An application of spherical geometry to hyperkähler slices
aa r X i v : . [ m a t h . S G ] F e b AN APPLICATION OF SPHERICAL GEOMETRY TOHYPERK ¨AHLER SLICES
PETER CROOKS AND MAARTEN VAN PRUIJSSEN
Abstract.
This work is concerned with Bielawski’s hyperk¨ahler slices in the cotangent bundles ofhomogeneous affine varieties. One can associate such a slice to the data of a complex semisimple Liegroup G , a reductive subgroup H ⊆ G , and a Slodowy slice S ⊆ g := Lie( G ), defining it to be thehyperk¨ahler quotient of T ∗ ( G/H ) × ( G × S ) by a maximal compact subgroup of G . This hyperk¨ahlerslice is empty in some of the most elementary cases (e.g. when S is regular and ( G, H ) = (SL n +1 , GL n ), n ≥ S = S reg is a regular Slodowy slice, proving that non-emptiness is equivalent to the so-called a -regularity of ( G, H ). This a -regularity condition is formulated in several equivalent ways, one beinga concrete condition on the rank and complexity of G/H . We also provide a classification of the a -regular pairs ( G, H ) in which H is a reductive spherical subgroup. Our arguments make essentialuse of Knop’s results on moment map images and Losev’s algorithm for computing Cartan spaces. Contents
1. Introduction 21.1. Context 21.2. Description of results 21.3. Organization 3Acknowledgements 32. Preliminaries 32.1. Symplectic varieties and quotients 32.2. Hyperk¨ahler manifolds 43. The hyperk¨ahler geometry of T ∗ ( G/H ) 53.1. The cotangent bundle of G T ∗ G T ∗ ( G/H ) 74. The hyperk¨ahler slice construction 84.1. The slice as a symplectic variety 84.2. Bielawski’s construction 104.3. The regular Slodowy slice 115. The spherical geometry of
G/H a -regularity 125.3. The Cartan space of a homogeneous affine variety 145.4. Preliminaries for the classifications 155.5. The classifications 17References 24 Mathematics Subject Classification.
Key words and phrases. hyperk¨ahler quotient, Slodowy slice, spherical geometry. Introduction
Context.
A smooth manifold is called hyperk¨ahler if it comes equipped with three K¨ahlerstructures that determine the same Riemannian metric, and whose underlying complex structuressatisfy certain quaternionic identities. Such manifolds are known to be holomorphic symplecticand Calabi–Yau, and they are ubiquitous in modern algebraic and symplectic geometry. Prominentexamples include the cotangent bundles [22] and (co-)adjoint orbits [6,19,23,24] of complex semisimpleLie groups, moduli spaces of Higgs bundles over compact Riemann surfaces [14], and Nakajima quivervarieties [29, 30]. Many examples arise via the hyperk¨ahler quotient construction [15], an analogue ofsymplectic reduction for a hyperk¨ahler manifold endowed with a structure-preserving Lie group actionand a hyperk¨ahler moment map. However, one always has the preliminary problem of determiningwhether the given hyperk¨ahler quotient is non-empty.While the above-described emptiness problem is likely intractable in the generality described above,one might hope to solve it for particular classes of hyperk¨ahler quotients. It is in this context thatone might consider Bielawski’s hyperk¨ahler slices [4, 5], which require fixing a compact, connected,semisimple Lie group K with complexification G := K C . Each sl -triple τ = ( ξ, h, η ) in g := Lie( G )determines a Slodowy slice S τ := ξ + ker(ad η ) ⊆ g , and hence also an affine variety G × S τ . Thisvariety is a hyperk¨ahler manifold carrying a tri-Hamiltonian action of K , and its symplectic geometryis reasonably well-studied (see [1,4,8,9]). Now suppose that K acts in a tri-Hamiltonian fashion on ahyperk¨ahler manifold M , and that this action extends to a holomorphic, Hamiltonian G -action withrespect to the holomorphic symplectic structure on M . The hyperk¨ahler slice for M and τ is thendefined to be ( M × ( G × S τ )) ///K , the hyperk¨ahler quotient of M × ( G × S τ ) by K . Several well-knownhyperk¨ahler manifolds are realizable as hyperk¨ahler slices, as discussed in the introduction of [5].In light of the preceding discussion, one might consider the following special case of the emptinessproblem: classify those pairs ( M, τ ) for which the hyperk¨ahler slice ( M × ( G × S τ )) ///K is non-empty.An initial objection is that no particular assumptions have been made about M and τ , so that thisproblem likely remains too general to be tractable. We thus note that the best studied Slodowy slicesare those associated to regular sl -triples τ (see [18]), i.e. those τ = ( ξ, h, η ) for which ξ is a regularelement of g . At the same time, some of the best understood hyperk¨ahler manifolds take the formof T ∗ ( G/H ) for H ⊆ G a closed, reductive subgroup (see [10]). We therefore study the emptinessproblem for hyperk¨ahler slices when τ is regular and M = T ∗ ( G/H ).Having decided to study hyperk¨ahler slices in T ∗ ( G/H ), we are naturally led to examine theHamiltonian geometry of T ∗ ( G/H ). The works of Knop [16, 17] encode this Hamiltonian geometryin the spherical geometry of
G/H , by which we mean the B -orbit structure of G/H for a Borelsubgroup B ⊆ G . Fix such a subgroup B ⊆ G and a maximal torus T ⊆ B having Lie algebra t ⊆ g . Knop uses the Cartan space a ∗ G/H ⊆ t ∗ to describe the (closure of the) moment map image of T ∗ ( G/H ). This is complemented by Losev’s work [25], which gives an algorithm for calculating theCartan space of any given affine homogeneous G -variety [25]. It is thus reasonable to imagine thatspherical-geometric ideas are relevant to our specific emptiness problem.1.2. Description of results.
Let all notation be as set in the previous subsection, and write S reg for the Slodowy slice determined by a regular sl -triple τ in g . Use the Killing form to identify g ∗ with g , and let µ : T ∗ ( G/H ) → g be the moment map of the Hamiltonian G -action on T ∗ ( G/H ).We note the existence of a non-negative, K -invariant potential function for the first K¨ahler triple on T ∗ ( G/H ) (Proposition 4), which by Bielawski’s results [4] implies that ( T ∗ ( G/H ) × ( G × S reg )) ///K and µ − ( S reg ) are canonically isomorphic as holomorphic symplectic manifolds. This isomorphismis subsequently used to prove that ( T ∗ ( G/H ) × ( G × S reg )) ///K = ∅ if and only if h ⊥ contains aregular element of g (Proposition 11), where h ⊥ ⊆ g denotes the annihilator of h := Lie( H ) under theKilling form. The emptiness problem for ( T ∗ ( G/H ) × ( G × S reg )) ///K thus reduces to classifying the YPERK ¨AHLER SLICES AND SPHERICAL GEOMETRY 3 pairs (
G, H ) for which h ⊥ contains a regular element. This is the stage at which spherical geometrybecomes relevant, as we explain below.Inside of G , fix a maximal torus T and a Borel subgroup B satisfying T ⊆ B . These choices allowus to form the Cartan space of G/H , denoted a G/H ⊆ t := Lie( T ). We refer to the pair ( G, H ) asbeing a -regular if a G/H contains a regular element of g , and we use Knop’s description of the momentmap image µ ( T ∗ ( G/H )) to prove the following equivalences (see Proposition 15, Corollary 17, andCorollary 19): (
G, H ) is a -regular ⇐⇒ h ⊥ contains a regular element ⇐⇒ Z G ( a G/H ) = T ⇐⇒ the identity component of H ∗ is abelian ⇐⇒ c G ( G/H ) + rk G ( G/H ) + dim H = dim B, (1)where Z G ( a G/H ) is the subgroup consisting of all elements in G that fix a G/H pointwise, H ∗ is thegeneric stabilizer for the H -representation h ⊥ (see 5.2), c G ( G/H ) is the complexity of
G/H , andrk G ( G/H ) is the rank of
G/H . The first equivalence further reduces our emptiness problem to one ofclassifying the a -regular pairs ( G, H ), thereby connecting our work to Losev’s results [25]. We thenclassify all such pairs (
G, H ) (i.e. we solve the emptiness problem for ( T ∗ ( G/H ) × ( G × S reg )) ///K )in each of the following three cases: • G is semisimple and H is a Levi subgroup of G (5.5.1); • G is semisimple and H is a symmetric subgroup of G (5.5.2); • G is semisimple and H is a reductive, spherical, non-symmetric subgroup of G (5.5.3).In each case, we reduce to the study of strictly indecomposable (see 5.3) pairs ( G, H ). It is in thelast two cases that we obtain the most explicit results, and where we provide tables of all a -regularpairs ( G, H ) that are strictly indecomposable.1.3.
Organization.
Section 2 establishes some of our conventions regarding symplectic and hy-perk¨ahler geometry. Section 3 then uses [10], [22], and [27] to develop the hyperk¨ahler-geometricfeatures of T ∗ ( G/H ) needed for the subsequent discussion of hyperk¨ahler slices. This leads to Sec-tion 4, which reviews Bielawski’s hyperk¨ahler slice construction and reduces the non-emptiness of( T ∗ ( G/H ) × ( G × S reg )) ///K to the condition that h ⊥ contain a regular element. Section 5 then formsthe spherical-geometric part of our paper, where we prove the equivalences (1) and subsequentlyobtain our classification results. Acknowledgements.
The central themes of this paper were developed at the Hausdorff ResearchInstitute for Mathematics (HIM), while both authors took part in the HIM–sponsored program
Sym-plectic geometry and representation theory . We gratefully acknowledge the HIM for its hospitalityand stimulating atmosphere. We also wish to recognize Steven Rayan and Markus R¨oser for enlight-ening conversations. The first author is supported by the Natural Sciences and Engineering ResearchCouncil of Canada [516638–2018]. 2.
Preliminaries
Symplectic varieties and quotients.
Let (
X, ω ) be a symplectic variety, which for us shallalways mean that X is a smooth affine algebraic variety over C equipped with an algebraic symplecticform ω ∈ Ω ( X ). Suppose that X is acted upon algebraically by a connected complex reductivealgebraic group G having Lie algebra g . We recall that this action is called Hamiltonian if it preserves ω and admits a moment map, i.e. a G -equivariant variety morphism µ : X → g ∗ satisfying thefollowing condition: d ( µ z ) = ι ˜ z ω PETER CROOKS AND MAARTEN VAN PRUIJSSEN for all z ∈ g , where µ z : X → C is defined by µ z ( x ) := ( µ ( x ))( z ), x ∈ X , and ˜ z is the fundamentalvector field on X associated to z . If the G -action is also free, then(2) X//G := µ − (0) /G := Spec max ( C [ µ − (0)] G )is a smooth affine variety whose points are precisely the G -orbits in µ − (0). The quotient variety X//G then carries a symplectic form ω that is characterized by the condition π ∗ ( ω ) = j ∗ ( ω ), where π : µ − (0) → X//G is the quotient map and j : µ − (0) → X is the inclusion. The symplectic variety( X//G, ω ) is called the symplectic quotient of X by G .2.2. Hyperk¨ahler manifolds.
