Wonderful Varieties of Type B and C

We show that the proof of Luna’s conjecture about the classification of general wonderful G-varieties can be reduced to the analysis of finitely many families of primitive cases. We work out all primitive cases arising with any classical group G. Luna’s conjecture states that, for any semisimple complex algebraic group G, wonderful G-varieties are classified by spherical systems of G ([12]). This has been proved for anyG adjoint with simply laced Dynkin diagram ([12, 5, 1]) or under other special hypotheses ([13, 3]): one reduces to the case of primitive spherical systems, which can be listed, and shows that any primitive spherical system can be realized as the spherical system of a (unique) wonderful variety. Afterwards, the uniqueness part of the conjecture has been proved in general, with different methods ([10]). Here we follow the approach of [12]. The argument of reduction to the primitive spherical systems is adapted to the general adjoint (not necessarily simply laced) case. In other words, we show that, in order to complete the proof of the conjecture, it is enough to prove that there exists a wonderful variety with any given primitive spherical system. The list of primitive spherical systems is given in [2]. Although it would be possible, we do not prove the existence of all the corresponding wonderful varieties. The list is very long and other general proofs of the conjecture seem now possible ([7, 8]). We rather make use of further ideas coming from [10] and [4] to explain that it is possible to reduce the analysis to a smaller class of spherical systems: without quotients of higher defect. We also give a further special reduction argument to spherical systems without tails. Finally, we analyze some relevant primitive spherical systems, mainly of type B and C. Together with [12, 5, 3] this completes the explicit case-by-case checking of Luna’s conjecture for any G of classical type. 1. Wonderful varieties and spherical systems 1.1. Basic definitions. Let G be a semisimple complex algebraic group, T a maximal torus, B a Borel subgroup containing T , S the corresponding set of simple roots of the root system of (G, T ). For all S ⊂ S, PS′ denotes the corresponding parabolic subgroup containing B (namely, S is a set of simple roots of the root system of (L, T ), where L = LS′ is the Levi subgroup of PS′ containing T ). Similarly, P−S′ denotes the opposite parabolic subgroup of PS′ , with respect to the maximal torus T . 1 2 P. BRAVI AND G. PEZZINI Definition 1.1.1. A G-variety is called wonderful of rank r if it is smooth, complete, with r smooth prime G-divisors D1, . . . , Dr with normal crossings, such that the G-orbit closures are exactly all the intersections ∩i∈IDi, for any I ⊆ {1, . . . , r}. We will always assume that the center of G acts trivially on a wonderful Gvariety. This is justified by the results of Luna in [12], where the classification is reduced to the case where G is adjoint. A wonderful G-variety is known to be projective and spherical, see [11]. Let X be a wonderful G-variety. There exists a unique point z ∈ X stabilized by B− (where B− is the Borel subgroup opposite to B with respect to T , i.e. B− ∩ B = T ). The orbit G.z is the (unique) closed orbit, ∩i=1Di; the parabolic subgroup opposite to Gz will be denoted by PX . Set S p X such that PX = PSp X . The T -weights occurring in the normal space TzX/TzG.z of G.z at z are called spherical roots, their set is denoted by ΣX . Spherical roots are in bijective correspondence with prime G-divisors, say σ 7→ D, and form a basis of the lattice of B-weights in C(X) (we are identifying B-weights with T -weights). The not G-stable prime B-divisors of X are called colors, their set is denoted by ∆X . The colors are representatives of a basis of PicX , so one can define a Z-bilinear pairing, called Cartan pairing, cX : Z∆X × ZΣX → Z such that [D] = ∑ D∈∆X cX(D,σ)[D]. For all α ∈ S, set ∆X(α) = {D ∈ ∆X : P{α}.D 6= D}. One has ∪α∈S∆X(α) = ∆X and, for all α ∈ S, card(∆X(α)) ≤ 2. Clearly, α ∈ S p X if and only if ∆X(α) = ∅. Moreover, if D ∈ ∆X(α) and