \(|W'_{I_j}| \leqslant |W_{I_j}|\). Since \(\varPsi \) is linearly independent, \(\varPhi _{I_h} \cap \varPhi _{I_k} = \varnothing \) for all \(h \ne k\); hence, \(W_{I_h} \cap W_{I_k} = \{e\}\) for all \(h \ne k\), and it follows that
$$\begin{aligned} |W'_I| = |W'_{I_1} \times \ldots \times W'_{I_m}| \leqslant |W_{I_1} \times \ldots \times W_{I_m}| \leqslant |W_I|. \end{aligned}$$
Therefore, by Corollary 6.10 we get
$$\begin{aligned} |\mathscr {B}(G/H)| = \sum _{I \subset \varPsi } |W/W_I| \leqslant \sum _{I \subset \varPsi } |W/W'_I| \leqslant \sum _{I' \subset \Delta } |W/W_{I'}| = |\mathscr {B}(G/H')|. \end{aligned}$$
\(\square \)

Proof of Theorem 7.1

We reduce the proof of the theorem to the case of a strongly solvable spherical subgroup of G, which is treated in the previous lemma.

Suppose that H is a solvable spherical subgroup and denote by \(H^\circ \subset H\) the identity component. Clearly \(G/H^\circ \) has a larger number of B-orbits than G / H, and since it is solvable and connected, \(H^\circ \) is contained inside a Borel subgroup of G. Therefore, we may assume that H is strongly solvable.

