Quaternionic Structures in Mathematics and Physics SISSA - Trieste, 5-9 September 1994

QUATERNIONIC KÄHLER METRICS AND NILPOTENT ORBITS

ANDREW SWANN

Abstract. The geometry of quaternionic Kähler metrics constructed from nilpotent orbits is surveyed. This includes the relationship with homogeneous twistor spaces, finite-dimensional quotient constructions and the problem of compactification.

1. Automorphisms and Homogeneity

Given a geometric structure on a manifold it is natural to ask what are the symmetries of that structure and what are the examples of such structures with a big group of symmetries. In this talk I wish to start by discussing such questions for quaternionic Kähler manifolds.

Let M be a quaternionic Kähler manifold, that is a manifold whose holonomy is contained in Sp(n) Sp(1), the subgroup of SO(4n) acting on ℝ^{4n} ≅ ℍ^{n} via (A, q) · ξ = Aξ\overline{q}. I will assume that the scalar curvature of M is non-zero, thus ruling out the case when M is locally hyperKähler, but will not necessarily assume that M is compact.

Fibred over M, is the twistor space Z [36, 5]. This is a complex-contact manifold with a real structure σ and a (possibly indefinite) Kähler metric. The fibres of Z → M are rational curves ℂP(1) ≅ S^{2} called twistor lines which are preserved by σ and have normal bundle ν ≅ 2nO(1). I will regard the complex-contact structure as a holomorphic one-form θ taking values in a holomorphic line bundle L → Z such that θ ∧ (dθ)^{n} vanishes nowhere and σ^{∗}θ = \overline{θ}. Writing D = ker θ, we have an exact sequence

0 ⟶ D ⟶ T Z ⟶^{θ} L ⟶ 0.

The data D, σ on the complex manifold Z is enough to determine the quaternionic Kähler manifold M via the inverse twistor construction [27, 34]. Thus an automorphism ϕ̌ of M preserving the quaternionic Kähler structure induces a holomorphic automorphism ϕ of Z preserving D and σ. Salamon [36] and Nitta & Takeuchi [33] showed that there is a correspondence

𝔤 ⊗ ℂ { Infinitesimal isometries of M preserving the quaternionic structure } ⊗ ℂ ⟷ { Infinitesimal holomorphic complex-contact automorphisms of Z } ≅ H^{0}(Z, L)

induced by the map Y̌ ↦ θ(Y), where Y̌ is a vector field on M with lift Y to Z. I have written 𝔤 for the Lie algebra of infinitesimal isometries of M and will use G to denote the corresponding connected, simply-connected Lie group.

This leads to the following two notions of homogeneity:

(A) M is homogeneous under G; (B) Z is homogeneous under G^{ℂ}. Case (A) has been well-studied. If G is compact, then in order for it to be the

isometry group of a homogeneous quaternionic Kähler manifold, G must be simple and M is then a compact symmetric space uniquely determined by the universal cover of G [40, 1, 2]. We will refer to these spaces M as Wolf spaces. For example: G = Sp(n + 1) gives M = ℍP(n), quaternionic projective space; G = SU(n) gives M = Gr_{2}(ℂ^{n}), the Grassmannian of complex two-planes in ℂ^{n}; G = SO(n) gives M = G̃r_{4}(ℝ^{n}), the Grassmannian of oriented four-planes in ℝ^{n}; G = G_{2} gives M = G_{2}/ SO(4). On the other hand, if G is completely solvable, then M = G and the possible groups G are now known [10, 11, 3, 13].

Case (B) does not coïncide with case (A) and is the subject of the next section. Note that the example of the non-compact dual of ℍP(n) shows that even if M is homogeneous under G, the group G^{ℂ} need not act on Z.

2. Homogeneous Twistor Spaces

Let us consider the case when M is a quaternionic Kähler manifold whose twistor space Z is homogeneous under the group G^{ℂ} of complex-contact automorphisms. Homogeneous complex-contact manifolds have been studied by Lichnerowicz [30] and what is relevant is a complexified version of the symplectification procedure discussed by Arnol’d [4]. Let π: U = L^{∗} \ 0 → Z be the total space of the dual of L with the zero section removed. Define α = π^{∗}θ and let ω = dα. Then ω is a complexsymplectic form on U and infinitesimal complex-contact automorphisms Y of Z lift to infinitesimal holomorphic automorphisms Ỹ of U preserving ω.