Recall that a smooth manifold M is called hyperk¨ahler if it comesequipped with three (integrable) complex structures I , I , and I , three (real) symplectic forms ω , ω , and ω , and a single Riemannian metric b , subject the following conditions: • ( I ℓ , ω ℓ , b ) is a K¨ahler triple for each ℓ = 1 , ,
3, i.e. ω ℓ ( · , · ) = b ( I ℓ ( · ) , · ); • I , I , and I satisfy the quaternionic identities I I = I = − I I , I I = − I = − I I , I I = I = − I I .One may construct new examples from existing ones via the hyperk¨ahler quotient construction, whichwe now recall. Let K be a compact connected Lie group acting freely on a hyperk¨ahler manifold M , and let k be the Lie algebra of K . Assume that the K -action is tri-Hamiltonian , meaning that K preserves each K¨ahler triple ( I ℓ , ω ℓ , b ) and acts in a Hamiltonian fashion with respect to eachsymplectic form ω ℓ . One thus has a hyperk¨ahler moment map , i.e. a map µ HK = ( µ , µ , µ ) : M → k ∗ ⊕ k ∗ ⊕ k ∗ with the property that µ ℓ : M → k ∗ is a moment map for the K -action with respect to ω ℓ , ℓ = 1 , ,
3. The smooth manifold
M ///K := µ − (0) /K = ( µ − (0) ∩ µ − (0) ∩ µ − (0)) /K is then canonically hyperk¨ahler (see [15, Theorem 3.2]), and it is called the hyperk¨ahler quotient of M by K . We shall let ( I ℓ , ω ℓ , b ), ℓ = 1 , ,
3, denote the three K¨ahler triples that constitute thehyperk¨ahler structure on
M ///K . It will be advantageous to note that(3) π ∗ ( ω ℓ ) = j ∗ ( ω ℓ ) , ℓ = 1 , , , where π : µ − (0) → M ///K is the quotient map and j : µ − (0) → M is the inclusion.Let M be a hyperk¨ahler manifold and consider the complex symplectic 2-form ω C := ω + iω .One can verify that ω C is holomorphic with respect to I , and we will refer to ( M, I , ω C ) as the underlying holomorphic symplectic manifold . This leads to the following definition, which will applyto many situations of interest in our paper. Definition 1.
Let K be a compact connected Lie group with complexification G := K C . We definea ( G, K ) -hyperk¨ahler variety is a to be a hyperk¨ahler manifold M satisfying the following conditions:(i) the underlying holomorphic symplectic manifold is a symplectic variety (as defined in 2.1),and this variety is equipped with a Hamiltonian action of G ;(ii) the G -action restricts to a tri-Hamiltonian action of K on M .Consider the hyperk¨ahler moment map µ HK = ( µ , µ , µ ) : M → k ∗ ⊕ k ∗ ⊕ k ∗ on a ( G, K )-hyperk¨ahler variety M . Define the complex moment map by µ C := µ + iµ : M → k ∗ ⊗ R C = g ∗ , which turns out to be the moment map for the Hamiltonian G -action on M . Now assume that this G -action is free. The inclusion µ − (0) ⊆ µ − C (0) then induces a map(4) ϕ : M ///K → M //G, where we recall that
M //G is defined via (2). This map defines a diffeomorphism from
M ///K to itsimage, the open subset ( G · µ − (0)) /G of µ − C (0) /G = M //G . Furthermore, ϕ is an embedding of YPERK ¨AHLER SLICES AND SPHERICAL GEOMETRY 5 holomorphic symplectic manifolds with respect to the underlying holomorphic symplectic structureon
M ///K . 3.
The hyperk¨ahler geometry of T ∗ ( G/H )It will be convenient to standardize some of the Lie-theoretic notation used in this paper. Let K be a compact connected semisimple Lie group, and fix a closed subgroup L ⊆ K . We will alsolet G := K C and H := L C denote the complexifications of K and L , respectively, noting that H is a closed reductive subgroup of G . Let k , l , g , and h be the Lie algebras of K , L , G , and H ,respectively, so that g = k ⊗ R C and h = l ⊗ R C . Each of these Lie algebras comes equipped withthe adjoint representation of the corresponding group, e.g. Ad : G → GL( g ), g Ad g . The symbol“Ad” will be used for all of the aforementioned adjoint representations, as context will always clarifyany ambiguities that this abuse of notation may cause.Let h· , ·i : g ⊗ C g → C denote the Killing form on g , which is G -invariant and non-degenerate. Itfollows that(5) g → g ∗ , x x ∨ := h x, ·i , x ∈ g defines an isomorphism between the adjoint and coadjoint representations of G . With this in mind,we will sometimes take the moment map for a Hamiltonian G -action to be g -valued.3.1. The cotangent bundle of G . Note that left and right multiplication give the commutingactions g · h := gh, g, h ∈ G (6a) g · h := hg − , g, h ∈ G (6b)of G on itself, and that these lift to commuting Hamiltonian actions of G on T ∗ G . To be moreexplicit about this point, we shall use the left trivialization of T ∗ G and the Killing form to identify T ∗ G with G × g . The lifts of (6a) and (6b) then become g · ( h, x ) = ( gh, x ) , g ∈ G , ( h, x ) ∈ G × g , (7a) g · ( h, x ) = ( hg − , Ad g ( x )) , g ∈ G , ( h, x ) ∈ G × g , (7b)respectively, while the induced symplectic form on G × g is defined on each tangent space T ( g,x ) ( G × g ) = T g G ⊕ g as follows (see [26, Section 5, Equation (14L)]):(8) (Ω L ) ( g,x ) (cid:18) ( d e L g ( y ) , z ) , ( d e L g ( y ) , z ) (cid:19) = h y , z i − h y , z i + h x, [ y , y ] i , y , y , z , z ∈ g , where L g : G → G denotes left multiplication by g and d e L g : g → T g G is the differential of L g atthe identity e ∈ G (see . One can then verify that φ L : G × g → g , ( g, x ) Ad g ( x ) , ( g, x ) ∈ G × g , (9a) φ R : G × g → g , ( g, x )
7→ − x, ( g, x ) ∈ G × g (9b)are moment maps for (7a) and (7b), respectively.3.2. Kronheimer’s hyperk¨ahler structure on T ∗ G . Let H denote the quaternions, to be iden-tified as a vector space with R via the usual basis { , i, j, k } . Now consider the real vector space C ∞ ([0 , , k ) of all smooth maps [0 , → k . A choice of K -invariant inner product h· , ·i k on k makes M := C ∞ ([0 , , k ) ⊗ R H = C ∞ ([0 , , k ) ⊕ into a Banach space with an infinite-dimensional hy-perk¨ahler manifold structure. This space carries the following hyperk¨ahler structure-preserving ac-tion of G := C ∞ ([0 , , K ), the gauge group of smooth maps [0 , → K with pointwise multiplication PETER CROOKS AND MAARTEN VAN PRUIJSSEN as the group operation:(10) γ · ( T , T , T , T ) := (Ad γ ( T ) − θ R ( ˙ γ ) , Ad γ ( T ) , Ad γ ( T ) , Ad γ ( T )) , γ ∈ G , ( T , T , T , T ) ∈ M , where θ R ∈ Ω ( K ; k ) is the right-invariant Mauer–Cartan form on K . The subgroup G := { γ ∈ G : γ (0) = e = γ (1) } ⊆ G then acts freely on M with a hyperk¨ahler moment map that can be written in the form Φ : M → C ∞ ([0 , , k ) ⊕ . It turns out that Φ − (0) consists of the solutions to N¨ahm’s equations (as definedin [10, Proposition 1], for example), and that Kronheimer constructed an explicit diffeomorphism(11) G × g ∼ = M /// G = Φ − (0) / G (cf. [22, Proposition 1]). The smooth manifold G × g thereby inherits a hyperk¨ahler structure ( I ℓ , ω ℓ , b ), ℓ = 1 , ,
3. We note that ω + iω equals the form Ω L from (8), while I is the usual complex structureon G × g (see [22, Section 2]).Kronheimer’s diffeomorphism 11 has some important equivariance properties that we now discuss.Note that G is the kernel of G → K × K, γ ( γ (0) , γ (1)) , γ ∈ G , so that we may identify G / G and K × K as Lie groups. The G -action on M induces a residual actionof G / G = K × K on M /// G , and this residual action is known to be tri-Hamiltonian (see [10, Lemma2]). Under (11), the action of K = { e } × K ⊆ K × K on M /// G corresponds to the K -action (7a)on G × g . The diffeomorphism also intertwines the action of K = K × { e } ⊆ K × K on M /// G withthe K -action 7b on G × g .The group SO ( R ) also has a natural manifestation in our setup. Given a point ( T , T , T , T ) ∈ M = C ∞ ([0 , , k ) ⊕ and a matrix A = ( a pq ) ∈ SO ( R ), let us set T ′ p := X q =1 a pq T q , p = 1 , , A · ( T , T , T , T ) := ( T , T ′ , T ′ , T ′ ) . This action of SO ( R ) on M descends to an isometric action on the hyperk¨ahler quotient M /// G . Onecan use (11) to interpret this as an isometric action of SO ( R ) on the hyperk¨ahler manifold G × g ,and it is not difficult to check that this action commutes with the K -actions (7a) and (7b). It isimportant to note that SO ( R ) does not preserve all of the hyperk¨ahler structure on G × g , in contrastto the K -actions. However, one can find a circle subgroup of SO ( R ) that preserves the K¨ahler triple( I , ω , b ) on G × g . A more explicit statement is that one can find an element θ ∈ so ( R ) whosefundamental vector field ˜ θ on G × g satisfies the following properties: L ˜ θ ω = ω , L ˜ θ ω = − ω , and ˜ θ generates a circle action on G × g that preserves ( I , ω , b ). This circle subgroup acts by rotations onspan R { ω , ω } , and the following is (the θ -component of) a moment map for its Hamiltonian actionon ( G × g , ω ):(12) ρ : G × g → R , [( T , T , T , T )] Z (cid:18) h T , T i k + h T , T i k (cid:19) dt, where [( T , T , T , T )] denotes the point in the Φ − (0) / G ∼ = G × g represented by ( T , T , T , T ) ∈ Φ − (0) (see [10, Section 4]). This leads to the following lemma. Lemma 2.
