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APPEARED IN BULLETIN OF THE

arXiv:math/9204227v1 [math.RT] 1 Apr 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 26, Number 2, April 1992, Pages 269-275NILPOTENT ORBITS, NORMALITY,AND HAMILTONIAN GROUP ACTIONSRanee Brylinski and Bertram KostantAbstract. Let M be a G-covering of a nilpotent orbit in g where G is a complexsemisimple Lie group and g = Lie(G).

We prove that under Poisson bracket the spaceR[2] of homogeneous functions on M of degree 2 is the unique maximal semisimpleLie subalgebra of R = R(M) containing g. The action of g′ ≃R[2] exponentiates toan action of the corresponding Lie group G′ on a G′-cover M′ of a nilpotent orbit ing′ such that M is open dense in M′. We determine all such pairs (g ⊂g′).The theory of coadjoint orbits of Lie groups is central to a number of areas inmathematics.

A list of such areas would include (1) group representation theory,(2) symmetry-related Hamiltonian mechanics and attendant physical theories, (3)symplectic geometry, (4) moment maps, and (5) geometric quantization.Frommany points of view the most interesting cases arise when the group G in questionis semisimple. For semisimple G the most familar of the orbits are of semisimpleelements.

In that case the associated representation theory is pretty much under-stood (Borel-Weil-Bott and noncompact analogues, e.g., Zuckerman functors). Theisotropy subgroups are reductive and the orbits are in one form or another relatedto flag manifolds and their natural generalizations.A totally different experience is encountered with nilpotent orbits of semisimplegroups.

Here the associated representation theory (unipotent representations) ispoorly understood and there is a loss of reductivity of isotropy subgroups.Tomake matters worse (or really more interesting) orbits are no longer closed andthere can be a failure of normality for orbit closures. In addition simple connectivityis generally gone but more seriously there may exist no invariant polarizations.The interest in nilpotent orbits of semisimple Lie groups has increased sharplyover the last two decades.

This perhaps may be attributed to the reoccuring experi-ence that sophisticated aspects of semisimple group theory often leads one to theseorbits (e.g., the Springer correspondence with representations of the Weyl group).In this note we announce new results concerning the symplectic and algebraicgeometry of the nilpotent orbits O and the symmetry groups of that geometry. Thestarting point is the recognition (made also by others) that the ring R of regularfunctions on any G-cover M of O is not only a Poisson algebra (the case for anycoadjoint orbit) but that R is also naturally graded.

The key theme is that the1991 Mathematics Subject Classification. Primary 22E46; Secondary 58F05, 58F06, 32M05,14L30.Received by the editors March 29, 1991 and, in revised form, May 29, 1991The first author is an Alfred P. Sloan FellowThe work of the second author was supported in part by NSF Grant DMS-8703278c⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2RANEE BRYLINSKI AND BERTRAM KOSTANTsame nilpotent orbit may be “shared” by more than one simple group, and thekey result is the determination of all pairs of simple Lie groups having a sharednilpotent orbit. Furthermore there is then a unique maximal such group and thisgroup is encoded in the symplectic and algebraic geometry of the orbit.

Remarkablya covering of nilpotent orbit of a classical group may “see” an exceptional Lie groupas the maximal symmetry group of this sympectic manifold. A beautiful instanceof this is that G2 is the symmetry group of the simply connected covering of themaximal nilpotent orbit of SL(3, C) and that this six-dimensional space “becomes”the minimal nilpotent orbit of G2.Our work began with a desire to thoroughly investigate a striking discovery ofLevasseur, Smith, and Vogan.

They found that the failure of the closure of theeight-dimensional nilpotent orbit of G2 to be a normal variety may be “remedied”by refinding this orbit as the minimal nilpotent orbit of SO(7, C). The failure hasa lot to do with the seven-dimensional representation of G2.

In general given Mwe have found that there exists a unique minimal representation π (containing theadjoint) wherein M may be embedded with normal closure. It was the study of πthat led to the discovery of the maximal symmetry group G′.

Using a new generaltransitivity result for coadjoint orbits we prove that, modulo a possible normalHeisenberg subgroup (and that occurs in only one case), G′ is semisimple.Past experience has shown that the action of a subgroup H on a coadjoint orbitof G is a strong prognostigator as to how the corresponding representation L of Gdecomposes under H. If this continues to hold for unipotent representations ourclassification result should yield all cases where L remains irreducible (or decom-poses finitely) under a semisimple subgroup.1. The maximal symmetry group and “shared” orbitsLet G be a simply connected complex semisimple Lie group and g the Lie algebraof G. Let e ∈g be nilpotent and assume (for simplicity of exposition but with noreal loss) that e has nonzero projection in every simple component of g. Let O bethe adjoint orbit of e and let ν : M −→O be a G-covering.