The map μ: U → 𝔤_{ℂ}^{∗} defined by

⟨μ(b), Ỹ ⟩ = −b(θ(Y)_{πb})

is a complex moment map for the action of G^{ℂ} on U. As Z is a twistor space, we have L|_{ℂP(1)} ≅ O(2) for each twistor line ℂP(1) and hence θ(Y) ∈ H^{0}(Z, L) always has a zero. In Lichnerowicz’ terminology, this implies that Z is a proper complex-contact manifold, which implies that the image O = μ(U) is a coadjoint orbit generated by rays and that μ is a local diffeomorphism.

If we assume that 𝔤 is semi-simple, then we may identify 𝔤^{ℂ} with 𝔤_{ℂ}^{∗} equivariantly via an Ad-invariant inner product ⟨·, ·⟩. Realising 𝔤^{ℂ} as a matrix algebra, the orbit O becomes an orbit of nilpotent matrices, that is X^{r} = 0 for some r. Moreover, via the projectivisation of μ, the twistor space Z is locally diffeomorphic to the complex projectivisation ℙO ⊂ ℙ𝔤^{ℂ} and θ is given by μ^{∗}θ_{O}, where θ_{O}[X, A] = ⟨X, A⟩ is the standard complex-contact form on ℙO induced by the Kirillov-Kostant-Souriau complex-symplectic form ω_{O}([X, A], [X, B]) = −⟨X, [A, B]⟩ on O.

The question now arises: for which nilpotent orbits O is ℙO a twistor space? Since Z is fibred over M we have the following diagram:

\begin{aligned}U & ⟶^{ℂ^{∗}} & Z \\ & [figure] & ↓_{S^{2}} \\ & & M \end{aligned}

and may regard U as a bundle over M with fibre ℍ^{∗}/{±1}. The fact that M is quaternionic Kähler implies that U admits a hyperKähler metric invariant not only under the action of G, but also under the action of SO(3) ⊂ ℍ^{∗}/{±1} [39]. Furthermore, this hyperKähler metric is positive definite if and only if M has positive scalar curvature.

By studying appropriate moduli spaces of G-instantons, Kovalev [21] extended work of Kronheimer [26, 25] to show that every adjoint orbit O of a complex semisimple Lie algebra 𝔤^{ℂ} admits a G-invariant hyperKähler metric, where G is the compact real from of 𝔤^{ℂ} (see also [6]). An examination [39] of Kronheimer’s proof [26] shows that if O is nilpotent, then O admits a free action of ℍ^{∗}/{±1} such that O/ℍ^{∗} is a quaternionic Kähler manifold of positive scalar curvature with twistor space ℙO.

Example 2.1. Let G be a compact, simple, simply-connected Lie group. Let O_{min} be the smallest non-trivial nilpotent orbit. Then O_{min}/ℍ^{∗} is the Wolf space with isometry group G (see case (A) above). For example, if G = SU(n), then O_{min} = { X ∈ 𝔰𝔩(n, ℂ) : X^{2} = 0, rank X = 1 }, the ℍ^{∗}-orbit through X is the intersection of O_{min} with the three-dimensional subspace of 𝔰𝔩(n, ℂ) spanned by \{X, \overline{X}^{t}, [X, \overline{X}^{t}] \} and the map X ↦ span_{ℂ^{n}}{Im X, (ker X)^{⊥}} identifies O/ℍ^{∗} with Gr_{2}(ℂ^{n}).

Note that the orbits O_{min} are the only nilpotent orbits for which the quaternionic Kähler manifolds O/ℍ^{∗} are compact.

3. Nilpotent Orbits for the Special Linear Group

The construction of the Kronheimer metric is somewhat indirect. For nilpotent orbits in classical complex Lie algebras there is a more straightforward approach, obtained by interpreting constructions of Kraft & Procesi [24, 23, 22]as hyperKähler quotients.

Here, I will outline the construction for orbits in 𝔰𝔩(n, ℂ). For the other classical algebras I will only give an example in the next section. Details may be found in [20].

Let X ∈ O ⊂ 𝔰𝔩(n, ℂ) be nilpotent. Then there is a natural number k such that X^{k} = 0 and X^{k−1} ≠ 0. I define unitary vector spaces V_{0}, …, V_{k} by V_{k} = ℂ^{n} and V_{k−i} = Im X^{i}, for i > 1 and let n_{j} := dim_{ℂ} V_{j}. Consider M given by

M = ⨁_{i=0}^{k−1} Hom(V_{i}, V_{i+1}) ⊕ Hom(V_{i+1}, V_{i}).

