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Schottky groups acting on homogeneous rational manifolds

Christian Miebach and Karl Oeljeklaus

Abstract

We systematically study Schottky group actions on homogeneous rational manifolds and find two new families besides those given by Nori’s well-known construction. This yields new examples of non-Kähler compact complex manifolds having free fundamental groups. We then investigate their analytic and geometric invariants such as the Kodaira and algebraic dimension, the Picard group and the deformation theory, thus extending results due to Lárusson and to Seade and Verjovsky. As a byproduct, we see that the Schottky construction allows to recover examples of equivariant compactifications of SL(2,)/Γ for Γ a discrete free loxodromic subgroup of SL(2,), previously obtained by A. Guillot.

Funding source: Agence Nationale de la Recherche

Award Identifier / Grant number: ANR-10-BLAN-0118

Funding statement: The authors would like to thank A. T. Huckleberry and P. Heinzner for invitations to the Ruhr-Universität Bochum, Germany, where part of the work was done. The first author is grateful for the hospitality of the Institut de Mathématiques de Marseille (I2M) as well as for an invitation to the Fakultät für Mathematik of the Universität Duisburg-Essen by D. Greb. The second author is partially supported by the ANR project MNGNK, decision #ANR-10-BLAN-0118.

A Minimal orbits of hypersurface type

Let G be a simply-connected semisimple complex Lie group, let Q be a parabolic subgroup of G, and let G0 be a non-compact simple real form of G. We say that two triplets (G,G0,Q) and (G,G~0,Q~) are equivalent if there exist g1,g2G such that G~0=g1G0g1-1 and Q~=g2Qg2-1. In this appendix we outline the classification (up to equivalence) of all triplets (G,G0,Q) such that the minimal G0-orbit is a real hypersurface in X=G/Q.

Throughout we write σ:𝔤𝔤 for conjugation with respect to 𝔤0. Let θ:𝔤𝔤 be a Cartan involution that commutes with σ. Then we have the corresponding Cartan decomposition 𝔤0=𝔨0𝔭0. The analytic subgroup K0 of G0 having Lie algebra 𝔨0 is a maximal compact subgroup of G0.

The following theorem summarizes the outcome of the appendix.

Theorem 1.

Up to equivalence, the homogeneous rational manifolds X=G/Q and the real forms G0 having a compact hypersurface orbit in X are the following:

  1. G0=SU(p,q) acting on X=p+q-1,

  2. G0=Sp(p,q) acting on X=2(p+q)-1,

  3. G0=SU(1,n) acting on X=Grk(n+1),

  4. G0=SO*(2n) acting on X=Q2n-2,

  5. G0=SO(1,2n) acting on X=IGrn(2n+1),

  6. G0=SO(2,2n) acting on X=IGrn+1(2n+2)0.

A.1 Root-theoretic description of the minimal G0-orbit in X=G/Q

Let 𝔞0 be a maximal Abelian subspace of 𝔭0 and let

𝔤0=𝔪0𝔞0λΛ(𝔤0)λ

be the corresponding restricted root space decomposition of 𝔤0, where 𝔪0=𝒵𝔨0(𝔞0) and where Λ=Λ(𝔤0,𝔞0)𝔞0*{0} is the restricted root system. Choosing a system Λ+ of positive restricted roots we obtain the nilpotent subalgebra 𝔫0:=λΛ+𝔤λ. From this we get the Iwasawa decomposition G0=K0A0N0, where A0 and N0 are the analytic subgroups of G0 having Lie algebras 𝔞0 and 𝔫0, respectively.

Let 𝔱0 be a maximal torus in 𝔪0. Then 𝔥0:=𝔱0𝔞0 is a maximally non-compact Cartan subalgebra of 𝔤0. Let

𝔤=𝔥αΔ𝔤α

be the root space decomposition of 𝔤 with respect to the Cartan subalgebra 𝔥:=𝔥0 with root system Δ=Δ(𝔤,𝔥)𝔥*{0}, where 𝔥:=i𝔱0𝔞0. Let R:𝔥*𝔞0* be the restriction operator and let Δ𝗂:={αΔ:R(α)=0} be the set of imaginary roots.

Remark 2.

We have

𝔪0=𝔱0αΔ𝗂𝔤α,

i.e., Δ𝗂 is the root system of 𝔪0 with respect to its Cartan subalgebra 𝔱0.