Suppose now that H is a strongly solvable spherical subgroup, assume that \(H \subset B\) and that \(T \cap H\) contains a maximal torus \(T_H \subset H\). Recall the restriction \(\tau : \varPsi \rightarrow \mathcal X(T_H)\); following the discussion at the end of Sect. 3 we identify the maps \(\tau : \varPsi \rightarrow \overline{\varPsi }\) and \(\delta : \varPsi \rightarrow \mathscr {D}\). If \(D \in \mathscr {D}\), denote \(\varPsi _D = \delta ^{-1}(D)\). As in [2, §2], define a partial order on \(\mathscr {D}\) as follows: \(D \leqslant D'\) if there are \(\beta \in \varPsi _D\) and \(\beta ' \in \varPsi _{D'}\) with \(\beta \leqslant \beta '\). Equivalently, by [2, Proposition 1], we have \(D \leqslant D'\) if and only if there exists \(\alpha \in \varPhi ^+\) such that \(\varPsi _D + \alpha \subset \varPsi _{D'}\). Pick a representative \(\beta _D \in \varPsi _D\) for all \(D \in \mathscr {D}\). Set \(\mathscr {D}^1 = \mathscr {D}\) and let \(\mathscr {D}_1\) be the set of the maximal elements in \(\mathscr {D}\). We define inductively
$$\begin{aligned} \mathscr {D}^{i+1} = \mathscr {D}^i {\smallsetminus }\bigcup _{D \in \mathscr {D}_i}\delta (F(\beta _D)), \end{aligned}$$
and we define \(\mathscr {D}_{i+1}\) as the set of the maximal elements in \(\mathscr {D}^{i+1}\). Let p be the maximum such that \(\mathscr {D}_p\) is not empty and set \(\mathscr {D}_* = \bigcup _{k \leqslant p} \mathscr {D}_k\). Finally, define
$$\begin{aligned} \varPsi ' = \bigcup _{D \in \mathscr {D}_*} F(\beta _D).\end{aligned}$$
We devote the following paragraphs to show that \(\delta _{|\varPsi '} : \varPsi ' \rightarrow \mathscr {D}\) is bijective. Indeed, it is surjective by construction. Suppose by contradiction that it is not injective and let \(\alpha , \alpha ' \in \varPsi '\) be such that \(\alpha \ne \alpha '\) and \(\delta (\alpha ) = \delta (\alpha ')\). Let \(D, D' \in \mathscr {D}_*\) and let \(\gamma , \gamma ' \in \varPsi \) be maximal elements such that
$$\begin{aligned} \alpha \leqslant \beta _D \leqslant \gamma \qquad \text { and } \qquad \alpha ' \leqslant \beta _{D'} \leqslant \gamma '. \end{aligned}$$
Since the maps \(\tau : \varPsi \rightarrow \overline{\varPsi }\) and \(\delta : \varPsi \rightarrow \mathscr {D}\) are canonically identified and since \(\delta (\alpha ) = \delta (\alpha ')\), by [2, Lemma 5] it follows that \(\beta _D \ne \beta _{D'}\), hence \(D \ne D'\). Similarly, by [2, Lemma 8 and Lemma 11] it follows that \(\delta (\gamma ) = \delta (\gamma ')\) and \(\gamma -\alpha = \gamma ' - \alpha '\). By Proposition 4.15 we have \(\beta _D - \alpha \in \varPsi ^\mathrm {\sharp }\) and \(\beta _{D'}-\alpha ' \in \varPsi ^\mathrm {\sharp }\). Setting moreover \(\tilde{\beta }_D = \alpha ' + \beta _D - \alpha \) and \(\tilde{\beta }_{D'} = \alpha + \beta _{D'} - \alpha '\), by the same proposition we see that \(\tilde{\beta }_D\) and \(\tilde{\beta }_{D'}\) are active roots, and they satisfy the equalities
$$\begin{aligned} \delta (\tilde{\beta }_D) = \delta (\beta _D) = D, \qquad \qquad \delta (\tilde{\beta }_{D'}) = \delta (\beta _{D'}) = D'. \end{aligned}$$
Since \(\gamma -\alpha = \gamma ' - \alpha '\), we have the inequalities
$$\begin{aligned} \alpha \leqslant \tilde{\beta }_{D'} \leqslant \gamma \qquad \text { and } \qquad \alpha ' \leqslant \tilde{\beta }_D \leqslant \gamma '. \end{aligned}$$
Notice that \(\beta _D\) and \(\tilde{\beta }_{D'}\) are comparable by Lemma 4.29: indeed \(\alpha \in F(\beta _D) \cap F(\tilde{\beta }_{D'})\), thus \({{\mathrm{supp}}}(\beta _D) \cap {{\mathrm{supp}}}(\tilde{\beta }_{D'})\) is not empty, and \({{\mathrm{supp}}}(\beta _D) \cup {{\mathrm{supp}}}(\tilde{\beta }_{D'})\) is connected. Therefore we have either that \(\tilde{\beta }_{D'} \in F(\beta _D)\) or that \(\tilde{\beta }_D \in F(\beta _{D'})\). It follows that either \(D' \in \delta (F(\beta _D))\) or \(D \in \delta (F(\beta _{D'}))\), and by the definition of \(\mathscr {D}_*\) we get \(D = D'\), a contradiction. Therefore we proved the bijectivity of the map \(\delta _{|\varPsi '} : \varPsi ' \rightarrow \mathscr {D}\), and it follows that \(\tau _{|\varPsi '} : \varPsi ' \rightarrow \overline{\varPsi }\) is bijective as well. By making use of the properties of \(\varPsi '\), we now construct a strongly solvable spherical subgroup \(H'\) containg T such that \(|\mathscr {B}(G/H)| \leqslant |\mathscr {B}(G/H')|\). Notice that the elements of \(\varPsi '\) are linearly independent: indeed \(\tau _{|\varPsi '} : \varPsi ' \rightarrow \overline{\varPsi }\) is bijective, and \(\overline{\varPsi }\) is linearly independent in \(\mathcal X(T_H)\) by [2, Theorem 1]. Let \(\alpha , \alpha ' \in \varPhi ^+\) and suppose that \(\alpha + \alpha ' \in \varPsi '\). Since \(\mathfrak h\subset \mathfrak g\) is a subalgebra and since by definition \(\mathfrak u_{\alpha +\alpha '} \not \subset \mathfrak h\), it follows that either \(\mathfrak u_\alpha \not \subset \mathfrak h\) or \(\mathfrak u_{\alpha '} \not \subset \mathfrak h\). It follows that either \(\alpha \in \varPsi \) or \(\alpha ' \in \varPsi \), hence either \(\alpha \in \varPsi '\) or \(\alpha ' \in \varPsi '\). Therefore
$$H' = T \prod _{\alpha \in \varPhi ^+ {\smallsetminus }\varPsi '} U_\alpha $$
is a strongly solvable subgroup of G, and by [2, Theorem 1] it is spherical because \(\varPsi '\) is linearly independent.
We claim that \(|\mathscr {B}(G/H)| \leqslant |\mathscr {B}(G/H')|\). Denote indeed \(\mathscr {D}' = {{\mathrm{Div}}}_T(B/H')\). Then by Corollary 3.9 we have a bijection \(\varPsi ' \rightarrow \mathscr {D}'\), and by the discussion above the restriction gives a bijection \(\varPsi ' \rightarrow \overline{\varPsi }\). Thus we have a bijection \(\mathscr {D}' \rightarrow \mathscr {D}\), and by Corollary 6.7 we get that W acts on \(\mathscr {B}(G/H)\) and on \(\mathscr {B}(G/H')\) with the same number of orbits. On the other hand, if \(I \subset \mathscr {D}\)and \(\varPhi '_I = \mathbb Z\varPsi '_I \cap \varPhi \) is the root system associated to the corresponding T-orbit in \(B/H'\), then by construction we have \(\varPsi '_I \subset \varPsi _I\), hence \(\varPhi '_I \subset \varPhi _I\). Denoting by \(W'_I\) be the Weyl group of \(\varPhi '_I\), it follows that \(W'_I \subset W_I\), thus by Corollary 6.10 we get
$$\begin{aligned} |\mathscr {B}(G/H)| = \sum _{I \subset \mathscr {D}} |W/W_I| \leqslant \sum _{I \subset \mathscr {D}} |W/W'_I| = |\mathscr {B}(G/H')|. \end{aligned}$$
On the other hand by construction \(H'\) is a strongly solvable spherical subgroup of G of maximal rank, therefore \(|\mathscr {B}(G/H')| \leqslant |\mathscr {B}(G/TU')|\) by Lemma 7.2.

Footnotes

  1. 1.

    Notice that when \({{\mathrm{supp}}}(\beta )\) is of type \(\mathsf F_4\), our enumeration of the set \({{\mathrm{supp}}}(\beta )\) in Table 1 differs from the one in [2]: following [4] for us \(\alpha _1\) and \(\alpha _2\) are the long simple roots.

Notes

Acknowledgements

We thank P. Bravi, M. Brion, F. Knop and A. Maffei for useful conversations on the subject, and especially R.S. Avdeev for numerous remarks and suggestions on previous versions of the paper which led to significant improvements. This work originated during a stay of the first named author in Friedrich-Alexander-Universität Erlangen-Nürnberg during the fall of 2012 partially supported by a DAAD fellowship, and he is grateful to F. Knop and to the Emmy Noether Zentrum for hospitality. Both the authors were partially supported by the DFG Schwerpunktprogramm 1388—Darstellungstheorie.

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Scuola Normale SuperiorePisaItaly
  2. 2.Dipartimento di Matematica SapienzaUniversità di RomaRomaItaly

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