Let R = R(M) bethe ring of regular functions on M and let R[g] be the copy of g in R defining themoment map M −→g∗. Identify g∗≃g in a G-equivariant way.Now R carries a G-invariant ring grading R = Lk≥0 R[k] (k ∈Z) such thatR[g] ⊂R[2].

Then R[0] = C. The Poisson bracket satisfies [R[k], R[l]] ⊂R[k+l−2],for all k, l (see also [7]). Hence R[2] is a finite-dimensional Lie subalgebra of R underPoisson bracket.

Our first main result isTheorem 1. (i) R[2] is a semisimple Lie algebra, call it g′.

If g is simple then g′ is simple . (ii) The condition R[g] ⊂R[2] determines the G-invariant ring grading on Runiquely.

(iii) R[2] + R[1] + R[0] is the unique maximal finite-dimensional Lie subalgebraof R containing R[g].In Table 1 all the Lie algebras are complex simple; particularly, in (2) and (3)n ≥2 and in (6) n ≥3. The last column V is a representation of g, written asa sum of its irreducible components (only fundamental dominant representationsoccur here).

In (3) V ⊕L = V2 C2n where L ≃C is sp(2n)-invariant. In (8) V isthe sum of the standard and the two half-spin representations.

NILPOTENT ORBITS, NORMALITY, AND HAMILTONIAN GROUP ACTIONS3Table 1gg′V(1)G2so(7)C7(2)so(2n + 1)so(2n + 2)C2n+1(3)sp(2n)sl(2n)(∧2C2n)/C(4)F4E6C26(5)sl(3)G2C3 ⊕∧2C3(6)so(2n)so(2n + 1)C2n(7)so(9)F4C16(8)so(8)F4C8 ⊕C8 ⊕C8(9)G2so(8)2C7Theorem 2. Table 1 gives a complete list of the simple Lie algebra pairs (g ⊂g′)that arise in Theorem 1 with g ̸= g′.

Furthermore g′ = g ⊕V as g-modules.Now the Poisson bracket on R defines an alternating bilinear form β on R[1] anda Lie algebra homomorphism R[2] →sp β.Theorem 3. β is a symplectic form on R[1] so that R[1] ⊕R[0] is a HeisenbergLie algebra. If g is simple then R[1] ̸= 0 in one and only one case, namely, wheng is of type Cn for some n and M is the simply connected (double covering) of theminimal nontrivial nilpotent orbit of g. In that case R[2] ≃sp(2n) and R[1] ≃C2ngenerates R freely.Now the functions in g′ ≃R[2] define a map φ: M −→g′ (again identify (g′)∗≃g′ ).

Let G′ be the simply connected Lie group with Lie algebra g′. The next resultsays that up to birationality φ is a moment map for G′.Theorem 4.

The image φ(M) lies in a nilpotent orbit O′ of G′ and φ(M) isZariski open dense in O′. There exists a unique G′-covering ν′ : M ′ −→O′ suchthat M ′ contains M and ν′ extends φ.

Moreover M and M ′ have the same regularfunctions, that is, R(M) = R(M ′), and also the same fundamental groups, that is,π1(M) ≃π1(M ′).We can construct M ′ in the following way. Given M, let X = Spec R be themaximal ideal spectrum of the finitely generated C-algebra R. Then X is a normalaffine variety.

Furthermore X contains M as an open dense subset. We call X thenormal closure of M. Indeed if M = O and O is normal then X = O.Our construction is: G′ acts on X and M ′ is the unique Zariski open orbit ofG′ on X.

Note that M is the unique Zariski open orbit of G on X so that clearlyM ⊂M ′ ⊂X.Thus the pair (M ⊂M ′) constitutes an orbit (cover) “shared” between G andG′ . Moreover M and M ′ have the same normal closure so that X is exactly sharedbetween G and G′.Now even though X may be singular we will say an isomorphism of X is symplec-tic if the corresponding automorphism of R preserves the Poisson bracket structure.Theorem 5.

Assume R[1] = 0. Then X is a singular variety.

The action of anyconnected Lie group of holomorphic symplectic isomorphisms of X that extends theaction of G is given by a subgroup of G′.

4RANEE BRYLINSKI AND BERTRAM KOSTANTHence, assuming g is simple, the example of Theorem 3 is the one and only onechoice of M such that X is smooth, and in that case one has X ≃C2n.2. Explanation of the tableWe now describe for each of the nine cases in Table 1 a choice of M such that(M, g) gives rise to (M ′, g′).