Let us define a system Δ+ of positive roots with respect to the lexicographic ordering given by a basis of 𝔥 whose first r elements form a basis of 𝔞0. Then, for every αΔΔ𝗂, we have αΔ+ if and only if R(α)Λ+ and R(Δ+Δ𝗂)=Λ+, see [31, p. 156].

Since the anti-involution σ stabilizes 𝔥, we obtain an induced involution on 𝔥* which we denote again by σ. One checks directly that σ leaves Δ invariant and that

Δ𝗂={αΔ:σ(α)=-α}.

A root αΔ is called real if σ(α)=α, and Δ𝗋 is the set of real roots. Since R(α)=R(σ(α)) for all αΔ, we get

σ(Δ+Δ𝗂)=Δ+Δ𝗂.

In other words, Δ+ is a σ-order in the terminology of [5].

Before we can state the main result of this subsection, we have to review the description of parabolic subalgebras of 𝔤 in terms of the root system Δ. Recall that a root αΔ+ is called simple if it cannot be written as the sum of two positive roots. Let ΠΔ+ be the subset of simple roots. The elements of Π form a basis of 𝔥* and every positive root can be uniquely written as a linear combination of simple roots with non-negative integer coefficients.

For an arbitrary subset Γ of Π we set Γ𝗋:=ΓΔ and Γ𝗇:=Δ+Γ𝗋. Then

𝔮Γ:=𝔥αΓ𝗋𝔤ααΓ𝗇𝔤α

is a parabolic subalgebra of 𝔤. The subalgebra αΓ𝗇𝔤α is the nilradical of 𝔮Γ, while the reductive subalgebra 𝔥αΓ𝗋𝔤α is a Levi subalgebra of 𝔮Γ. Let QΓ be the analytic subgroup of G having Lie algebra 𝔮Γ. Then QΓ is a parabolic subgroup of G and every parabolic subgroup of G is conjugate to QΓ for a suitable choice of ΓΠ.

After replacing the triplet (G,G0,Q) by an equivalent one we may assume that G0eQ is compact in G/Q. Due to [32, Lemma 3.1] this means that there exist a maximally non-compact Cartan subalgebra 𝔥0 of 𝔤0 and a σ-order Δ+ of Δ=Δ(𝔤,𝔥) such that Q=QΓ for a suitable subset ΓΠ. By [32, Theorem 2.12] the real codimension of G0eQ in X is given by |Γ𝗇σ(Γ𝗇)|. Therefore the minimal G0-orbit is a hypersurface if and only if Γ𝗇σ(Γ𝗇)={α0} for some α0Δ+.

Suppose that the minimal G0-orbit in X=G/Q is a hypersurface. Then σ(α0)=α0, i.e., Δ𝗋+ cannot be empty. This implies that the Lie algebra 𝔤 must be simple, too, and that there are at least two conjugacy classes of Cartan subalgebras in 𝔤0. Furthermore, it is not hard to see that, if 𝔤0 is a split real form, then G0SL(2,) and X1.

The strategy of the classification is as follows. For every complex simple Lie algebra 𝔤 and for every real form 𝔤0 we determine explicitly the corresponding involution σ of 𝔥* and a σ-order Δ+ of Δ=Δ(𝔤,𝔥). Then we enumerate all subsets ΓΠΔ+ such that Γ𝗇σ(Γ𝗇)={α0}. This procedure will result in the list given in the beginning of Section 4.

In closing let us note that, if the compact G0-orbit in X=G/Q is a hypersurface, then X is K-spherical. Since the triplets (G,G0,Q) such that X=G/Q is K-spherical are classified in [12, Table 2], the number of possibilities of Γ that have to be checked is further reduced.

The necessary information about root systems and Satake diagrams can be found in [13, Section X.3.3 and Table VI].

A.2 The series 𝖠n

Let 𝔤=𝔰𝔩(n+1,) with n1. The root system of 𝔤 is given by

Δ={±(ek-el):1k<ln+1},

where (e1,,en+1) is the standard basis of n+1 and Δ is contained in the hypersurface {xn+1:x1++xn+1=0}. For Δ+:={ek-el:1k<ln+1} we have

Π={α1=e1-e2,α2=e2-e3,,αn=en-en+1}.