The function ρ is invariant under each of the K -actions (7a) and (7b) on G × g .Proof. Since h· , ·i k is a K -invariant inner product, the function M → R , ( T , T , T , T ) Z (cid:18) h T , T i k + h T , T i k (cid:19) dt YPERK ¨AHLER SLICES AND SPHERICAL GEOMETRY 7 is invariant under the action (10) of G . This function therefore descends to a G / G -invariant functionon the hyperk¨ahler quotient M /// G . The descended function is exactly ρ once we identify M /// G with G × g via 11. Now recall that the G / G -action on M /// G corresponds to a ( K × K )-action on G × g , meaning that ρ is a ( K × K )-invariant function on G × g . It just remains to recall that the K -action (7a) (resp. (7b)) is the action of K = { e } × K ⊆ K × K (resp. K = K × { e } ⊆ K × K ). (cid:3) The hyperk¨ahler structure on T ∗ ( G/H ) . Let G act on G/H via left multiplication, andconsider the canonical lift to a Hamiltonian action of G on T ∗ ( G/H ). Note also that ( g / h ) ∗ is arepresentation of H , and let G × H ( g / h ) ∗ denote the quotient of G × ( g / h ) ∗ by the following actionof H : h · ( g, φ ) := ( gh − , h · φ ) h ∈ G, ( g, φ ) ∈ G × ( g / h ) ∗ . We then have a canonical G -equivariant isomorphism T ∗ ( G/H ) ∼ = G × H ( g / h ) ∗ , where G acts onthe latter variety via left multiplication on the first factor. At the same time, the H -representation( g / h ) ∗ is canonically isomorphic to the annihilator h ⊥ ⊆ g of h under the Killing form. We thus havea G -equivariant isomorphism(13) T ∗ ( G/H ) ∼ = G × H h ⊥ , with G × H h ⊥ defined analogously to G × H ( g / h ) ∗ .Now consider the restriction of (7b) to an action of H ⊆ G on G × g , noting that this restrictedaction is Hamiltonian with respect to Ω L . The moment map for this H -action is obtained by com-posing the g ∗ -valued version of φ R : G × g → g with the projection g ∗ → h ∗ . It follows that thepreimage of 0 under the new moment map is G × h ⊥ ⊆ G × g . The symplectic quotient of G × g by H is therefore given by ( G × g ) //H = G × H h ⊥ . It is straightforward to verify that the induced symplectic structure on G × H h ⊥ renders (13) a G -equivariant isomorphism of symplectic varieites. It is also straightforward to check that(14) ν H : G × H h ⊥ → g , [( g, x )] Ad g ( x ) , ( g, x ) ∈ G × h ⊥ is a moment map for the Hamiltonian action of G on G × H h ⊥ .The above-defined holomorphic symplectic structure and Hamiltonian G -action on G × H h ⊥ turnout to come from a ( G, K )-hyperk¨ahler variety structure (see Definition 1), which we now discuss.Accordingly, recall that (7a) and (7b) define commuting, tri-Hamiltonian actions of K on G × g .Let us restrict the latter action to the subgroup L ⊆ K fixed in the introduction to Section 3, andthen consider the associated hyperk¨ahler quotient ( G × g ) ///L . Note that (7a) then descends to atri-Hamiltonian action of K on ( G × g ) ///L . At the same time, (4) takes the form of a K -equivariantmap(15) ( G × g ) ///L → ( G × g ) //H = G × H h ⊥ . One can then invoke [10, Section 2] and/or [27, Theorem 3.1] to deduce the following fact.
Theorem 3.
The map (15) is a K -equivariant isomorphism of holomorphic symplectic manifolds. Let ( I Hℓ , ω Hℓ , b H ), ℓ = 1 , ,
3, denote the hyperk¨ahler manifold structure on G × H h ⊥ for which (15)is an isomorphism of hyperk¨ahler manifolds, which by the preceding discussion makes G × H h ⊥ intoa ( G, K )-hyperk¨ahler variety. To help investigate this (
G, K )-hyperk¨ahler structure, we use Lemma2 to see that ρ descends to a K -invariant function ρ H : ( G × g ) ///L → R . Note that since (15) is K -equivariant, we may regard ρ H as a K -invariant function on G × H h ⊥ . Proposition 4.
The function ρ H : G × H h ⊥ → R is a K -invariant potential for the K¨ahler manifold ( G × H h ⊥ , I H , ω H , b H ) , i.e. ω H = 2 i∂∂ρ H for the Dolbeault operators ∂ and ∂ associated with I H . PETER CROOKS AND MAARTEN VAN PRUIJSSEN
Proof.
Let ( µ , µ , µ ) : G × g → ( k ∗ ) ⊕ denote the hyperk¨ahler moment map for the tri-Hamiltonian K -action (7b), and let ( µ H , µ H , µ H ) : G × g → ( l ∗ ) ⊕ be the induced hyperk¨ahler moment map forthe action of L ⊆ K . Consider the action of SO ( R ) on G × g , recalling our description of a specificsubgroup S ⊆ SO ( R ) and its action on G × g (see 3.2). This description implies that S preserves µ and acts by rotations on span R { µ , µ } . We conclude that S preserves µ H and acts by rotationson span R { µ H , µ H } , so that the submanifold ( µ H ) − (0) ∩ ( µ H ) − (0) ∩ ( µ H ) − (0) ⊆ G × g is necessarily S -invariant. Observe that the actions of S and L on this submanifold commute, owing to the factthat the action of SO ( R ) on G × g commutes with the K -action (7b). The quotient (cid:0) ( µ H ) − (0) ∩ ( µ H ) − (0) ∩ ( µ H ) − (0) (cid:1) /L = ( G × g ) ///L therefore carries a residual S -action, so that we may use the hyperk¨ahler isomorphism (15) to equip G × H h ⊥ with a corresponding S -action. The relations (3) then imply that S preserves ω H . Nowconsider the element θ ∈ so ( R ) discussed in 3.2, recalling that ρ is the θ -component of a momentmap for the S -action on G × g . It is then straightforward to check that ρ H is the θ -component ofa moment map for the S -action that preserves ω H . Note also that the identities L ˜ θ ω = ω and L ˜ θ ω = − ω give L ˜˜ θ ω H = ω H and L ˜˜ θ ω H = − ω H , where ˜˜ θ is the fundamental vector field on G × H h ⊥ associated to θ . These last two sentences giveexactly the ingredients needed to reproduce a calculation from [15, Section 3(E)], to the effect that ρ H is a K¨ahler potential for ( I H , ω H , b H ). (cid:3) The hyperk¨ahler slice construction
The slice as a symplectic variety.
Recall the notation established in the introduction toSection 3, and let ad : g → gl ( g ) , x ad x , x ∈ g denote the adjoint representation of g . One calls τ = ( ξ, h, η ) ∈ g ⊕ an sl - triple if [ ξ, η ] = h ,[ h, ξ ] = 2 ξ , and [ h, η ] = − η , in which case there is an associated Slodowy slice S τ := ξ + ker(ad η ) ⊆ g . We will make extensive use of the affine variety G × S τ , some geometric features of which we nowdevelop.Consider the isomorphisms T ∗ G ∼ = G × g ∗ ∼ = G × g induced by the right trivialization of T ∗ G andthe Killing form. The symplectic form on T ∗ G thereby corresponds to such a form Ω R on G × g ,described as follows on the tangent space T ( g,x ) ( G × g ) = T g G ⊕ g (see [26, Section 5, Equation (14R)]:(16) (Ω R ) ( g,x ) (cid:18) ( d e R g ( y ) , z ) , ( d e R g ( y ) , z ) (cid:19) = h y , z i − h y , z i − h x, [ y , y ] i for all y , y , z , z ∈ g , where R g : G → G is right multiplication by g , d e R g is its differential at e . Itturns out that G × S τ is a symplectic subvariety of ( G × g , Ω R ). The G -action g · ( h, x ) = ( hg − , x ) , g ∈ G, ( h, x ) ∈ G × S τ is then Hamiltonian and µ τ : G × S τ → g , ( g, x )
7→ − Ad g − ( x ) , ( g, x ) ∈ G × S τ is a moment map. YPERK ¨AHLER SLICES AND SPHERICAL GEOMETRY 9
Remark 5.
Bielawski’s paper [4] uses Ω R to realize G × S τ as a symplectic subvariety G × g , asopposed to using the other symplectic form Ω L (see (8)). It is for the sake of consistency withBielawski’s work that we are using the same convention. However, this is the only case in which weuse Ω R preferentially to Ω L .Now let X be a symplectic variety endowed with a Hamiltonian G -action and moment map µ : X → g . The diagonal action of G on X × ( G × S τ ) is then Hamiltonian and admits a moment mapof ˜ µ : X × ( G × S τ ) → g , ( x, ( g, y )) µ ( x ) + µ τ ( g, y ) , x ∈ X, ( g, y ) ∈ G × S τ . Noting that this diagonal action is free, one has the symplectic quotient( X × ( G × S τ )) //G = ˜ µ − (0) /G. Proposition 6.
Let ( X, ω ) be a symplectic variety on which G acts in a Hamiltonian fashion withmoment map µ : X → g , and let τ be an sl -triple. The following statements then hold. (i) There is a canonical isomorphism of affine varieties µ − ( S τ ) ∼ = ( X × ( G × S τ )) //G . (ii) Under the isomorphism from (i) , the symplectic form on ( X × ( G × S τ )) //G corresponds tothe restriction of ω to µ − ( S τ ) . (iii) µ − ( S τ ) is a symplectic subvariety of X .Proof. To prove (i), note that ( x, ( e, µ ( x ))) ∈ ˜ µ − (0) for all x ∈ µ − ( S τ ). We may therefore considerthe morphism(17) ϕ : µ − ( S τ ) → ˜ µ − (0) , x ( x, ( e, µ ( x ))) , x ∈ µ − ( S τ ) , and its composition with the quotient map π : ˜ µ − (0) → ˜ µ − (0) /G = ( X × ( G × S τ )) //G , i.e. ϕ : µ − ( S τ ) → ( X × ( G × S τ )) //G, x [( x, ( e, µ ( x )))] , x ∈ µ − ( S τ ) . At the same time, it is straightforward to check that g · x ∈ µ − ( S τ ) for all ( x, ( g, y )) ∈ ˜ µ − (0). Wethus have the morphism ψ : ˜ µ − (0) → µ − ( S τ ) , ( x, ( g, y )) g · x, ( x, ( g, y )) ∈ ˜ µ − (0) , which is easily seen to be G -invariant. It follows that ψ descends to the quotient ˜ µ − (0) /G =( X × ( G × S τ )) //G , thereby giving a morphism ψ : ( X × ( G × S τ )) //G → µ − ( S τ ). Furthermore, itis a straightforward calculation that ϕ and ψ are inverses. This proves (i).In preparation for (ii), let ω denote the symplectic form on ( X × ( G × S τ )) //G and consider theinclusions j : G × S τ → G × g and k : ˜ µ − (0) → X × ( G × S τ ). Note that π ∗ ( ω ) is the restriction to˜ µ − (0) of the symplectic form on X × ( G × S τ ). This last symplectic form is ω ⊕ j ∗ (Ω R ), so that wehave(18) π ∗ ( ω ) = k ∗ ( ω ⊕ j ∗ (Ω R )) . Our objective is to prove that ϕ ∗ ( ω ) = ℓ ∗ ( ω ), where ℓ : µ − ( S τ ) → X is the inclusion. Accordingly,note that ϕ ∗ ( ω ) = ϕ ∗ ( π ∗ ( ϕ )) [since ϕ = π ◦ ϕ ]= ( k ◦ ϕ ) ∗ ( ω ⊕ j ∗ (Ω R )) [by (18)] . It follows that(19) (cid:0) ϕ ∗ ( ω ) (cid:1) x ( v , v ) = (cid:0) ω x ⊕ (Ω R ) ( e,µ ( x )) (cid:1) ( d x ϕ ( v ) , d x ϕ ( v ))for all x ∈ µ − ( S τ ) and v , v ∈ T x ( µ − ( S τ )). At the same time, (17) implies the identity d x ϕ ( v i ) = ( v i , (0 , d x µ ( v i ))) for i = 1 , T ( x, ( e,µ ( x ))) (˜ µ − (0)) ⊆ T ( x, ( e,µ ( x ))) ( X × ( G × S τ )) = T x X ⊕ ( g ⊕ T µ ( x ) S τ ) . By incorporating this into (19), we obtain (cid:0) ϕ ∗ ( ω ) (cid:1) x ( v , v ) = ω x ( v , v ) + (Ω R ) ( e,µ ( x )) (cid:0) (0 , d x µ ( v )) , (0 , d x µ ( v ) (cid:1) = ω x ( v , v ) [by (16)] . We conclude that ϕ ∗ ( ω ) = ℓ ∗ ( ω ), proving (ii).It remains only to prove (iii), i.e. the claim that ℓ ∗ ( ω ) is non-degenerate. However, this followsimmediately from (i), (ii), and the fact that ω is non-degenerate. (cid:3) Bielawski’s construction.