card(∆X(α)) = 1 then cX(D,−) is uniquely determined by α. One has card(∆X(α)) = 2 if and only if α ∈ S ∩ΣX , in this case, say ∆X(α) = {D, D}, cX(D,−) and cX(D,−) are not always determined by α, but their sum is. Let AX denote the subset of colors D ∈ ∪α∈S∩ΣX∆X(α) endowed with the Z-linear functionals cX(D,−). Definition 1.1.2. The datum of (S X ,ΣX ,AX), also denoted by SX , is called the spherical system of X . Any G-orbit closure X ′ of X is a wonderful G-variety itself. Its set of spherical roots Σ = ΣX′ is a subset of ΣX , and X ′ is called the localization of X with respect to Σ and denoted by X ′ = XΣ′ . The spherical system of X ′ is given by S X = S p X , ΣX′ = Σ ′ and AX′ , which can be identified with the subset of colors D ∈ ∪α∈S∩Σ′∆X(α); one has cX′(D,σ) = cX(D,σ) for all σ ∈ Σ. In particular, any spherical root of X is the spherical root of a wonderful Gvariety of rank 1. The wonderful G-varieties of rank 1 are well known, for all G. The finite set of spherical roots of wonderful G-varieties of rank 1 is denoted by Σ(G). The spherical system of X is determined by the spherical systems of all the localizations XΣ′ of rank 2 (actually, it is enough to restrict to the localizations of rank 1 and those of rank 2 with a simple spherical root). Furthermore, the wonderful G-varieties of rank 2 are known, for all G. 1.2. Spherical systems. A spherical system of G of rank r ≤ 2 is, by definition, the spherical system of a wonderful G-variety of rank r. Wonderful G-varieties of rank r ≤ 2 are classified by spherical systems of G of rank r. Definition 1.2.1. A spherical system of G (of rank equal to cardΣ > 2) is a triple S ⊂ S, Σ ⊂ Σ(G) and A = ∪α∈S∩ΣA(α), endowed with a Z-bilinear pairing WONDERFUL VARIETIES OF TYPE B AND C 3 c : ZA × ZΣ → Z, such that for all Σ ⊂ Σ with cardΣ = 2 the triple S, Σ and AΣ′ = ∪α∈S∩Σ′A(α), with the restriction of c, is the spherical system of a wonderful G-variety of rank 2. Conjecture 1.2.1 ([12]). Wonderful G-varieties are classified by spherical systems of G. 2. Geometric constructions 2.1. The set of all colors. Given an abstract spherical system S = (S,Σ,A) for a given adjoint group G, it is possible to define a set ∆ ⊇ A of its colors (and a corresponding extension of c), which plays the role of ∆X . Following [12] and its notations, we identify each element in ∆ \A with the simple roots it is moved by. This gives a disjoint union: ∆ = A ∪∆ ′ ∪∆ The set ∆ ′ is the set of simple roots α such that 2α is a spherical root. For such a color D, we have c(D,−) = 12 〈α ,−〉. The set ∆ is: ( S \ ( Σ ∪ 1 2 Σ ∪ S ))/ ∼ where α ∼ β if α = β, or if α ⊥ β and α + β ∈ Σ. For such a color D, we have c(D,−) = 〈α,−〉, for α any representative of the ∼-equivalence class associated to D. 2.2. Localization and morphisms. Let X be a wonderful variety and S a subset of S. Consider a Levi subgroup LS′ of the parabolic subgroup PS′ ; the variety X contains a well defined wonderful LS′-subvariety XS′ , called the localization of X in S (see [12] for details). Its spherical system satisfies: S XS′ = S p X ∩ S ; the set ΣXS′ is the set of spherical roots of X whose support is contained in S ; the set AXS′ is the union of ∆X(α) for all α ∈ S ′ ∩ ΣX . The localization of an abstract spherical system S in S is defined analogously. Certain classes of morphisms between wonderful varieties can be represented by subsets of the colors of the domain. Let f : X → Y be a surjective G-equivariant morphism between two wonderful G-varieties, and define ∆f = {D ∈ ∆X | f(D) = Y }. Let now ∆ be a subset of ∆X ; we say that ∆ ′ is distinguished if there exists a linear combination δ of elements of ∆ with positive coefficients, such that cX(δ, σ) ≥ 0 for all spherical roots σ. We define the quotient spherical system S /∆ in the following way: (1) S X/∆ ′ = {α ∈ S | ∆(α) ⊆ ∆}; (2) ΣX/∆ ′ = the indecomposable elements (or equivalently the minimal generators) of the semigroup {σ ∈ NΣX | cX(δ, σ) = 0 ∀δ ∈ ∆}; (3) AX/∆ ′ = ⋃ α∈S∩ΣX/∆′ A(α). The bilinear pairing for S /∆ is induced from cX in a natural way. We say that ∆ is (*)-distinguished, if it is distinguished and the