A direct calculation shows

ek-el=αk++αl-1

for all 1k<ln+1.

The non-compact real forms of 𝔤 are 𝔰𝔩(n+1,), 𝔰𝔩((n+1)/2,) if n+1 is even, and 𝔰𝔲(p,q) with 1pq and p+q=n+1. Since 𝔰𝔩(n,) is a split real form and since 𝔰𝔩(n,) contains only one Cartan subalgebra up to conjugation, see [18, Appendix C.3], we can restrict attention to 𝔤0:=𝔰𝔲(p,q). The real rank of 𝔤0 is rk𝔤0:=dim𝔞0=p and the restricted root system Λ is (𝖡𝖢)p for p<q and 𝖢p for p=q. The action of σ on Δ is given by

σ(ek)={-en+2-k,1kp or q+1kn+1,-ek,p+1kq.

This follows from the fact that 𝔥0=𝔱0𝔞0 is conjugate to the Abelian Lie algebra consisting of all matrices of the form

diag(itp+s1,itp-1+s2,,it1+sp,ir1,,irq-p,it1-sp,,itp-s1),

where tk,sl,rm such that 2(t1++tp)+r1++rq-p=0. One verifies directly that Δ+ is a σ-order.

Remark 3.

Let Γk:=Π{αk} for 1kn. Then we have

Γk𝗇={e1-ek+1,,e1-en+1,,ek-ek+1,,ek-en+1}.

The cardinality of Γk𝗇 is k(n+1-k). The corresponding homogeneous rational manifold is X=G/QΓkGrk(n+1).

Claim.

If the minimal G0-orbit in X=G/Q is a hypersurface, then Q is a maximal parabolic subgroup of G, i.e., Q=QΓk for some 1kn.

Proof.

Exclude the trivial case p=q=1 and suppose that QΓ is not maximal, i.e., that ΓΠ{αk,αl} for some 1k<ln. Then Γ𝗇 contains

Γk𝗇Γl𝗇={e1-ek+1,,e1-en+1,,ek-ek+1,,ek-en+1,
ek+1-el+1,,ek+1-en+1,,el-el+1,,el-en+1}.

If p is arbitrary and l=n, then Γ𝗇 contains e1-ej and ej-en+1 for all k+1jn. Since we have excluded p=q=1, either we have 1=p<q or 2pq. In the first case Γ𝗇σ(Γ𝗇) contains e1-ep+1 and ep+1-en+1, while in the second case Γ𝗇σ(Γ𝗇) contains e1-e2 and en-en+1. Hence, in both cases the minimal G0-orbit is not a hypersurface.

If p=1 and 1k<ln-1, then Γ𝗇σ(Γ𝗇) contains again e1-e2 and e2-en+1 so that the minimal G0-orbit is not a hypersurface.

Suppose finally that p2 and 1k<ln-1. Then Γ𝗇σ(Γ𝗇) contains e1-en+1 and e2-en, which finishes the proof of the claim. ∎

Claim.

The minimal orbit of G0=SU(1,n) is a hypersurface in X=G/QΓk for every 1kn.

Proof.

Since p=1, we have σ(ej)=-ej for all 2jn. Therefore, the only roots in Γk𝗇 which are not imaginary are e1-ek+1,,e1-en+1 and e2-en+1,,ek-en+1. But for 2jn only one of the roots e1-ej and σ(e1-ej)=ej-en+1 can belong to Γk𝗇, which proves Γk𝗇σ(Γk𝗇)={e1-en+1}. ∎

Claim.

The minimal G0-orbit in X=G/QΓ1Pn and X=G/QΓnPn is a hypersurface for any G0=SU(p,q).

Proof.

If Γ=Π{α1}, then Γ𝗇={e1-e2,,e1-en+1}. For every 2jn+1 we have σ(e1-ej)=-σ(ej)-en+1 and σ(ej)=-e1 occurs only for j=n+1, which proves Γ𝗇σ(Γ𝗇)={e1-en+1}.

The case Γ=Π{αn} can be treated similarly. ∎

In order to take care of the remaining cases we show the following:

Claim.

Suppose that p2 and 2kn-1. The minimal G0-orbit in X=G/QΓk is not a hypersurface.

Proof.