We now review the pertinent hyperk¨ahler-geometric features of µ − ( S τ ), which are largely due to Bielawski’s work [4]. The following ( G, K )-hyperk¨ahler variety willplay an essential role.
Theorem 7. If τ is an sl -triple, then G × S τ is canonically a ( G, K ) -hyperk¨ahler variety. TheHamiltonian G -action and underlying holomorphic symplectic structure on G × S τ associated withthis ( G, K ) -hyperk¨ahler structure are precisely those described in 4.1. Now let X be any ( G, K )-hyperk¨ahler variety. Given an sl -triple τ , note that product manifold X × ( G × S τ ) is naturally hyperk¨ahler and carries a free, diagonal G -action. It is then not difficultto check that X × ( G × S τ ) is a ( G, K )-hyperk¨ahler variety, with underlying holomorphic symplecticstructure equal to the natural product holomorphic symplectic structure on X × ( G × S τ ). With thisin mind, we can define hyperk¨ahler slices as follows. Definition 8.
Given a (
G, K )-hyperk¨ahler variety X and an sl -triple τ , we refer to ( X × ( G × S τ )) ///K as the hyperk¨ahler slice for X and τ .This construction can be used to produce a number of well-studied hyperk¨ahler manifolds, some ofwhich are mentioned in the introduction of [5]. For several of these examples, there is a particularlyconcrete description of the underlying holomorphic symplectic manifold. Indeed, let X and τ be asdescribed in the definition above. Note that (4) manifests as a map(20) ( X × ( G × S τ )) ///K → ( X × ( G × S τ )) //G, which features in the following rephrased version of [4, Theorem 1]. Theorem 9 (Bielawski) . Let τ be an sl -triple, and let ( X, ( I ℓ , ω ℓ , b ) ℓ =1 ) be a ( G, K ) -hyperk¨ahlervariety with complex moment map µ : X → g . Consider the map (21) ( X × ( G × S τ )) ///K → µ − ( S τ ) obtained by composing (20) with the isomorphism ( X × ( G × S τ )) //G ∼ = −→ µ − ( S τ ) from Proposition6(i). If the K¨ahler manifold ( X, I , ω , b ) has a K -invariant potential that is bounded from below oneach G -orbit, then (21) is an isomorphism of holomorphic symplectic manifolds. Remark 10.
Bielawski speaks of hyperk¨ahler slices only when the hypotheses of Theorem 9 aresatisfied (see [5, Section 1]). He then defines a hyperk¨ahler slice to be a hyperk¨ahler manifold ofthe form µ − ( S τ ), where µ − ( S τ ) is equipped with the hyperk¨ahler structure induced through theisomorphism (21). In particular, Definition 8 mildly generalizes Bielawski’s original notion.Let us briefly consider the hyperk¨ahler slice construction for ( G, K )-hyperk¨ahler varieties of theform ( G × H h ⊥ , ( I Hℓ , ω Hℓ , b H ) ℓ =1 ), as introduced in 3.3. Accordingly, recall the notation adopted in3.3. The function ρ H is bounded from below on all of G × H h ⊥ (see (12)), while we recall that ρ H is YPERK ¨AHLER SLICES AND SPHERICAL GEOMETRY 11 a K -invariant potential for the K¨ahler manifold ( G × H h ⊥ , ( I H , ω H , b H )) (see Proposition 4). It thenfollows from Theorem 9 that(22) (cid:16) ( G × H h ⊥ ) × ( G × S τ ) (cid:17) ///K ∼ = ν − H ( S τ )as holomorphic symplectic manifolds for all sl -triples τ in g . We exploit this fact in what follows.4.3. The regular Slodowy slice.
Recall that dim(ker(ad x )) ≥ r for all x ∈ g , and that x is called regular if equality holds. Let g reg ⊆ g denote the set of all regular elements, which is known to bea G -invariant, open, dense subvariety of g . This leads to the notion of a regular sl -triple, i.e. an sl -triple τ = ( ξ, h, η ) in g for which ξ ∈ g reg . Fix one such triple τ for the duration of this paper,and let S reg := S τ denote the associated Slodowy slice. The slice S reg is known to be contained in g reg , and to be a fundamental domain for the action of G on g reg (see [18, Theorem 8]). Note thatthis last sentence may be rephrased as follows: x ∈ g belongs to g reg if and only if x is G -conjugateto a point in S reg , in which case x is G -conjugate to a unique point in S reg .As discussed in the 1.2, we wish to study the emptiness problem for hyperk¨ahler slices of the form (cid:0) ( G × H h ⊥ ) × ( G × S reg ) (cid:1) ///K . The following result is a crucial first step. Proposition 11.
The hyperk¨ahler slice (cid:0) ( G × H h ⊥ ) × ( G × S reg ) (cid:1) ///K is non-empty if and only if h ⊥ ∩ g reg = ∅ .Proof. Using (22), we conclude that (cid:0) ( G × H h ⊥ ) × ( G × S reg ) (cid:1) ///K = ∅ if and only if the image of ν H meets S reg . This image is precisely G · h ⊥ ⊆ g (see (14)), reducing our task to one of proving that G · h ⊥ ∩ S reg = ∅ if and only if h ⊥ ∩ g reg = ∅ . To prove this, we simply appeal to the discussion of S reg above and note that x ∈ h ⊥ belongs to g reg if and only if x is G -conjugate to a point in S reg . (cid:3) The spherical geometry of
G/H
The image of the moment map.
Let us continue with the notation set in the introductionof Section 3. Choose opposite Borel subgroups
B, B − ⊆ G , declaring the former to be the positiveBorel and the latter to be the negative Borel. It follows that T := B ∩ B − is a maximal torus of G ,and we shall let b , b − , and t denote the Lie algebras of B , B − , and T , respectively. We thus havea weight lattice Λ ⊆ t ∗ and canonical group isomorphisms Λ ∼ = Hom( T, C × ) ∼ = Hom( B, C × ), whereHom is taken in the category of algebraic groups. We also have sets of roots ∆ ⊆ Λ, positive roots∆ + ⊆ ∆, negative roots ∆ − ⊆ ∆, and simple roots Π ⊆ ∆ + . Note that by definition b = t ⊕ M α ∈ ∆ + g α and b − = t ⊕ M α ∈ ∆ − g α , where g α is the root space associated to α ∈ ∆.We now establish two important conventions. To this end, recall the isomorphism (5) between theadjoint and coadjoint representations of G . Our first convention is to use ( · ) ∨ for both (5) and itsinverse, so that the inverse will presented as g ∗ ∼ = −→ g , φ φ ∨ , φ ∈ g ∗ . As for our second convention, note that the map g ∗ → t ∗ restricts to an isomorphism from the imageof t under (5) to t ∗ . We will use this isomorphism to regard t ∗ as belonging to g ∗ .Now let Y be a smooth, irreducible G -variety having field of rational functions C ( Y ), noting that C ( Y ) is then a G -module. A non-zero f ∈ C ( Y ) is called a B -semi-invariant rational function ofweight λ ∈ Λ if b · f = λ ( b ) f for all b ∈ B . Those λ admitting such an f form the weight lattice of Y , i.e. Λ Y := { λ ∈ Λ : ∃ a B -semi-invariant rational function on Y of weight λ } . The weight lattice of Y can also be viewed as the character lattice of a quotient of T , once we appealto Knop’s local structure theorem [34, Theorem 4.7]. This theorem gives a parabolic subgroup P ⊆ G that contains B , has a Levi decomposition P = P u L with T ⊆ L , and satisfies the followingproperty: there exists a locally closed affine P -stable subvariety Z ⊆ Y such that P u × Z → Y mapssurjectively onto an open affine subset Y of Y. One also has [ L, L ] ⊆ L ⊆ L , where L is the kernelof the L -action on Z . The quotient A Y := L/L is a torus that acts freely on Z , and there existsan affine variety C with a trivial L -action such that Z ∼ = A Y × C as L -varieties. It follows thatΛ Y = Hom( A Y , C × ).The subspace a ∗ Y := Λ Y ⊗ Z C ⊆ t ∗ is sometimes called the Cartan space of the G -variety Y . LetΛ ∨ Y ⊆ t and a Y ⊆ t denote the preimage and image of Λ Y and a ∗ Y under (5), respectively, noting that(23) e A Y := Λ ∨ Y ⊗ Z C × is a subtorus of T with Lie algebra a Y . We shall also refer to a Y as the Cartan space of Y . Example 12.
In what follows, we compute the Cartan space of
G/T . Let Λ + ⊆ t ∗ denote the setof dominant weights of G , and let V λ be the irreducible G -module of highest weight λ ∈ Λ + . Recallthe following classical fact about C [ G/T ], the coordinate ring of
G/T : C [ G/T ] ∼ = M Λ ∈ Λ + ( V ∗ λ ) ⊕ d λ as G -modules, where d λ := dim(( V λ ) T ) and ( V λ ) T is the subspace of T -fixed vectors in V λ . Note that d λ = 0 if and only if λ lies in the root lattice Q ⊆ t ∗ . Note also that ( V λ ) ∗ ∼ = V − w λ as G -modules,where w is the longest element of the Weyl group W := N G ( T ) /T . It follows that for λ ∈ Λ + , V λ is an irreducible summand of C [ G/T ] if and only if λ ∈ − w (Λ + ∩ Q ) = Λ + ∩ Q . Since C [ G/T ] is a G -submodule of C ( G/T ), this implies that Λ + ∩ Q is contained in Λ G/T . Now observe that Λ + ∩ Q generates t ∗ over C , yielding a ∗ G/T = Λ
G/T ⊗ Z C = t ∗ . We also conclude that a G/T = t .We now recall a key geometric feature of the Cartan space construction. Let Y be any smooth,irreducible G -variety and consider the canonical lift of the G -action on Y to a G -action on T ∗ Y . Thelatter action is Hamiltonian with respect to the standard symplectic form on T ∗ Y , and there is adistinguished moment map µ Y : T ∗ Y → g . Lemma 3.1 and Corollary 3.3 from [17] then combine togive the following equality of closures in g . Theorem 13 (Knop) . If Y is a smooth, irreducible, quasi-affine G -variety, then µ Y ( T ∗ Y ) = G · a Y . a -regularity. Recall the notation set in the introduction to Section 3, which we now use togetherwith the notation of 5.1. It is then not difficult to prove that a G/H depends only on the pair ( g , h ).For this reason, we set a ( g , h ) ∗ := a ∗ G/H and a ( g , h ) := a G/H . We will sometimes denote a ( g , h ) (resp. a ( g , h ) ∗ ) by a (resp. a ∗ ) when the underlying pair ( g , h ) is clear from context. Definition 14.