In each case M ′ = O′ is the orbit of the highest rootvector in the simple Lie algebra g′, this is the minimal nontrivial nilpotent orbit(cf. the next section).In cases (1)–(4) g is any one of the four simple Lie algebras that are doublylaced (i.e., having two root lengths).

Choose O to be the orbit of a short rootvector and let M be its simply connected cover. Then M = O in (1) while M isa two-fold cover of O in (2)–(4).

Then g′ is simply laced . Furthermore V = Vαis the irreducible g-representation with highest weight α equal to the highest shortroot of g. Case (1) is a restatement in our language of a result proved by Levasseurand Smith [4] in answer to a conjecture of Vogan [6] (see introduction).In (5) choose O to be the six-dimensional (maximal) nilpotent orbit of all princi-pal nilpotent elements and let M be the three-fold simply connected covering spaceof O.

This case was discovered by us in collaboration with Vogan. A noncommu-tative analog of this example is given in a result of Zahid [8] .In (6) choose O to be the nilpotent orbit of Jordan type (see e.g., [3]) (3, 12n−3)and let M be the simply connected double cover of O.

In (7) choose O to be ofJordan type (24, 1) and let M be the simply connected (double cover) of O. In(8) choose O to be of Jordan type (3, 22, 1) and let M be the simply connected(four-fold) cover of O.In (9) choose O to be the unique ten-dimensional nilpotent orbit and let M bethe simply connected six-fold cover of O (π1(O) ≃S3).

In this example Levasseurand Smith already showed in [4] that G has an open dense orbit on O′, again inresponse to a question of Vogan.Moreover Vogan has constructed a unipotentrepresentation theoretic analogue of the example. If π denotes the minimal unitaryrepresentation of SO(4, 4) (see e.g., [2]) then π extends to the outer automorphismgroup A of g and in particular to a group S ≃S3 that induces A. Vogan showsthat a split form Go of G2 and S behave like a Howe pair with respect to π andthat π|Go decomposes into six irreducible components.

Furthermore McGovern in[5, Theorem 4.1] has constructed a Dixmier algebra analogue of this exampleRemarkably three of the four nilpotent orbits of G2 have now appeared as“shared” orbits (the principal orbit does not appear).Cases (5)–(8) are precisely the the pairs (g ⊂g′) in the table with rank g=rank g′. Each pair (g ⊂g′) is of the form (s0 ⊂s) where s is a doubly lacedsimple Lie algebra and s0 is a subalgebra of s containing a Cartan subalgebra ofs and all associated long root vectors.

Moreover every such pair (s0 ⊂s) wheres0 is simple arises in (5)–(8). On the other hand the pairs (s0 ⊂s) where s0 isnonsimple occur precisely when s ≃sp(2n) and s0 ≃sp(2n1) ⊕· · · ⊕sp(2nk) wheren1 + · · · + nk = n and k ≥2.

These pairs do arise in Theorem 1 (when M is chosenso that in each simple component one has the example of Theorem 3) and thesetogether with (5)–(8) exhaust all equal rank pairs (g ⊂g′) arising in Theorem 1such that g′ is simple.A general result regarding the ranks of g and g′ is that rankg′ > rank g wheneverM = O and g ̸= g′.

NILPOTENT ORBITS, NORMALITY, AND HAMILTONIAN GROUP ACTIONS5Two instances of a triple of Lie algebras having a “shared” orbit can be foundamong these examples, namely, (a) so(8) ⊂so(9) ⊂f4 (where f4 is of type F4) and(b) g2 ⊂so(7) ⊂so(8). This is not unexpected by the theory since we in fact provethat if h is any Lie subalgebra between g and g′ then h is semisimple and also h issimple if g is simple.

Moreover if H is the simply connected group correspondingto h then the statements made in Theorem 4 and immediately afterward for G′ andX apply equally well to H and X. In particular one has a unique open H-orbit M hin X, M ⊂M h ⊂M ′, and M h covers H-equivariantly a nilpotent H-orbit Oh ⊂h.Furthermore the whole graded Poisson ring structure on R arising from M and Ois the same as would arise from M h and Oh.

In particular the maximal semisimpleLie algebra R[2] remains the same.3. Methods of proofTwo key ideas are used in proving the classification of pairs.The first is arepresentation theoretic.

We are able to compute g′ ≃R[2] as a g-module (forarbitrary g and M). Let ε be a point of M lying over e and let (h, e, f ) be astandard basis of an sl(2) subalgebra a of g. For any g-module V let V [2] be the2-eigenspace of h in the fixed space (V ∗)Gε.