Since 2kn-1, the set Γk𝗇 contains the two roots e1-en+1 and e2-en. Moreover, due to p2, these roots are real, hence the claim follows. ∎

In summary, we have established the first and third entry in the list given in the beginning of Section 4.

A.3 The series 𝖡n

Let 𝔤=𝔰𝔬(2n+1,). The root system of 𝔤 is given by

Δ={±ek:1kn}{±ek±el:1k<ln},

where (e1,,en) is the standard basis of n. For Δ+:={ek,ek±el:1k<ln} we have

Π={α1=e1-e2,α2=e2-e3,,αn-1=en-1-en,αn=en}.

A direct calculation shows

ek=αk++αn,
ek-el=αk++αl-1,
ek+el=αk++αl-1+2(αl++αn)

for all 1k<ln.

The only non-compact real forms of 𝔤 are 𝔤0:=𝔰𝔬(p,q) with 1pq, p+q=2n+1. The real rank of 𝔤0 is rk𝔤0=p and the restricted root system Λ coincides with 𝖡p. The action of σ on Δ is given by

σ(ek)={ek,1kp,-ek,p+1kp+[q-p2]=n.

Therefore the simple roots α1,,αp-1 are real and αp+1,,αn are imaginary, while σ(αp)=ep+ep+1.

According to [12, Table 2] the only ΓΠ such that the minimal G0-orbit in X=G/QΓ might be a hypersurface are the following: if p=1, then ΓΠ is arbitrary; if p=2, then Γ=Π{αj} for 1jn; if p3, then Γ is either Π{α1} or Π{αn}.

Let us assume first p2. If Γ=Π{α1}, then Γ𝗇 contains the real roots e1 and e1±e2 so that the minimal G0-orbit cannot be a hypersurface. If 2jn and Γ=Π{αj}, then Γ𝗇 contains the real roots e1 and e2 so that the minimal G0-orbit cannot be a hypersurface.

Assume now that p=1. If Γ does not contain αj for some 1jn-1, then Γ𝗇 contains e1±en, so that the minimal G0-orbit cannot be a hypersurface. On the other hand, for Γ=Π{αn} we have Γ𝗇={e1,,en,ek+el:1k<ln}, hence Γ𝗇σ(Γ𝗇)={e1}. In this case the minimal G0-orbit is a hypersurface.

Remark 4.

Let Γ=Π{αn}. Then G0=SO(1,2n) has a compact hypersurface orbit in X=G/QΓ. We have dimX=|Γ𝗇|=n(n+1)/2. Note that XIGrn(2n+1).

A.4 The series 𝖢n

Let 𝔤=𝔰𝔭(n,) with n2. The root system of 𝔤 is given by

Δ={2ek:1kn}{±ek±el:1k<ln},

where (e1,,en) is the standard basis of n. For Δ+:={2ek,ek±el:1k<ln} we have

Π={α1=e1-e2,α2=e2-e3,,αn-1=en-1-en,αn=2en}.

A direct calculation shows that

2ek=2(αk++αn-1)+αn,
ek-el=αk++αl-1,
ek+el=αk++αl-1+2(αl++αn-1)+αn

for all 1k<ln.

The non-compact real forms of 𝔤 are 𝔰𝔭(n,) and 𝔰𝔭(p,q) with 1pq, p+q=n. Since 𝔰𝔭(n,) is a split real form, it is sufficient to consider 𝔤0:=𝔰𝔭(p,q). The real rank of 𝔤0 is p and the restricted root system coincides with (𝖡𝖢)p for p<q and 𝖢p for p=q. The action of σ on Δ is given by

σ(ek)={ek+1,if 1k2p is odd,ek-1,if 1k2p is even,-ek,2p+1kn.

According to [12, Table 2] the only ΓΠ such that the minimal G0-orbit in X=G/QΓ might be a hypersurface are the following: if p=1, then Γ=Π{αk,αl} for all 1kln (with k=l allowed); if p=2, then Γ=Π{αk} for all 1kn; if p3, then the only possibilities for Γ are Π{αk} for k=1,2,3,n or Π{α1,α2}.

Let p be arbitrary. For Γ=Π{α1} we have Γ𝗇={e1±e2,,e1±en,2e1} and thus Γ𝗇σ(Γ𝗇)={e1+e2}. Hence, G0=Sp(p,q) has a compact hypersurface orbit in X=G/QΓ2n-1. Now suppose that Γ does not contain the root αk for some k2. Then Γ𝗇 contains 2e1 and 2e2=σ(2e1), so that the minimal G0-orbit in X is not a hypersurface.