We say that the pair (
G, H ) or the corresponding pair ( g , h ) of Lie algebras is a - regular if a ( g , h ) contains a regular element of g .We now give a few characterizations of a -regularity. In what follows, e A G/H is the subtorus of T defined by setting Y = G/H in (23) and Z G ( e A G/H ) consists of all g ∈ G that commute with everyelement of e A G/H . We also let Z G ( a ) be the subgroup of all g ∈ G that fix a pointwise, and we let z g ( a ) be the subspace of all x ∈ g that commute with every element of a . Proposition 15.
With all notation as described above, the following conditions are equivalent. (i) (
G, H ) is a -regular; (ii) h ⊥ ∩ g reg = ∅ ; (iii) Z G ( a ) = T . YPERK ¨AHLER SLICES AND SPHERICAL GEOMETRY 13
Proof.
We begin by proving that h ⊥ ∩ g reg = ∅ if and only if ( G, H ) is a -regular. To show the forwardimplication, assume that h ⊥ ∩ g reg = ∅ . Identifying T ∗ ( G/H ) with G × H h ⊥ and recalling the momentmap ν H (see (14)), Theorem 13 implies that ν H ( G × H h ⊥ ) = G · a . This amounts to the statement that G · h ⊥ = G · a . Since h ⊥ ∩ g reg = ∅ by hypothesis, we must have G · a ∩ g reg = ∅ . Note also that G · a is a constructiblesubset of g , so that G · a intersects every non-empty open subset of G · a . These last two sentencesimply that G · a ∩ g reg = ∅ , which is equivalent to a ∩ g reg = ∅ . We conclude that ( G, H ) is a -regular.In an analogous way, one argues that ( G, H ) being a -regular implies h ⊥ ∩ g reg = ∅ .We are reduced to establishing that ( G, H ) is a -regular if and only if Z G ( a ) = T . Accordingly,recall that an element of t is regular if and only if it does not lie on any root hyperplane. It followsthat ( G, H ) is not a -regular if and only if a belongs to the union of all root hyperplanes. Since a isirreducible, this is equivalent to a being contained in a particular root hyperplane, i.e. a ⊆ ker( α )for some α ∈ ∆. This holds if and only if g α ⊆ z g ( a ) for some α ∈ ∆. Now note that z g ( a ) is a T -invariant subspace of g containing t , meaning that z g ( a ) = t ⊕ M α ∈ S g α for some subset S ⊆ ∆. It follows that g α ⊆ z g ( a ) for some α ∈ ∆ if and only if z g ( a ) = t . The secondof these conditions is equivalent to having Z G ( a ) = T , if one knows Z G ( a ) to be connected and havea Lie algebra of z g ( a ). Connectedness follows from the observation that Z G ( a ) = Z G ( e A G/H ) (see [33,Theorem 24.4.8]), together with the fact that centralizers of tori are connected (see [33, Proposition28.3.1]). At the same time, it is clear that z g ( a ) is the Lie algebra of Z G ( a ) (cf. [33, Proposition24.3.6]). This completes the proof. (cid:3) Let H act on a complex algebraic variety X . A subgroup e H ⊆ H is called a generic stabilizer for this action if there exists a non-empty open dense subset U ⊆ X with the following property:the H -stabilizer of every x ∈ U is conjugate to e H . A generic stabilizer is known to exist if X is alinear representation of H [32]. We therefore have a generic stabilizer for the H -action on h ⊥ , andwe denote it by H ∗ . This group is known to be reductive (see [34, Theorem 9.1]). Remark 16.
A generic stabilizer is unique up to conjugation, meaning that H ∗ more appropriatelydenotes a conjugacy class of subgroups in H . However, we shall always take H ∗ to be a fixed subgroupin this conjugacy class.Now recall our discussion of the the local structure theorem for a smooth, irreducible G -variety Y ,as well as the notation introduced in that context (see 5.1). If Y = G/H , then the group L turnsout to be precisely H ∗ (see [16, Section 8]). Corollary 17.
The pair ( G, H ) is a -regular if and only if the connected component of the identityin H ∗ is abelian.Proof. Proposition 15 and the fact that H ∗ = L reduce our task to one of proving that Z G ( a ) = T if and only if the identity component in L is abelian. To this end, consider [34, Definition 8.13]and [34, Proposition 8.14]. Since G/H is an affine variety, these two statements imply that L = Z G ( a ).Our task is therefore to prove that L = T if and only if the identity component in L is abelian. Theforward implication follows immediately from the inclusion L ⊆ L , so that we only need to verifythe opposite implication. Note that L is a Levi factor of a parabolic subgroup of G , as discussed in 5.1. This means that L isconnected and reductive, forcing the derived subgroup [ L, L ] to be connected as well. The inclusion[
L, L ] ⊆ L thus shows [ L, L ] to be contained in the identity component in L . If we now assumethat this component is abelian, then [ L, L ] must also be abelian. It follows that L is itself abelian.Together with the inclusion T ⊆ L (see 5.1) and the fact that L is a connected, reductive subgroupof G , this last sentence implies that L = T . The proof is complete. (cid:3) Corollary 17 can be used to easily assess a -regularity in several examples. To see this, we notethat [20] fully describes the H -representation h ⊥ in many cases. Each of these descriptions can becombined with the tables of `Elaˇsvili [12, 13] to compute H ∗ , after which Corollary 17 can be applied.We illustrate this in the following example. Example 18.
Consider the pair (
G, H ) = (SL p + q , S(GL p × GL q )) with 1 ≤ p ≤ q . The vectorspace h ⊥ is isomorphic to ( C p ⊗ ( C q ) ∗ ) ⊕ (( C p ) ∗ ⊗ C q ) as an H -representation. The Lie algebra of thegeneric stabilizer for this action is isomorphic to C p ⊕ sl ( q − p ) if p < q and to C p − if p = q . Hence( G, H ) is a -regular if and only if q − p ≤ a -regularity in terms of spherical-geometric invariants.Recall that the rank rk G ( Y ) of a G -variety Y is the dimension of a Y . The complexity c G ( Y ) of Y isthe codimension of a generic B -orbit in Y . We then have the following equalities, which are due toKnop [16]: 2 c G ( G/H ) + rk G ( G/H ) = dim G − H + dim H ∗ ;(24) rk G ( G/H ) = dim T − dim T ∗ , (25)where T ∗ is a maximal torus of H ∗ . Corollary 19.
The pair ( G, H ) is a -regular if and only if c G ( G/H ) + rk G ( G/H ) + dim H = dim B .Proof. Corollary 17 shows that (
G, H ) is a -regular if and only if the identity component in H ∗ isabelian. This is in turn equivalent to dim H ∗ = dim T ∗ , and the result then follows from (24) and(25). (cid:3) The criteria established in Corollaries 17 and 19 become effective once we are able to eitherdetermine the Cartan space a ( g , h ) or the generic stabilizer H ∗ . The latter is difficult to accomplishin full generality, but Losev’s work [25] makes the former achievable in a systematic way. Losev’smethod features prominently in the next subsection.5.3. The Cartan space of a homogeneous affine variety.
Continuing with the notation usedin 5.2, we recall Losev’s algorithm [25] for determining the Cartan space of (
G, H ). We begin withthe following definition (cf. [34, Section 10]).
Definition 20.
The pair (
G, H ) or the corresponding pair ( g , h ) is called:(i) decomposable if there exist non-zero proper ideals g , g in g and any ideals h , h in h suchthat g = g ⊕ g , h = h ⊕ h , h ⊆ g , and h ⊆ g ;(ii) indecomposable if it is not decomposable;(iii) strictly indecomposable if ( g , [ h , h ]) is indecomposable.We note that the Cartan space of a decomposable pair ( g ⊕ g , h ⊕ h ) is a ( g , h ) ⊕ a ( g , h ). Atthe same time, observe that ( x , x ) ∈ g ⊕ g is a regular element if and only if x and x are regularelements of g and g , respectively. These last two sentences imply that ( g ⊕ g , h ⊕ h ) is a -regularif and only if ( g , h ) and ( g , h ) are a -regular. Recognizing its relevance to later arguments, werecord this conclusion as follows. YPERK ¨AHLER SLICES AND SPHERICAL GEOMETRY 15
Lemma 21.
Consider a collection of indecomposable pairs ( g i , h i ) , i = 1 , . . . , n , and suppose thatour pair ( g , h ) is given by (26) ( g , h ) = (cid:18) n M i =1 g i , n M i =1 h i (cid:19) . Then ( g , h ) is a -regular if and only if ( g i , h i ) is a -regular for all i = 1 , . . . , n . Remark 22.
Note that our pair ( g , h ) is necessarily expressible in the form (26), i.e. there existindecomposable pairs ( g i , h i ), i ∈ { , . . . , n } , such that g i (resp. h i ) is an ideal in g (resp. h ) forall i and (26) holds. This observation follows from Definition 20 via a straightforward inductionargument, and it will be used implicitly in some of our arguments.We now resume the main discussion. Note that for a subalgebra j ⊆ h , we have an inclusion a ( g , h ) ⊆ a ( g , j ) of Cartan spaces. It follows that a ( g , h ) ⊆ a ( g , j ) for all ideals j ≤ h , which leads tothe following definition (cf. [25, Definition 1.1]). Definition 23.
A reductive subalgebra j ⊆ g is called essential if for every proper ideal i ≤ j , theinclusion a ( g , j ) ⊆ a ( g , i ) is strict.Now consider the Lie algebra h ∗ of H ∗ , where H ∗ is the generic stabilizer for the H -action on h ⊥ (see 5.2). Losev shows that h ∗ generates an ideal h ess ≤ h that is an essential subalgebra of g . Thisessential subalgebra is reductive and has the following properties: • h ess ≤ h is the unique ideal of h for which a ( g , h ) = a ( g , h ess ); • h ess is maximal (for inclusion) among the ideals of h that are essential subalgebras of g .In principle, this reduces the computation of a ( g , h ) to the task of determining h ess and a ( g , h ess ).The preceding discussion allows us to sketch the main results of [25]. Losev classifies the essentialsubalgebras j ⊆ g that are semisimple, and in each such case he presents a ( g , j ) as the span of certainlinear combinations of fundamental weights. This information may also be used to determine theCartan space when j is non-semisimple, provided that one knows the center of j . To this end, Losevgives an algorithm for calculating the centers of non-semisimple essential subalgebras.5.4. Preliminaries for the classifications.
We now discuss four items that are crucial to theclassifications in 5.5. Our first item is the following elementary observation.
Observation 24.
Let r be a complex reductive Lie algebra with a reductive ideal i ≤ r . If j is areductive ideal in i , then j is also an ideal in r . This follows immediately from the decomposition of areductive Lie algebra into a direct sum of its center and simple ideals, and it will be used implicitlyin some of what follows.We also need the following definition, which serves to formalize a standard idea. Definition 25.
Let r and r be complex Lie algebras with respective subalgebras s and s . We referto ( r , s ) and ( r , s ) as being conjugate if r = r and s = φ ( s ) for some Lie algebra automorphism φ : r → r .With this in mind, we have the following lemma. Lemma 26.