set ΣX/∆ ′ is a basis of the Z-module {σ ∈ ZΣX | cX(σ, δ) = 0 ∀δ ∈ ∆}. Proposition 2.2.1 ([12]). Let X be a wonderful G-variety. The map f 7→ ∆f induces a bijection between the set of (*)-distinguished subsets of ∆X and the set of equivalence classes of couples (f, Y ) where f : X → Y is a G-equivariant, surjective 4 P. BRAVI AND G. PEZZINI map with connected fibers, and where the equivalence relation is induced naturally by G-equivariant isomorphisms Y → Y . In principle, a distinguished but not (*)-distinguished set of colors corresponds to a G-equivariant morphism X → Y too. In this case Y satisfies the definition of a wonderful variety, except for the smoothness condition on Y and its prime Gdivisors. Such a variety is also called the canonical embedding of its open G-orbit. On the other hand, there is no known example of a distinguished set of colors that is not (*)-distinguished, and we conjecture indeed that none exists in general. A (*)-distinguished subset ∆ ⊆ ∆X is smooth if ΣX/∆ ′ ⊆ ΣX ; homogeneous (or parabolic) if ΣX/∆ ′ = ∅. Proposition 2.2.2 ([12]). A G-equivariant, surjective map with connected fibers between wonderful varieties f : X → Y is smooth if and only if ∆f is smooth. Moreover, Y is homogeneous if and only if ∆f is homogeneous. All the above definitions can be obviously given also for an abstract spherical system S . Definition 2.2.1. Let S be a spherical system with set of colors ∆, and ∆1, ∆2 two subsets of ∆. We say that ∆1 and ∆2 decompose S if: (1) ∆1 and ∆2 are non-empty and disjoint; (2) ∆1, ∆2 and ∆3 = ∆1 ∪∆2 are (*)-distinguished; (3) (Σ \ (Σ/∆1)) ∩ (Σ \ (Σ/∆2)) = ∅; (4) (S/∆1) \ S and (S/∆2) \ S are orthogonal; (5) ∆1 or ∆2 are smooth. Definition 2.2.2. Let S be a spherical system. A color δ ∈ A is projective if c(δ, σ) ≥ 0 for all σ ∈ Σ. Definition 2.2.3. A spherical system S is primitive if supp(Σ) = S, there is no projective color, and there is no pair ∆1,∆2 ⊂ ∆ decomposing S . 2.3. Reduction to primitive spherical systems. To prove Conjecture 1.2.1 it is enough to show that there exists a (unique) wonderful G-variety with any given primitive spherical system. This is clear from the following: Proposition 2.3.1. Let S = (S,Σ,A) be a spherical system of G. If one of the following conditions is fullfilled then there exists a (unique) wonderful G-variety with S as spherical system. (1) There exist a proper subset S of simple roots containing supp(Σ) and a (unique) wonderful L S -variety Y with spherical system equal to the localization of S with respect to S. (2) There exist two (*)-distinguished subsets of colors ∆1 and ∆2 decomposing S and for i = 1, 2, 3 there exists a (unique) wonderful G-variety Xi with spherical system equal to the quotient of S by ∆i, where ∆3 = ∆1 ∪∆2. (3) There exist a projective element δ ∈ A and a (unique) wonderful G-variety Xδ with spherical system equal to the quotient of S by {δ}. In the first two cases Proposition 2.3.1 has been proved in [12, Section 3]. In the first case X is G-isomorphic to the parabolic induction G ∗PS′ Y . In the second case X is G-isomorphic to the fiber product X1 ×X3 X2. WONDERFUL VARIETIES OF TYPE B AND C 5 In loc.cit. the third case has also been proved, but with some additional hypothesis: the support of δ is requested to intersect only the support of simple spherical roots. The latter is always true if G has a simply laced Dynkin diagram. In §3.2 we prove it in full generality. In loc.cit. it is also shown in general that, if X is a wonderful G-variety with (S,Σ,A) as spherical system with a projective element δ ∈ A, then the morphism associated to δ is a projective fibration, namely smooth with fibers isomorphic to a projective space. 3. Wonderful subgroups 3.1. Finding the wonderful subgroup. 