A.5 The series 𝖣n

Let 𝔤=𝔰𝔬(2n,) with n4.[1] The root system of 𝔤 is given by

Δ={±ek±el:1k<ln},

where (e1,,en) is the standard basis of n. For Δ+:={ek±el:1k<ln} we have

Π={α1=e1-e2,α2=e2-e3,,αn-1=en-1-en,αn=en-1+en}.

Remark 5.

There exists an automorphism of Π that exchanges αn-1 and αn. Consequently, there exists an outer automorphism of G=SO(2n,) that maps QΠ{αn-1} onto QΠ{αn} although these parabolic groups are not conjugate in G. In particular, the corresponding homogeneous rational manifolds are isomorphic. As hermitian symmetric spaces they are isomorphic to SO(2n)/U(n).

A direct calculation shows that

ek-el=αk++αl-1,
ek+en-1=αk++αnfor all 1kn-2,
ek+en=αk++αn-2+αnfor all 1kn-2,

and

ek+el=αk++αl-1+2(αl++αn-2)+αn-1+αn

for all 1k<ln-2.

The non-compact real forms of 𝔤 are 𝔰𝔬*(2n) and 𝔰𝔬(p,q) with 1pq, p+q=2n.

Consider first 𝔤0=𝔰𝔬*(2n). The real rank of 𝔤0 is [n/2] and the restricted root system is (𝖡𝖢)m for n=2m+1 and 𝖢m if n=2m.

We start with the case that n=2m is even. The corresponding involution of Δ is induced by

σ(ek)={ek+1,1kn is odd,ek-1,1kn is even.

One verifies directly that Δ+ is a σ-order.

For Γ=Π{α1} we have Γ𝗇={e1±e2,,e1±en} and hence

Γ𝗇σ(Γ𝗇)={e1+e2}.

Consequently, the minimal G0-orbit in X=G/QΓ is a hypersurface. If Γ does not contain α2, then Γ𝗇 contains e1-e3 and e2-e4=σ(e1-e3), i.e., the minimal G0-orbit in X=G/QΓ is not a hypersurface. If n6 and if Γ does not contain αk for 3kn, then Γ𝗇 contains the two real roots e1+e2 and e3+e4. On the other hand, for n=4 and Γ=Π{α3} we obtain Γ𝗇σ(Γ𝗇)={e1+e2}, hence the minimal orbit of SO*(8) in X=G/QΓ is a hypersurface in this case. One checks directly that in the remaining cases SO*(8) does not have a compact hypersurface orbit.

Suppose now that n=2m+15 is odd. In this case the involution of Δ is given by

σ(ek)={ek+1,1kn-1 is odd,ek-1,1kn-1 is even,-ek,k=n.

As above we see that for Γ=Π{α1} we have Γ𝗇σ(Γ𝗇)={e1+e2}, while the minimal G0-orbit in X=G/QΓ is not a hypersurface if Γ does not contain αk for 2kn.

In summary, the only cases in which the minimal orbit of G0=SO*(2n) in X=G/QΓ is a hypersurface are Γ=Π{α1} as well as n=4 and Γ=Π{α3}.

Remark 6.

The exceptional case n=4 is explained by 𝔰𝔬*(8)𝔰𝔬(6,2) which corresponds to the fact that SO(8)/U(4) is isomorphic to the 3-dimensional quadric.

In the rest of this subsection we treat the case 𝔤0=𝔰𝔬(p,q) with 1pq, p+q=2n. The real rank of 𝔤0 is p and the restricted root system is 𝖡p for p<q and 𝖣p for p=q.

Remark 7.

The Lie algebra 𝔰𝔬(n,n) is a split real form of 𝔤. The Lie algebra 𝔰𝔬(1,2n-1) contains only one conjugacy class of Cartan subalgebras, see [18, Appendix C.3].

The involution of Δ is induced by

σ(ek)={ek,1kp,-ek,p+1kp+[q-p2]=n.

According to [12] the minimal G0-orbit in X=G/QΓ may be a hypersurface only in the following cases. If p=1, then ΓΠ is arbitrary; if p=2, then Γ coincides with Π{αk} or Π{αk,αn-1} or Π{αk,αn} for any k; if p3, then the only possibilities for Γ are Π{α1} or Π{αn-1} or Π{αn}.