Assume that g is simple and let h ⊆ g be a reductive subalgebra. (i) If ( g , i ) is not conjugate to a pair in Tables 1 or 2 from [25] for any ideal i ≤ h , then h ess = { } .In this case, a ( g , h ) = t and ( g , h ) is a -regular. (ii) If h ess = { } , then ( g , h ) is a -regular if and only if ( g , [ h ess , h ess ]) is conjugate to a pair inTable I below. g i sl k sl k ⊕ sl k sl k − sl k ⊕ sl k − sp sp ⊕ sl ⊕ sl sp sl ⊕ sl ⊕ sl e sl sl n +1 sl n +1 sl n +1 sp n TABLE I.
For each line, the embedding i ⊆ g is as described in [25, Section 6]. Proof.
We begin by proving (i), and thus assume that ( g , i ) is not conjugate to a pair in Tables 1 or2 from [25] for any ideal i ≤ h . Noting the particular classification that each table gives, we concludethat [ h ess , h ess ] cannot contain a non-zero semisimple ideal. Hence [ h ess , h ess ] = { } , i.e. h ess is abelian.Since h ess is also reductive, one can find a Lie algebra automorphism of g that sends h ess into t . Thisimplies that ( g , h ) is conjugate to a pair ( g , e h ) satisfying e h ess ⊆ t . We may therefore assume that h ess ⊆ t .Note that the inclusions { } ⊆ h ess ⊆ t yield a ( g , t ) ⊆ a ( g , h ess ) ⊆ a ( g , { } ), which by Example 12amounts to the statement t ⊆ a ( g , h ess ) ⊆ a ( g , { } ). At the same time, the inclusion a ( g , { } ) ⊆ t follows from how we defined Cartan spaces in 5.1. We conclude that t = a ( g , h ess ) = a ( g , { } ) . Recalling the properties of h ess discussed in 5.3, the first equality implies that t = a ( g , h ) and thesecond equality gives h ess = { } . The a -regularity of ( g , h ) now follows from the fact that t ∩ g reg = ∅ ,completing our proof of (i).To prove (ii), we first assume that ( g , [ h ess , h ess ]) is conjugate to a pair in Table I. If h ess is semisim-ple, i.e. [ h ess , h ess ] = h ess , then a ( g , [ h ess , h ess ]) = a ( g , h ess ) = a ( g , h ). This observation and aninspection of [25, Table 1] reveal that ( g , h ) is a -regular. If h ess is not semisimple, then ( g , [ h ess , h ess ])is conjugate to one of items 6 and 7 in Table I and h ess has a non-trivial center z ( h ess ). The re-sult [25, Theorem 1.3(c)] then shows that z ( h ess ) ⊆ z := z ( z g ([ h ess , h ess ])) , where z g ([ h ess , h ess ]) is the subalgebra of all elements in g that commute with every element of[ h ess , h ess ] and z ( z g ([ h ess , h ess ])) is the center of this subalgebra. Noting again that ( g , [ h ess , h ess ])is conjugate to item 6 or 7 in Table I, one uses [25, Table 2] to see that a ( g , [ h ess , h ess ] + z ) has regularelements. Note also that h ess = [ h ess , h ess ] + z ( h ess ) ⊆ [ h ess , h ess ] + z implies a ( g , [ h ess , h ess ] + z ) ⊆ a ( g , h ess ) = a ( g , h ) . The previous two sentences together show that ( g , h ) is a -regular.For the converse we suppose that h ess ≤ h is not the trivial ideal. The discussion above impliesthat h ess cannot be abelian, so that [ h ess , h ess ] ≤ h is a semisimple and non-trivial ideal. It thenfollows from Losev’s setup in [25] that [ h ess , h ess ] is conjugate to a pair in [25, Table 1] or [25, Table2]. Hence there are three mutually exclusive possibilities: ( g , [ h ess , h ess ]) is conjugate to a pair in:(a) [25, Table 1], but not to one in [25, Table 2];(b) [25, Table 1] and [25, Table 2];(c) [25, Table 2], but not to one in [25, Table 1]. YPERK ¨AHLER SLICES AND SPHERICAL GEOMETRY 17
In each instance, we simply use Losev’s tables to inspect all possible Cartan spaces a ( g , h ) anddetermine whether each has a regular element.We first suppose that (a) holds. Then ( g , h ) is a -regular precisely when ( g , h ess ) is conjugate toone of the items 2 (with k = n/ , ( n + 1) / n = 4), 7 or 21 from [25, Table 1]. These pairsconstitute the first five lines of Table I.Now suppose that (b) holds. Then ( g , [ h ess , h ess ]) is conjugate to one of the items 1, 2 (with n/ Lemma 27. If H is any closed, reductive subgroup of G , then a ( g , h ) ∗ is spanned by Λ + ( G, H ) . The classifications. We maintain the notation used in 5.3, and now address the classificationof a -regular pairs ( G, H ) (equivalently, a -regular pairs ( g , h )) in each of the following three cases: H is a Levi subgroup 5.5.1, H is symmetric 5.5.2, and H is simultaneously reductive, spherical, and non-symmetric 5.5.3. In each case, we reduce to the classification of strictly indecomposable, a -regularpairs. We list all conjugacy classes of such pairs in each of the cases 5.5.2 and 5.5.3, where the notionof conjugacy class comes from Definition 25. Remark 28. We emphasize that the classification of strictly indecomposable pairs works differentlyin each of the above-mentioned cases. In the case of Levi subgroups H ⊆ G , the classification isalmost entirely based on Losev’s work [25]. This is in contrast to the case of symmetric subgroups, inwhich we appeal to representation-theoretic results about symmetric spaces. Several of these resultsare not applicable to the case of a reductive spherical H ⊆ G , for which we instead harness the worksof Brion [7], Kr¨amer [21], and Mikityuk [28]. Remark 29. Note that every symmetric subgroup of G is reductive and spherical (see [34, Theorem26.14]). The techniques and arguments in 5.5.3 thereby imply the classification results in 5.5.2.Despite this, we believe that the representation-theoretic approach taken in 5.5.2 is independentlyinteresting and worthwhile. Further distinctions between 5.5.2 and 5.5.3 are discussed in Remark 32and Example 36.5.5.1. Levi subgroups. Assume that H is a Levi subgroup of G , by which we mean that H is a Levifactor of a parabolic subgroup P ⊆ G . It follows that h is a Levi factor of a parabolic subalgebra p ⊆ g . Now let g = g ⊕ · · · ⊕ g n be the decomposition of g into its simple ideals g , . . . , g n . Theparabolic subalgebra p is then a sum of parabolic subalgebras p i ⊆ g i for i = 1 , . . . , n , implying that h is a sum of Levi factors h i ⊆ p i , i = 1 , . . . , n . An application of Lemma 21 then shows that ( g , h )is a -regular if and only if ( g i , h i ) is a -regular for all i = 1 , . . . , n . It therefore suffices to assume that g is simple. Our classification then takes the following form. Proposition 30. Assume that g is simple and that h is a Levi subalgebra of g with h ess = { } . Thepair ( g , h ) is then a -regular if and only if it is conjugate to a pair in Table II. In this table, l is anyreductive subalgebra of sl n +1 that satisfies the following conditions: sl n +1 ∩ l = { } , l commuteswith sl n +1 , and sl n +1 ⊕ l is a Levi subalgebra of sl n +1 . We are implicitly using the embedding sl n +1 ⊆ sl n +1 from line 6 of Table I. g l sl k s ( gl k ⊕ gl k )2 sl k − s ( gl k ⊕ gl k − )3 e sl ⊕ C sl n +1 sl n +1 ⊕ l TABLE II. Line 3 is to be understood as follows. Up to Lie algebra automorphism, e contains precisely one subalgebra isomorphic to sl ⊕ sl (see [11, Theorem 5.5, Table12, and Theorem 11.1]). By choosing a Cartan subalgebra of sl and identifying it with C , one obtains a unique automorphism class of subalgebras in e that are isomorphicto sl ⊕ C . This turns out to be a class of Levi subalgebras in e , and the reader maytake any of these to be the subalgebra l in line 3. Proof. We first assume that ( g , h ) is conjugate to a pair in Table II. A case-by-case analysis revealsthat each pair in Table II is a -regular, implying that ( g , h ) is a -regular.Conversely, assume that ( g , h ) is a -regular. Lemma 26(ii) then implies the existence of an ideal i in h for which ( g , i ) is conjugate to a pair in Table I. We will therefore begin by finding the pairs in TableI for which this is possible. For each such pair ( r , j ), we will subsequently find the Levi subalgebras l ⊆ r that contain j as an ideal. Note that ( g , h ) will then be conjugate to one of the pairs ( r , l ) arisingin this way. It will then suffice to observe that the aforementioned pairs ( r , l ) appear in Table II.Let ( r , j ) be any of the pairs appearing in lines 3,4, and 7 of Table I. Observe that the Dynkindiagram of j is not a subdiagram in the Dynkin diagram of r . At the same time, the Dynkin diagramof any ideal in a Levi subalgebra of g must be a subdiagram in the Dynkin diagram of g . It followsthat ( g , i ) cannot be conjugate to ( r , j ) for any ideal i ≤ h .In light of the previous paragraph, we may restrict our attention to the pairs in lines 1,2,5, and 6 ofTable I. Let ( r , j ) be any such pair, recalling that the embedding of j into r is described in [25, Section6] (cf. the caption of Table I). This description is easily seen to imply that j is an ideal in a Levisubalgebra of r . If ( r , j ) is in one of lines 1,2, and 5 from Table I, then the Dynkin diagram of j uniquely determines a Levi subalgebra l ⊆ r that contains j as an ideal. The pair ( r , l ) is recorded inTable II. If ( r , j ) is in line 6 from Table I, i.e. r = sl n +1 and j = sl n +1 , then there are several Levisubalgebras l ⊆ r that contain j as an ideal. The Dynkin diagram of any such l is a subdiagram in theDynkin diagram of sl n +1 , and it contains the Dynkin diagram of sl n +1 as a connected component.It follows that l = sl n +1 ⊕ l for some reductive subalgebra l ⊆ sl n +1 that satisfies the desiredhypotheses. (cid:3) Symmetric subgroups. Using the notation established in 5.1 and the introduction of Section3, we assume that the subgroup H ⊆ G is symmetric. This means that H is an open subgroupof G θ , the subgroup of fixed points of an involutive algebraic group automorphism θ : G → G . Itfollows that ( g , h ) is a symmetric pair, i.e. h coincides with the set of θ -fixed vectors g θ ⊆ g for thecorresponding involutive Lie algebra automorphism θ : g → g . Lemma 31. If h is any reductive subalgebra of g , then ( g , h ) is a symmetric pair if and only if thereexist strictly indecomposable symmetric pairs ( g i , h i ) , i ∈ { , . . . , n } , such that g i (resp. h i ) is anideal in g (resp. h ) for all i and ( g , h ) = (cid:18) n M i =1 g i , n M i =1 h i (cid:19) . YPERK ¨AHLER SLICES AND SPHERICAL GEOMETRY 19 Proof. The