3.1.1. Minimal inclusions. Let S be a spherical system. In view of the following proposition, we say that S is reductive if there exists a linear combination σ of spherical roots with non-negative coefficients, such that c(δ, σ) > 0 for all colors δ. Proposition 3.1.1 ([12]). Let X be a wonderful variety, with generic stabilizer H. Then H is reductive if and only if S is reductive. Let S = (Σ, S,A) be a spherical system with set of colors ∆, and suppose S is not reductive. This should correspond to a wonderful subgroup H , where moreover H is contained in some proper parabolic subgroup. Here and until the end of §3.1 we suppose that H exists. Up to conjugating H if necessary, it follows that there exists a proper parabolic subgroup Q− ⊇ B− containing H and minimal with respect to this property. It is possible to choose Levi subgroups L of H and LQ− of Q− so that L ⊆ LQ− . From minimality of Q− it follows that L is very reductive in LQ− and H r ⊆ Q−: in particular, H ⊆ Q− and the connected center C of L is contained in the connected center CQ− of LQ− . There exists also a wonderful subgroup K such that Q− ⊇ K ⊇ H and K/H is connected, that is a wonderful subgroup properly containing H and minimal in the following sense (see [4], Section 2.3.5). Choose a Levi component LK of K in such a way that LQ− ⊇ LK ⊇ L. Then the following holds: (1) the Levi subgroups LK , L differ at most only for their connected centers; (2) we have K ⊃ H ⊇ (K,K); (3) the quotient LieK/LieH is simple as L-module. These subgroups Q− and K correspond resp. to two quotients S /∆Q− and S /∆K of S . Let us now abandon the hypothesis that H exists. We usually find ∆Q− and ∆K which play the same role: ∆Q− , a minimal homogeneous subset of colors; ∆K , a special (not any) minimal (*)-distinguished subset of colors contained in ∆Q− , such that the wonderful subgroup K corresponding to S /∆K is known. We then find H inside K, as above. Assuming (L,L) = (LK , LK), it remains to find C and H. 6 P. BRAVI AND G. PEZZINI 3.1.2. The connected center. The dimension of C is immediately given by the well known formula: dimC = card∆− cardΣ. This integer is also called d(S ), defect of S . Consider all B-proper rational functions on G/H having neither zeros nor poles along colors in ∆Q− . Their B-weights form a sublattice N of ZΣ: fix a basis B of it. For each color D in ∆\∆Q− define αD as the unique simple root moving D (this root is unique because ∆Q− is homogeneous). Define also λD to be the fundamental dominant weight associated to αD, and λ ∗ D = −w0(λD), where w0 is the longest element in the Weyl group. Finally, for all γ ∈ N , define λ(γ) to be the weight: ∑ D∈∆\∆Q − c(D, γ) · λD restricted to the subgroup CQ− . Lemma 3.1.1. The subgroup C is the connected part of the intersection of the kernels of the weights λ(γ), for γ ranging through N (or, equivalently, B). Proof. Let γ ∈ N and consider a B-proper rational function fγ on G/H having B-weight γ. Call Fγ the pull-back of fγ on G along the projection G → G/H . Since fγ has neither zeros nor poles on colors of ∆Q− , Fγ has neither zeros nor poles on the pull-backs on G of these colors. Its zeros or poles must then lie only on the pullbacks on G of colors of G/Q− along the projection G → G/Q−. This means that Fγ is Q−-proper under right translation, and it is immediate to see that the Q−-weight of Fγ is just λ(γ). But Fγ is also of course H-stable under right translation, and this implies that λ(γ) is constantly 1 on C. Vice versa, suppose to have any weight χ of CQ− which is constantly 1 on C: considering χ as a Q−-weight, we can write it as a linear combination with integer coefficients of weights of the form λD for D ranging through the colors of G/Q−. So χ is the Q−-weight (under right translation) of some rational function on G, call it F , which is B-proper under left translation and Q−-proper under right translation, whose zeros and poles lie on the pullbacks on G of colors of G/Q− along the projection G → G/Q−. The function F under right translation is Q−-stable and (LQ− , LQ−)-stable; it is CQ− -proper but C-stable. Thus F isH-stable, and it descends to a rational function f on G/H . Having neither zeros nor poles along colors in ∆Q− by construction, the function f is equal to fγ for some γ ∈ N . 