Let us begin with the case p3. If Γ=Π{αk} for k=1,n-1,n, then Γ𝗇 contains the real roots e1+e2 and e2+e3 so that the minimal G0-orbit in X=G/QΓ cannot be a hypersurface.

Suppose now that p=2. If Γ does not contain αk for some 1kn-2, then Γ𝗇 contains e1±en. Since σ(e1-en)=e1+en, the minimal G0-orbit is not a hypersurface in this case. If Γ=Π{αn-1}, we have Γ𝗇={e1-en,,en-1-en,ek+el(1k<ln-1)} and one verifies Γ𝗇σ(Γ𝗇)={e1+e2}. Hence, the minimal G0-orbit in X=G/QΓ is a hypersurface. For Γ=Π{αn} we have Γ𝗇={ek+el:1k<ln} and obtain again Γ𝗇σ(Γ𝗇)={e1+e2}, which leads to the same conclusion as above.

Remark 8.

For p=1 the above considerations show that G0 acts transitively on X=G/QΓ for Γ=Π{αk}, where k=n-1,n.

In summary, the only cases in which the minimal orbit of G0=SO(p,q) in X=G/QΓ is a hypersurface are p=2 and Γ=Π{αk} for k=n-1,n.

A.6 The exceptional Lie algebra 𝔤=E6

Combined with the general remarks in [5], the Satake diagrams yield explicit formulas of the involutions corresponding to the non-split non-compact real forms of the exceptional Lie algebras E6, E7, E8 and F4.

Let 𝔤=E6. Identifying 𝔥* with V={x8:x6=x7=-x8} a system of simple roots is given by

Π={α1=12(e1-e2-e3-e4-e5-e6-e7+e8,α2=e1+e2,αj=ej-1-ej-2(3j6)}.

The Lie algebra 𝔤=E6 has two non-split non-compact real forms, namely 𝐸𝐼𝐼 and 𝐸𝐼𝐼𝐼.

Suppose first that 𝔤0=𝐸𝐼𝐼. Since there is no imaginary simple root, the Satake diagram of 𝔤0 determines directly the involution σ:Δ+Δ+. More precisely, we have

σ(α1)=α6,σ(α3)=α5,σ(α2)=α2,σ(α4)=α4.

According to [12] we must only check Π{α1} and Π{α6}.

Let Γ=Π{αj} for j=1,6. In both cases Γ𝗇σ(Γ𝗇) contains the two real roots α1+α3+α4+α5+α6 and α1+α2+α3+α4+α5+α6. Hence the minimal G0-orbit in X=G/QΓ is not a hypersurface.

Suppose now that 𝔤0=𝐸𝐼𝐼𝐼. It can be seen from its Satake diagram that Π𝗂={α3,α4,α5} and that

σ(α1)=α6+C1,3α3+C1,4α4+C1,5α5,
σ(α2)=α2+C2,3α3+C2,4α4+C2,5α5,
σ(α6)=α1+C6,3α3+C6,4α4+C6,5α5.

Since σ is involutive, we obtain C6,j=C1,j for j=3,4,5. This gives

σ(α1+α3+α4+α5+α6)=α1+(2C1,3-1)α3+(2C1,4-1)α4+(2C1,5-1)α5+α6.

Comparison with the list of positive roots shows that α1+α3+α4+α5+α6 must be a real root, i.e., that C1,3=C1,4=C1,5=1. Similarly, the only possibilities for σ(α2) are α2, α2+α4, α2+α4+α5, α2+α3+α4, α2+α3+α4+α5 and α2+α3+2α4+α5. However, since we know that σ(α2)-α2 is not a root, we only have σ(α2)=α2 or σ(α2)=α2+α3+2α4+α5. In the first case we obtain σ(α2+α4)=α2-α4, which contradicts the fact that Δ+ is a σ-order. Therefore, we see that σ(α2)=α2+α3+2α4+α5.

Let now Γ=Π{αj} for 1j6. Then Γ𝗇σ(Γ𝗇) contains always the roots α1+α2+α3+α4+α5+α6 and

σ(α1+α2+α3+α4+α5+α6)=α1+α2+2α3+3α4+2α5+α6.