backward implication follows from the following simple observation: if ( g , h ) and ( g , h )are symmetric pairs, then ( g ⊕ g , h ⊕ h ) is also a symmetric pair.To prove the forward implication, assume that ( g , h ) is a symmetric pair and let θ : g → g bean involutive automorphism for which h = g θ . Note that each simple ideal of g is either θ -stable orinterchanged by θ with a different simple ideal. We may therefore identify g with g ⊕ · · · ⊕ g ⊕ s ⊕ g s +1 · · · ⊕ g s + t for simple Lie algebras g , . . . , g s + t , such that θ becomes the following map: ( x, y ) ( y, x ) on eachsummand g ⊕ i and x θ j ( x ) on each summand g j , where θ j : g j g j is an involutive automorphism.It follows that h = g θ = diag( g ) ⊕ · · · ⊕ diag( g s ) ⊕ g θ s +1 s +1 ⊕ · · · ⊕ g θ s + t s + t , where diag( g i ) := { ( x, y ) ∈ g ⊕ i : x = y } for all i ∈ { , . . . , s } .In light of the above, it suffices to prove that the symmetric pairs ( g ⊕ i , diag( g i )) and ( g j , g θ j j ) arestrictly indecomposable for all i ∈ { , . . . , s } and j ∈ { s + 1 , . . . , s + t } . The strict indecomposabilityof the latter pair follows from the fact that g j is simple. Now observe that the simplicity of g i ∼ =diag( g i ) implies that ( g ⊕ i , [diag( g i ) , diag( g i )]) = ( g ⊕ i , diag( g i )). It follows that ( g ⊕ i , diag( g i )) isstrictly indecomposable if and only if it is indecomposable. However, since diag( g i ) is simple, thedecomposability of ( g ⊕ i , diag( g i )) would entail diag( g i ) being contained in a proper ideal of g ⊕ i .This is not possible, meaning that ( g ⊕ i , diag( g i )) is indeed strictly indecomposable. The proof iscomplete. (cid:3) Remark 32. One immediate consequence is that every indecomposable symmetric pair ( g , h ) isstrictly indecomposable. This is not true of an arbitrary reductive spherical pair ( g , h ) (see Example36).Together with Lemma 21, Lemma 31 reduces the classification of a -regular symmetric pairs tothe classification of a -regular, strictly indecomposable symmetric pairs. Let ( g , h ) be a pair of thelatter sort, and let ( G, H ) denote an associated pair of groups. Let us also consider an involutiveautomorphism θ : g → g satisfying h = g θ . This forms part of the eigenspace decomposition g = h ⊕ q ,where q ⊆ g is the − θ . One can then find a maximal abelian subspace c ⊆ q , meaningthat c is a vector subspace of q that is maximal with respect to the following condition: c is abelianand consists of semisimple elements in g (cf. [33, Corollary 37.5.4]).Now recall our discussion of the generic stabilizer H ∗ ⊆ H and its Lie algebra h ∗ ⊆ h (see 5.2 and5.3). At the same time, let z h ( Y ) denote the subalgebra of all x ∈ h that commute with every vectorin a subset Y ⊆ g . Lemma 33. We have h ∗ = z h ( c ) .Proof. The H -module isomorphisms h ⊥ ∼ = g / h ∼ = q imply that H ∗ is a generic stabilizer for the H -action on q . Note also that H · c ⊆ q is dense (see [33, Lemma 38.7.1]) and constructible. It followsthat h ∗ = z h ( c ) for all c in an open dense subset c ⊆ c . At the same time, there exists an open densesubset c ⊆ c with the property that z h ( c ) = z h ( c ) for all c ∈ c (see the proof of [33, Proposition38.4.5]). Hence h ∗ = z h ( c ), as desired. (cid:3) Remark 34. With Remark 16 in mind, one can phrase Lemma 33 as follows: z h ( c ) represents theconjugacy class of Lie algebras of generic stabilizers for the H -action on h ⊥ .We now explain the classification of a -regular, strictly indecomposable symmetric pairs ( g , h ). Upto conjugation (see Definition 25), such pairs are parametrized by Satake diagrams (see [34, Section26.5]). The Satake diagram for a symmetric pair ( g , h ) is the Dynkin diagram of g , together withextra decorations that encode the associated involution θ : g → g . Part of this decoration consists of painting some of the nodes black; these are precisely the simple roots of z h ( c ). At the same time,recall that Lemma 33 identifies z h ( c ) with h ∗ . Appealing to Corollary 17, we see that the a -regularityof ( g , h ) is equivalent to the Satake diagram of ( g , h ) having none of its nodes painted black. Thisleads to the following result. Proposition 35. A strictly indecomposable symmetric pair ( g , h ) is a -regular if and only if it isconjugate to one of the pairs in Table III. In this table, s denotes any simple Lie algebra. g h sl n so n sl n +1 sl n +1 ⊕ sl n ⊕ C sl n sl n ⊕ sl n ⊕ C so n +1 so n +1 ⊕ so n so n so n ⊕ so n so n so n − ⊕ so n +1 sp n gl n e sp e sl ⊕ sl e sl e so f sp ⊕ sl g sl ⊕ sl s ⊕ s diag( s ) TABLE III. The embeddings h ⊆ g are obtained from [21, Table 1], which describeseach embedding on the level of algebraic groups. Proof. Following the discussion above, we only need to list the symmetric pairs whose Satake diagramshave no black nodes. These diagrams can be found in [34, Table 26.3], and the result follows froman inspection of this table. (cid:3) Reductive spherical subgroups. Using the notation in 5.1 and the introduction of Section 3,we additionally assume that ( G, H ) and ( g , h ) are reductive spherical pairs . This means that H isa reductive spherical subgroup of G , i.e. H is reductive and B has an open orbit in G/H . Notethat this is equivalent to h being a reductive subalgebra of g satisfying e b + h = g for some Borelsubalgebra e b ⊆ g (see [34, Section 25.1]). We shall sometimes also require ( G, H ) and ( g , h ) to benon-symmetric, noting that the classification in 5.5.2 renders this a harmless assumption. Example 36. In contrast to the situation considered in Remark 32, an indecomposable reductivespherical pair need not be strictly indecomposable. Set g = sl n +1 ⊕ sl and let h ⊆ g be the image of sl n ⊕ C → g , ( A, t ) (diag( A + tI n , − nt ) , diag( t, − t )) , ( A, t ) ∈ sl n ⊕ C . This is an indecomposable spherical pair, but it is not strictly indecomposable. Remark 37. The strictly indecomposable reductive spherical pairs ( G, H ) have been classified byKr¨amer [21] for G simple, and by Mikityuk [28] and Brion [7] for G semisimple. YPERK ¨AHLER SLICES AND SPHERICAL GEOMETRY 21 We begin by assuming that our reductive spherical pair ( G, H ) is strictly indecomposable. Nownote that Lemma 27 allows us to investigate a -regularity via Λ + ( G, H ), and the case-by-case analysesof [21] thereby become important. The aforementioned reference gives explicit semigroup generatorsof Λ + ( G, H ) if G is simple. If G is only semisimple, then a description of Λ + ( G, H ) can be obtainedfrom [3, Table 1] as follows. If h has a trivial center, then generators of Λ + ( G, H ) are given in [3, Table1]. If h has a non-trivial center, then [3, Table 1] provides a finite set { ( λ , χ ) , . . . , ( λ s , χ s ) } ofgenerators for the so-called extended weight semigroup of ( G, H ). The λ i are dominant weights for G and the χ i are characters of H . The weight semigroup Λ + ( G, H ) identifies with the collectionof all points in the extended weight semigroup that have the form ( λ, λ belongs to Λ + ( G, H ) if and only if λ = P si =1 n i λ i forsome non-negative integers n i satisfying P si =1 n i χ i = 0. Together with an inspection of [21, Table 1]and [3, Table 1], this discussion yields the following fact. Lemma 38. If ( G, H ) is a strictly indecomposable reductive spherical pair that is not symmetric,then ( G, H ) is a -regular if and only if ( g , h ) is conjugate to a pair in Table IV. g h sl n +1 sl n +1 ⊕ sl n sl n +1 sp n ⊕ C sl n +1 sp n so n +1 gl n sl n +1 ⊕ sl n sl n ⊕ C so n +1 ⊕ so n so n sl n ⊕ sp m gl n − ⊕ sl ⊕ sp m − ( n = 3 , , , m = 1 , sl n ⊕ sp m sl n − ⊕ sl ⊕ sp m − ( n = 3 , , m = 1 , sp m ⊕ sp n sp n − ⊕ sp ⊕ sp m − ( m, n = 1 , sp n ⊕ sp sp n − ⊕ sp ( n = 3 , sp ℓ ⊕ sp m ⊕ sp n sl ℓ − ⊕ sp m − ⊕ sp n − ⊕ sp ( ℓ, m, n = 1 , sp n ⊕ sp ⊕ sp m sp n − ⊕ sp ⊕ sp ⊕ sp n − ( n, m = 1 , TABLE IV. The embeddings h ⊆ g are obtained from [21, Table 1] and [7, Theorem0], which describe each embedding on the level of algebraic groups.Together with Proposition 35, this result classifies the a -regular, strictly indecomposable reductivespherical pairs. A natural next step is to study the indecomposable reductive spherical pairs thatare a -regular, for which we need the following lemma. Lemma 39. Let ( g , h ) be a strictly indecomposable reductive spherical pair. If ( g , [ h , h ]) is a -regular,then ( g , h ) is a -regular.Proof. The statement is obviously true if h is semisimple, so we assume h to be non-semisimple. Letus prove the statement by contraposition, assuming that ( g , h ) is a strictly indecomposable reductive spherical pair that is not a -regular. At the same time, h being non-semisimple and an inspectionof [34, Table 26.3], [21, Table 1], and [3, Table 1] reveal that ( g , h ) is conjugate to one of the pairsin Table V below. It therefore suffices to prove the following claim: if ( g , h ) is conjugate to a pair inTable V, then ( g , [ h , h ]) is not a -regular. g h sl p + q ( | p − q | > sl p ⊕ sl q ⊕ C so n gl n e so (10) ⊕ C e e ⊕ C sp n ( n > sp n − ⊕ C so so ⊕ so sl n ⊕ sp m ( n > m > gl n − ⊕ sl ⊕ sp m − TABLE V. The embeddings h ⊆ g are as described in [21, Table 1] and [7, Theorem0], where they are given as embeddings of the corresponding algebraic groups.Suppose that ( g , h ) is conjugate to a pair in lines 1,2,3, or 7 of Table V. It then follows that( g , [ h , h ]) is a strictly indecomposable reductive spherical pair, as it appears in at least one of theclassifications of Kr¨amer [21], Mikityuk [28], and Brion [7]. At the same time, one can verify that( g , [ h , h ]) is not conjugate to a pair in Table III or Table IV. Proposition 35 and Lemma 38 then implythat ( g , [ h , h ]) is not a -regular.Now assume that ( g , h ) is conjugate to one of the remaining pairs in Table V. Let ( G, H ) be acorresponding reductive spherical pair of groups, and let us take G to be simply-connected. We notethat [34, Table 10.2] then provides explicit generators of Λ + ( G, [ H, H ]). It is now straightforward toapply Lemma 27 and conclude that ( g , [ h , h ]) is not a -regular. (cid:3) We now study the