If we define j(α, β) = (−β^{∗}, α^{∗}), where^{∗} denotes the unitary adjoint, then M becomes isomorphic to the flat hyperKähler manifold ℍ^{N}, where N = n_{0}n_{1} + ⋯ + n_{k−1}n_{k}. It is convenient to think of a point of M as a diagram

{0} = V_{0} ⇄_{β_{0}}^{α_{0}} V_{1} ⇄_{β_{1}}^{α_{1}} V_{2} ⇄_{β_{2}}^{α_{2}} ⋯ ⇄_{β_{k−1}}^{α_{k−1}} V_{k} = ℂ^{n}.

The action of G := U(n_{1}) × ⋯ × U(n_{k−1}) on V_{1} ⊕ ⋯ ⊕ V_{k−1} induces an action of G on M preserving the hyperKähler structure and this action has hyperKähler moment map μ = iμ_{r} + 2kμ_{c}, where

μ_{c} = (α_{i}β_{i} − β_{i+1}α_{i+1}),
μ_{r} = (α_{i}α_{i}^{∗} − β_{i}^{∗}β_{i} + β_{i+1}β_{i+1}^{∗} − α_{i+1}^{∗}α_{i+1}).

The hyperKähler quotient construction [17] now implies that a dense open subset of μ^{−1}(0)/G is a hyperKähler manifold. In fact, in this case one can show that μ^{−1}(0)/G is a stratified hyperKähler manifold.

The map M → 𝔤𝔩(n, ℂ) given by (α, β) ↦ α_{k−1}β_{k−1} is G-invariant and induces a homeomorphism μ^{−1}(0)/G → \overline{O}, where \overline{O} is the closure of the nilpotent orbit containing X. Note that all the SL(n, ℂ)-orbits in \overline{O} are nilpotent: it is these nilpotent orbits that provide the stratification of μ^{−1}(0)/G. (Incidentally, general considerations [12] show that a hyperKähler quotient μ^{−1}(0)/G is always a union of hyperKähler manifolds, but this is weaker than saying that μ^{−1}(0)/G is stratified, cf. [15, 37].)

The above construction was used by Nakajima [32] with a non-degenerate value of the moment map to show that the cotangent bundle of the flag manifold associated to V_{1} ⩽ V_{2} ⩽ ⋯ ⩽ V_{k} = ℂ^{n} admits complete, U(n)-invariant hyperKähler metrics.

The use of the zero value of the moment map above implies that the above construction commutes with the action of ℍ^{∗} by multiplication on the right and so gives that \overline{O}/ℍ^{∗} is a stratified quaternionic Kähler manifold obtained as a quaternionic Kähler quotient of ℍP(N − 1), cf. [39].

Example 3.1. Let O = O_{min} ⊂ 𝔰𝔩(n, ℂ), so points of M ≅ ℍ^{n} are diagrams

ℂ ⇄ ℂ^{n}

and G = U(1). The above construction now describes Gr_{2}(ℂ^{n}) as a quaternionic Kähler quotient of ℍP(n − 1) by U(1) and coïncides with the construction given by Galicki [14].

4. Construction of Other Classical Nilpotent Orbits

For the cases of nilpotent orbits in 𝔬(n, ℂ) and 𝔰𝔭(n, ℂ), let us look at the following example.

Let O_{3,2^{2}}^{O(7)} be the orbit of elements in 𝔬(7, ℂ) ≅ Λ^{2}ℂ^{7} which are nilpotent and have Jordan type (3, 2^{2}). Consider the diagram

ℂ ⇄_{β_{1}}^{α_{1}} ℂ^{4} ⇄_{β_{2}}^{α_{2}} ℂ^{7}.  (4.1)

Now ℂ^{7} has a non-degenerate, symmetric, bilinear form B_{3} = ⟨·, ·⟩ such that O(7, ℂ) consists of the linear automorphisms of ℂ^{7} preserving ⟨·, ·⟩. For X ∈ O_{3,2^{2}}^{O(7)}, we have Im X = ℂ^{4} and that ⟨·, ·⟩ induces a complex symplectic form B_{2} = ω on Im X [38]. Similarly ω induces a symmetric bilinear form B_{1} = ⟨·, ·⟩ on Im X^{2} = ℂ. We now only consider diagrams (4.1) in which β_{i} = α_{i}^{†}, the adjoint of α_{i} with respect to the bilinear forms B_{j}, and restrict G to consist of those elements of U(1) × U(4) preserving B_{1} ⊕ B_{2}, thus G = O(1) × Sp(2). The hyperKähler quotient construction now describes O_{3,2^{2}}^{O(7)} as a hyperKähler quotient of ℍ^{16} by O(1) × Sp(2).