3.1.3. The unipotent radical. The unipotent radical H is a subgroup of K, containing (K,K) and such that LieK/LieH is a simple L-module. For computations in particular cases, it is useful to recall that H is also a subgroup of Q−, and that the quotient LieQ−/LieH u is a spherical L-module. The following lemma follows from the results of [10]. Lemma 3.1.2 ([10]). The spherical system S uniquely determines the structure of LieQ−/LieH u as an L-module. Let us describe how to calculate the L-module structure of LieQ−/LieH u starting from the spherical system S . A fiber F of the morphism G/H → Q− is affine WONDERFUL VARIETIES OF TYPE B AND C 7 and isomorphic to LQ− ×L V where V = LieQ u −/LieH . Restricting B-proper rational functions on G/H to F we obtain (B ∩ LQ−)-proper rational functions: this induces a bijection between the associated lattices of B(resp. (B ∩ LQ−)-) weights (a proof of this fact can be found in [6]). Restricting (B ∩ LQ−)-proper regular functions from F to V , we obtain all C[V ]. On the other hand, regular functions on F come exactly from rational functions on G/H having no poles on colors in ∆Q− . Therefore the weight monoid of V as a spherical L-module is given by restricting to B∩L all B-weights in Ξ that are non-negative on ∆Q− . Finally, the classification of spherical modules is used to determine the L-module structure of V . 3.2. Projective fibrations. We proceed completing the proof of Proposition 2.3.1. Let S = (S,Σ,A) be a spherical system with a projective δ ∈ A. Set supp δ = {α ∈ S : δ ∈ A(α)}. Lemma 3.2.1. Suppose supp δ ∩ supp(Σ \ S) 6= ∅. If there exists a wonderful variety Xδ with spherical system S /{δ}, then there exists a wonderful variety with spherical system S . Recall that (as said in §2.3) the assertion is true also if supp δ∩ supp(Σ\S) = ∅. Let us first prove that one can reduce to the case of card(supp δ) = 1, so we assume the assertion in this case. Let X be a wonderful variety with spherical system S with a projective δ with card(supp δ) = 1. Let K be the generic stabilizer of Xδ, with Levi decomposition K = LK K . Then a generic stabilizer H ⊂ K of X has the same Levi factor L = LK and unipotent radical H , such that LieH is a 1-codimensional L-submodule of LieK. Let S be a spherical system with a general projective element δ. Let Xδ be a wonderful variety with spherical system S /{δ}, and let K be its generic stabilizer with Levi decomposition K = LKK . Then for each α ∈ supp δ we can consider the spherical system obtained from S /{δ} by adding the simple spherical root α such that one of the two corresponding elements ofA(α), say δ α is projective, hence card(supp δ α ) = 1. Then there exists a wonderful variety with this spherical system and generic stabilizer given by Hα = LKH u α where LieH u α is a 1-codimensional LKsubmodule in LieK. Take the subgroup H = LH given as follows: LieH equal to a 1-codimensional (LK , LK)-submodule in LieK u and in general position among those containing ⋂ α∈supp δ LieH α; L equal to the normalizer of H in LK . Then H is the generic stabilizer of a wonderful variety with spherical system S . Let now S be a spherical system with a projective element δ with card(supp δ) = 1, say supp δ = {α}. If there exists a (*)-distinguished subset of colors ∆2 disjoint from A(α), then the subsets of colors ∆1 = {δ} and ∆2 decompose the spherical system S . Therefore, by using the second case of Proposition 2.3.1, we can suppose that there exists no (*)-distinguished subset of colors ∆2 disjoint from A(α). This implies that the spherical system is indecomposable. Clearly, we can also suppose that the spherical system is cuspidal. Furthermore, we assumed supp δ ∩ (Σ \ S) 6= ∅. 8 P. BRAVI AND G. PEZZINI A careful combinatorial analysis then shows that the only possibilities are the following “minimal” spherical systems, which will be treated case by case in §4.4. Here (and often in the rest of the paper) we describe spherical systems by providing their Luna diagrams, see [4] for their definition. (1) G of type Cn, n ≥ 2 q e e q q q q q pppppp ppppp e pp p p p p p p p p p p p p p p p p p p p (2) G of type G2 q e e q pppppp pppp e pp p p p p p p p p p p p p p p p p p p p (3) G of type F4 q q q q p p p p p p p p p p p e pp p p p p p p p p p p p p p p p p p p p e p p p p p p p p p p p p p p p p p p p p p e e