a -regular, indecomposable reductive spherical pairs. Let ( g , h ) be an indecom-posable reductive spherical pair and note that ( g , [ h , h ]) has the following form (cf. Remark 22):( g , [ h , h ]) = (cid:18) n M i =1 g i , n M i =1 e h i (cid:19) , where for all i ∈ { , . . . , n } , e h i is a semisimple ideal in [ h , h ], g i is a reductive ideal in g containing e h i ,and ( g i , e h i ) is indecomposable. Note that each pair ( g i , e h i ) is actually strictly indecomposable, owingto the fact that e h i is semisimple.Let π i : g → g i denote the projection onto the i th factor and set z i := π i ( z ( h )), where z ( h ) is thecenter of h . It is clear that z i is reductive and that it commutes with e h i , from which we concludethat h i := e h i ⊕ z i ⊆ g i is a reductive subalgebra. Now set h := n M i =1 h i ⊆ g . It follows by construction that [ h , h ] ⊆ h and z ( h ) ⊆ L ni =1 z i ⊆ h , implying that h ⊆ h and e b + h ⊆ e b + h for any Borel subalgebra e b ⊆ g . Since ( g , h ) is a reductive spherical pair, the previous sentence shows( g , h ) to be a reductive spherical pair. Our next result establishes that ( g i , h i ) is a reductive sphericalpair for all i ∈ { , . . . , n } . YPERK ¨AHLER SLICES AND SPHERICAL GEOMETRY 23 Lemma 40. Let ( g , h ) be an indecomposable reductive spherical pair and use the notation from above.Then ( g i , h i ) is a strictly indecomposable reductive spherical pair for all i ∈ { , . . . , n } .Proof. Since ( g , h ) is spherical, there exists a Borel subalgebra e b ⊆ g satisfying e b + h = g . Thedecomposition g = g ⊕ · · · ⊕ g n gives rise to a decomposition of the form e b = b ⊕ · · · ⊕ b n , where b i is a Borel subalgebra of g i for all i ∈ { , . . . , n } . Now note that b i + h i = g i for all i ∈ { , . . . , n } ifand only if e b + h = g . Recalling that ( g , h ) is a reductive spherical pair, the previous sentence impliesthat ( g i , h i ) is a reductive spherical pair for all i ∈ { , . . . , n } .To complete the proof, we observe that [ h i , h i ] = e h i for all i ∈ { , . . . , n } . The strict indecompos-ability of ( g i , h i ) thus follows from the indecomposability of ( g , e h i ). (cid:3) We may now relate the a -regularity of ( g , h ) to that of ( g , h ). Proposition 41. Let ( g , h ) be an indecomposable reductive spherical pair and use the notation fromabove. Then ( g , h ) is a -regular if and only if ( g , h ) is a -regular.Proof. The inclusion of subalgebras [ h , h ] ⊆ h ⊆ h implies the inclusion of Cartan spaces a ( g , h ) ⊆ a ( g , h ) ⊆ a ( g , [ h , h ]), from which we deduce the backward implication.For the forward implication, suppose that ( g , h ) is a -regular. The inclusion a ( g , h ) ⊆ a ( g , [ h , h ])then shows ( g , [ h , h ]) to be a -regular, which is equivalent to all of the strictly indecomposable pairs( g i , e h i ) being a -regular (see Lemma 21). Since ( g i , h i ) is a strictly indecomposable reductive sphericalpair (see Lemma 40) with [ h i , h i ] = e h i , Lemma 39 implies that ( g i , h i ) must be a -regular. It thenfollows from Lemma 21 that ( g , h ) is a -regular. (cid:3) We now connect this discussion of a -regularity for indecomposable reductive spherical pairs to theoverarching objective — a classification of a -regular reductive spherical pairs. The following lemmais a crucial step in this direction. Lemma 42. If h is any reductive subalgebra of g , then ( g , h ) is a reductive spherical pair if and onlyif there exist indecomposable reductive spherical pairs ( g i , h i ) , i ∈ { , . . . , n } , such that g i (resp. h i )is an ideal in g (resp. h ) for all i and ( g , h ) = (cid:18) n M i =1 g i , n M i =1 h i (cid:19) . Proof. By virtue of Remark 22, one can find indecomposable pairs ( g i , h i ) satisfying the above-advertised properties. The proof then becomes entirely analogous to that of Lemma 40. (cid:3) The classification of a -regular reductive spherical pairs is now described as follows. By virtueof Lemmas 21 and 42, it suffices to classify the indecomposable reductive spherical pairs that are a -regular. We thus suppose that ( g , h ) is any indecomposable reductive spherical pair. If ( g , h ) isstrictly indecomposable, then it is a -regular if and only if it is conjugate to a pair in Table III orTable IV. If ( g , h ) is not strictly indecomposable, then we consider the associated pair ( g , h ). The a -regularity of ( g , h ) is then equivalent to that of ( g , h ) (see Proposition 41). This is in turn equivalentto every strictly indecomposable pair ( g i , h i ) being a -regular (see Lemma 21), which can be assessedvia Tables III and IV. Remark 43. One might ask about the feasibility of classifying the a -regular reductive sphericalpairs ( G, H ) satisfying c G ( G/H ) > 0. The complexity-one case might be tractable, largely becausethe papers [2] and [31] classify all strictly indecomposable reductive spherical pairs ( G, H ) with c G ( G/H ) = 1. One can thereby determine which of the strictly indecomposable, complexity-onepairs are a -regular. In analogy with 5.5.2 and 5.5.3, this might imply a classification of all reductivespherical ( G, H ) with c G ( G/H ) = 1. The case of c G ( G/H ) > References [1] Abe, H., and Crooks, P. Hessenberg varieties, Slodowy slices, and integrable systems. arxiv:1807.07792 (2018),36pp. To appear in Math. Z .[2] Arzhantsev, I. V., and Chuvashova, O. V. Classification of affine homogeneous spaces of complexity one. Mat.Sb. 195 , 6 (2004), 3–20.[3] Avdeev, R. S. Extended weight semigroups of affine spherical homogeneous spaces of nonsimple semisimplealgebraic groups. Izv. Ross. Akad. Nauk Ser. Mat. 74 , 6 (2010), 3–26.[4] Bielawski, R. Hyperk¨ahler structures and group actions. J. London Math. Soc. (2) 55 , 2 (1997), 400–414.[5] Bielawski, R. Slices to sums of adjoint orbits, the Atiyah-Hitchin manifold, and Hilbert schemes of points. Complex Manifolds 4 (2017), 16–36.[6] Biquard, O. Sur les ´equations de N¨ahm et la structure de Poisson des alg`ebres de Lie semi-simples complexes. Math. Ann. 304 , 2 (1996), 253–276.[7] Brion, M. Classification des espaces homog`enes sph´eriques. Compositio Math. 63 , 2 (1987), 189–208.[8] Crooks, P. An equivariant description of certain holomorphic symplectic varieties. Bull. Aust. Math. Soc. 97 , 2(2018), 207–214.[9] Crooks, P., and Rayan, S. Abstract integrable systems on hyperk¨ahler manifolds arising from Slodowy slices.arxiv:1706.05819 (2017), 17pp. To appear in Math. Res. Lett .[10] Dancer, A., and Swann, A. Hyperk¨ahler metrics associated to compact Lie groups. Math. Proc. CambridgePhilos. Soc. 120 , 1 (1996), 61–69.[11] Dynkin, E. B. Semisimple subalgebras of semisimple Lie algebras. Transl., Ser. 2, Am. Math. Soc. 6 (1957),111–243.[12] `Elaˇsvili, A. G. Canonical form and stationary subalgebras of points in general position for simple linear Liegroups. Funkcional. Anal. i Priloˇzen. 6 , 1 (1972), 51–62.[13] `Elaˇsvili, A. G. Stationary subalgebras of points of general position for irreducible linear Lie groups. Funkcional.Anal. i Priloˇzen. 6 , 2 (1972), 65–78.[14] Hitchin, N. J. The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3) 55 , 1 (1987), 59–126.[15] Hitchin, N. J., Karlhede, A., Lindstr¨om, U., and Roˇcek, M. Hyperk¨ahler metrics and supersymmetry. Comm. Math. Phys. 108 , 4 (1987), 535–589.[16] Knop, F. Weylgruppe und Momentabbildung. Invent. Math. 99 , 1 (1990), 1–23.[17] Knop, F. The asymptotic behavior of invariant collective motion. Invent. Math. 116 , 1-3 (1994), 309–328.[18] Kostant, B. Lie group representations on polynomial rings. Amer. J. Math. 85 (1963), 327–404.[19] Kovalev, A. G. N¨ahm’s equations and complex adjoint orbits. Quart. J. Math. Oxford Ser. (2) 47 , 185 (1996),41–58.[20] Kr¨amer, M. Eine Klassifikation bestimmter Untergruppen kompakter zusammenh¨angender Liegruppen. Comm.Algebra 3 , 8 (1975), 691–737.[21] Kr¨amer, M. Sph¨arische Untergruppen in kompakten zusammenh¨angenden Liegruppen. Compositio Math. 38 , 2(1979), 129–153.[22] Kronheimer, P. A hyperk¨ahler structure on the cotangent bundle of a complex Lie group. arXiv:math/0409253(2004), 11 pp.[23] Kronheimer, P. B. A hyperk¨ahlerian structure on coadjoint orbits of a semisimple complex group. J. LondonMath. Soc. (2) 42 , 2 (1990), 193–208.[24] Kronheimer, P. B. Instantons and the geometry of the nilpotent variety. J. Differential Geom. 32 , 2 (1990),473–490.[25] Losev, I. V. Computation of Cartan spaces for affine homogeneous spaces. Mat. Sb. 198 , 10 (2007), 31–56.[26] Marsden, J. E., Ratiu, T., and Raugel, G. Symplectic connections and the linearisation of Hamiltoniansystems. Proc. Roy. Soc. Edinburgh Sect. A 117 , 3-4 (1991), 329–380.[27] Mayrand, M. Stratified hyperk¨ahler spaces from semisimple Lie algebras. arXiv:1709.09126 (2017), 19 pp. Toappear in Transformation Groups.[28] Mikityuk, I. V. Integrability of invariant Hamiltonian systems with homogeneous configuration spaces. Mat. Sb.(N.S.) 129(171) , 4 (1986), 514–534, 591.[29] Nakajima, H. Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras. Duke Math. J. 76 , 2 (1994),365–416.[30] Nakajima, H. Quiver varieties and Kac-Moody algebras. Duke Math. J. 91 , 3 (1998), 515–560.[31] Panyushev, D. I. Complexity and rank of actions in invariant theory. J. Math. Sci. (New York) 95 , 1 (1999),1925–1985. Algebraic geometry, 8.[32] Richardson, Jr., R. W. Principal orbit types for algebraic transformation spaces in characteristic zero. Invent.Math. 16 (1972), 6–14. YPERK ¨AHLER SLICES AND SPHERICAL GEOMETRY 25 [33] Tauvel, P., and Yu, R. W. T. Lie algebras and algebraic groups . Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2005.[34] Timashev, D. A. Homogeneous spaces and equivariant embeddings , vol. 138 of Encyclopaedia of MathematicalSciences . Springer, Heidelberg, 2011. Invariant Theory and Algebraic Transformation Groups, 8.(Crooks) Department of Mathematics, Northeastern University, 360 Huntington Ave., Boston, MA02115, USA E-mail address : [email protected] (van Pruijssen) Institut f¨ur Mathematik, University of Paderborn, Warburger Str. 100, 33098 Pader-born, Germany E-mail address ::