Consequently, the minimal G0-orbit in X=G/QΓ cannot be a hypersurface for any ΓΠ.

A.7 The exceptional Lie algebra 𝔤=E7

Let 𝔤=E7. Identifying 𝔥* with the space V={x8:x8=-x7} a system of simple roots is given by

Π={α1=12(e1-e2-e3-e4-e5-e6-e7+e8),α2=e1+e2,αj=ej-1-ej-2(3j7)}.

The Lie algebra 𝔤=E7 has two non-split non-compact real forms, namely 𝐸𝑉𝐼 and 𝐸𝑉𝐼𝐼.

Let 𝔤0=𝐸𝑉𝐼. Its Satake diagram shows Π𝗂={α2,α5,α7}. In a first step we determine the integers Ck,l such that

σ(αk)=αk+Ck,2α2+Ck,5α5+Ck,7α7

for k=1,3,4,6. One checks immediately that σ(α1)=α1 and σ(α3)=α3. For the remaining cases the only possibilities that respect σ(αk)-αkΔ are

σ(α4)=α4orσ(α4)=α2+α4+α5

and

σ(α6)=α6orσ(α6)=α5+α6+α7.

Since α4+α5,α5+α6Δ+ we obtain

σ(α4)=α2+α4+α5,
σ(α6)=α5+α6+α7.

According to [12] the only possibility for a minimal orbit of hypersurface type is Γ=Π{α7}. Since in this case Γ𝗇 contains the two real roots α2+α3+2α4+2α5+2α6+α7 and α1+α2+α3+2α4+2α5+2α6+α7, the minimal G0-orbit in X=G/QΓ cannot be a hypersurface.

Let 𝔤0=𝐸𝑉𝐼𝐼. Here we have Π𝗂={α2,α3,α4,α5} and we must determine

σ(αk)=αk+Ck,2α2+Ck,3α3+Ck,4α4+Ck,5α5

for k=1,6,7. One sees directly σ(α7)=α7. For the remaining cases the only possibilities that respect σ(αk)-αkΔ are

σ(α1)=α1orσ(α1)=α1+α2+2α3+2α4+α5

and

σ(α6)=α6orσ(α6)=α2+α3+2α4+2α5+α6.

Since α1+α3,α5+α6Δ+, we obtain

σ(α4)=α1+α2+2α3+2α4+α5,
σ(α6)=α2+α3+2α4+2α5+α6.

Let now Γ=Π{αk} for 1k7. Then Γ𝗇σ(Γ𝗇) contains always the roots α1+α2+α3+α4+α5+α6+α7 and

σ(α1+α2+α3+α4+α5+α6+α7)=α1+α2+2α3+3α4+2α5+α6+α7.

Consequently, the minimal G0-orbit in X=G/QΓ is never a hypersurface.

A.8 The exceptional Lie algebra 𝔤=E8

According to [12] no real form of G can have a compact hypersurface in any G-homogeneous rational manifold.

A.9 The exceptional Lie algebra 𝔤=F4

The rank of 𝔤=F4 is 4 and the root system is given by

Δ={ek:1k4}{±ek±el:1k<l4}{12(±e1±e2±e3±e4)}.

Choosing Δ+={ek}{ek±el}{12(e1±e2±e3±e4)}, we obtain

Π={α1=12(e1-e2-e3-e4),α2=e4,α3=e3-e4,α4=e2-e3}.

The non-compact real forms of 𝔤 are 𝐹𝐼 and 𝐹𝐼𝐼. Since 𝐹𝐼 is split, we concentrate on 𝔤0=𝐹𝐼𝐼. According to [5, p. 21] the simple roots α2, α3 and α4 are imaginary while

σ(α1)=α1+3α2+2α3+α4.

Equivalently, we have σ(e1)=e1 and σ(ek)=-ek for 2k4. One checks that Δ+ is a σ-order.

A direct calculation shows that the minimal G0-orbit in X=G/QΓ is never a hypersurface.

A.10 The exceptional Lie algebra 𝔤=G2

The only non-compact real form of 𝔤 is split.

Acknowledgements

The authors would also like to thank the referees for their helpful remarks.

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Received: 2015-10-05
Revised: 2016-10-25
Published Online: 2016-12-14
Published in Print: 2019-08-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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