Note that the group O(1) is isomorphic to ℤ_{2} so the hyperKähler quotient of ℍ^{16} by just Sp(2) has the same dimension as O_{3,2^{2}}^{O(7)}. This quotient corresponds to the diagram

ℂ^{4} ⇄ ℂ^{8},

with Sp(2) acting on ℂ^{4} ≅ ℍ^{2} and the hyperKähler quotient is the closure \overline{O_{2^{4}}^{O(8)}} of the O(8)-orbit of Jordan type 2^{4}. (One should regard the left-hand factor ℂ in (4.1) has having been moved to the right-hand side and added to ℂ^{7} to get ℂ^{8}.) We thus have a map \overline{O_{2^{4}}^{O(8)}} → \overline{O_{3,2^{2}}^{O(7)}} which is generically two-to-one. A closer examination leads to the following diagram:

\begin{aligned}O_{2^{4}}^{O(8)} & ⟶^{2:1} & O_{3,2^{2}}^{O(7)} \\ | & & | \\ O_{2^{2},1^{4}}^{O(8)} & ⟶ & O_{3,1^{4}}^{O(7)} \\ & [figure] & | \\ & & O_{2^{1},1^{3}}^{O(7)} \end{aligned}  (4.2)

Here a vertical line indicates that the lower orbit is contained in the closure of the upper orbit; the union of the left-hand column is \overline{O_{2^{4}}^{O(8)}} \ \{0\} and that of the right-

hand column \overline{O_{3,2^{2}}^{O(7)}} \ \{0\}. Note that the hyperKähler quotient construction shows that the uppermost map O_{2^{4}}^{O(8)} → O_{3,2^{2}}^{O(7)} is a local isometry. This map is also O(7, ℂ) equivariant, although only the compact group O(7) acts isometrically.

Springer & Steinberg [38] show that the nilpotent orbits for SL(n, ℂ), Sp(n, ℂ) and O(n, ℂ) are determined by their Jordan normal forms. This is the reason I have been discussing O(n, ℂ) instead of SO(n, ℂ). In the latter case, the Jordan normal form determines the nilpotent orbit unless it is very even, that is all the Jordan blocks are even in size and multiplicity. For each very even Jordan type there are precisely two SO(n, ℂ)-nilpotent orbits. For example, O_{2^{4}}^{O(8)} splits as O_{2^{4}+}^{SO(8)} ∪ O_{2^{4}−}^{SO(8)}. Under this splitting the upper map in (4.2) becomes two isometries O_{2^{4}±}^{SO(8)} ⟶^{≅} O_{3,2^{2}}^{O(7)} = O_{3,2^{2}}^{SO(7)}. More generally, we have:

Proposition 4.1. For each n ⩾ 1, the three nilpotent orbits O_{2^{2n}+}^{SO(4n)}, O_{2^{2n}−}^{SO(4n)} and O_{3,2^{2n−2}}^{SO(4n−1)} are diffeomorphic and carry isometric hyperKähler structures.

When n = 1, the above follows easily from the splitting 𝔰𝔬(4) = 𝔰𝔬(3) ⊕ 𝔰𝔬(3).

Remark 4.2. Levasseur & Smith [29] found a similar example of equivalence, this time associated to G_{2}. Let π: O_{2^{2},1^{3}}^{SO(7)} → 𝔤_{2}^{ℂ} be the map induced by projection to the first factor in the splitting 𝔰𝔬(7, ℂ) ≅ 𝔤_{2}^{ℂ} ⊕ V_{7}^{ℂ}, where V_{7} is the sevendimensional irreducible representation of G_{2}. (The representation V_{7} may be realised as the space Im 𝕆 of pure imaginary Cayley numbers.) Then π(O_{2^{2},1^{3}}^{SO(7)}) is the union of O_{16}^{G_{2}} ∪ O_{12}^{G_{2}} of the two smallest nilpotent orbits in 𝔤_{2}^{ℂ} (the lower indices indicating the real dimensions of these orbits) and π is a birational finite morphism. Note that here we do not have an equivalence of orbits, but rather an equivalence of their closures.

5. Compactifications via Minimal Orbits

Let us return the lower part of (4.2). The orbits O_{2,1^{4}}^{O(8)} and O_{2^{2},1^{3}}^{O(7)} are the smallest non-trivial nilpotent orbits in 𝔬(8, ℂ) and 𝔬(7, ℂ) respectively, and the associated quaternionic Kähler manifolds are G̃r_{4}(ℝ^{8}) and G̃r_{4}(ℝ^{7}). The lower part of diagram (4.2) shows that G̃r_{4}(ℝ^{8}) is a compactification of the double cover of the quaternionic Kähler manifold M = O_{3,1^{4}}^{O(7)}, obtained by adjoining a copy of G̃r_{4}(ℝ^{7}).

In [19] a similar example was given with an exceptional group. The group G_{2} is the group of linear automorphisms of the Cayley numbers 𝕆 which preserve Cayley multiplication. The Wolf space G_{2}/ SO(4) coïncides with the space of quaternionic subalgebras of 𝕆. The group SU(3) may be realised as the subgroup of G_{2} consisting of those elements fixing a given pure imaginary unit Cayley number e. A quaternionic subalgebra ℍ of 𝕆 containing 1 and e is determined by a pair of points {x, ex} orthogonal to 1 and e. However, left multiplication by e defines a complex structure such that 𝕆 = ⟨1⟩ ⊕ ⟨e⟩ ⊕ ℂ^{3}. As {x, ex} spans a complex line in ℂ^{3}, the set of quaternionic subalgebras containing 1 and e defines a subspace of G_{2}/ SO(4) isomorphic to ℂP(2). Now SU(3) has centre ℤ_{3} which acts on the complement of ℂP(2) freely and isometrically. The quotient ((G_{2}/ SO(4)) \ ℂP(2))/ℤ_{3} is isometric to the quaternionic Kähler manifold obtained from the largest nilpotent orbit in 𝔰𝔩(3, ℂ).

One application of such examples is to the study of harmonic maps in to quaternionic Kähler manifolds. We will not discuss this here, but examples may be found in [31, 18, 19].

The above examples are special cases of the following problem studied on the level of homogeneous complex manifolds by Brylinski & Kostant [8, 9]; namely, when does an open set in a nilpotent orbit O′ ⊂ 𝔤′^{ℂ} cover a nilpotent orbit O ⊂ 𝔤^{ℂ} G^{ℂ}- equivariantly. They prove that if (O, 𝔤^{ℂ}, O′, 𝔤′^{ℂ}) is a quadruple of such Lie algebras and orbits, then so is (O′′, 𝔤^{ℂ}, O_{min}, 𝔤′^{ℂ}), and then go on to find all such quadruples of the latter type, showing in particular how the Lie algebra 𝔤′^{ℂ} is determined by the ring of regular functions on O′′. In [20] it is shown that if 𝔤^{ℂ} and 𝔤′^{ℂ} are classical then the Brylinski-Kostant covering may be realised quaternionically. Note that this procedure is very different from the Springer resolution which gives a desingularisation of certain nilpotent orbits via cotangent bundles of flag manifolds (see for example [7, 32]).

The known compact examples of quaternionic Kähler manifolds with positive scalar curvature are precisely the Wolf spaces, that is precisely the quaternionic Kähler metrics constructed from the smallest non-trivial nilpotent orbits, Example 2.1. Brylinski & Kostant’s result may be used to show that most of the other quaternionic Kähler metrics constructed above from nilpotent orbits are not locally isometric to Wolf spaces. It is therefore intriguing to ask whether these other metrics may be compactified. It is particularly interesting to note that the smallest example not yet known to have a compactification, the quaternionic Kähler manifold associated to the largest nilpotent orbit in 𝔰𝔭(2, ℂ), is twelve dimensional—precisely the lowest dimension in which the classification of compact quaternionic Kähler manifolds of positive scalar curvature is not yet known [16, 35, 28].

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Department of Mathematics and Computer Science, Odense University, Campusvej 55, DK-5230 Odense M, Denmark

Current address: School of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, England

E-mail address: A.F.Swann@maths.bath.ac.uk