Then g′ ≃g ⊕niV1 ⊕· · · ⊕nsVs whereV1, . .

. , Vs is a complete list of inequivalent simple g-modules, excluding componentsof the adjoint representation, such that ni = dim Vi[2] is nonzero.

This follows byrecognizing that the grading on R comes from exponentiating a natural action of aCartan subalgebra in a .The second idea is due to David Vogan who observed that for a given pair (g, g′)arising in Theorem 1 we may change (if necessary) the choice of O and M so thatO′ is minimal . Vogan himself has determined the pairs (g, g′) in many of the caseslisted above.A principle used in setting up the theory is that one should study minimalembeddings of X in order to study M (again M is arbitary).

We prove the coveringmap ν extends to a finite G-morphism ν : X →O and then the fiber ν−1(0) overzero is a single point o. Then o is the unique G-fixed point in X and the maximalideal of R corresponding to o is m = L∞k=1 R[k].

Furthermore, X is singular ifand only if X is singular at o and then o is the “most” singular point of X (cf.Theorems 3 and 5).We say that the pair (v, V ) defines an embedding of X in case V is a G-moduleand v ∈V Gε (where ε ∈M) is such that the natural map M →G·v extends to a G-isomorphism X →G · v. We find that the Zariski tangent space To(X) = (m/m2)∗at o provides a minimal embedding for X.Theorem 6. There exists a vector u ∈To(X)Gε such that (i) (u, To(X)) definesan embedding of X and (ii) if (v, V ) is any pair that defines an embedding of Xthen there exists a surjective G-map τ : V →To(X) such that τ(v) = u.Clearly one has an injection R[2] →m/m2 when R[1] = 0.Regarding thenormality of O we find that O is a normal variety if and only if as G-modulesR[g] ≃R[2] ≃m/m2.The proofs of Theorems 1, 3, and 5 are all applications of the following generaltransitivity theorem for coadjoint orbits of a special kind of Lie algebra.

This isone of our main results.

6RANEE BRYLINSKI AND BERTRAM KOSTANTTheorem 7. Assume that s is a finite-dimensional Lie algebra over R or C andthat s is a semidirect sum s = r + u where u is an abelian ideal in s, r is a semisimpleLie subalgebra of s, and ur = 0.

One may regard s∗= r∗+ u∗in an obvious way.Let γ ∈s∗and write γ = µ + λ with µ ∈r∗and λ ∈u∗. Then one has s · γ = r · γif and only if λ = 0.Theorem 7 says that if λ ̸= 0 then the subgroup of Ad s corresponding to rcannot operate transitively (even infinitesmally) at γ on the coadjoint orbit.4.

An application to symmetry of flag varietiesFinally we give an application of our results to a well-known problem in geometry.If P is a parabolic Lie subgroup of a simple Lie group G it is a solved problem (see[1]) to determine the connected component of the full group F of holomorphicautomorphisms of the projective variety G/P. It is precisely for the G given incases (1), (2), and (3) in the table that P exists so that F is larger than that givenby the action of G. Furthermore in those cases F is in fact given by the action ofG′.

We obtain a stronger statement (and recover the known result) inTheorem 8. Let P be any parabolic subgroup of G and choose M to be the uniqueopen orbit in T ∗(G/P) so that we can take O to be the G-orbit in g of a Richard-son element in the nilradical of Lie P.Then one has a desingularization mapT ∗(G/P) →X and the pullback of R is the full ring of regular functions onT ∗(G/P).Furthermore the action of G′ on X lifts uniquely to an action as agroup of symplectic holomorphic automorphisms of T ∗(G/P) and as such G′ ismaximal.

G′ preserves the cotangent polarization of T ∗(G/P) so that G′ acts onG/P and hence there exists a parabolic subgroup P ′ ⊂G′ such that G/P = G′/P ′.The action of G′ on G/P is the connected component of the group of all holomor-phic automorphisms of G/P. Consequently any connected Lie group of symplecticholomorphic automorphisms of T ∗(G/P) containing the action of G automaticallypreserves the cotangent space polarization of T ∗(G/P) and consequently will act asa group of holomorphic automorphisms of G/P.AcknowledgmentsThe authors thank Madhav Nori and David Vogan for helpful discussions.

Thefirst author thanks the Laboratoire Math´ematiques Fondamentales of Universit´eParis VI for its hospitality.References1. M. Demazure, Automorphismes et d´eformations des vari´et´es de Borel, Invent.

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NILPOTENT ORBITS, NORMALITY, AND HAMILTONIAN GROUP ACTIONS76. D. A. Vogan, The orbit method and primitive ideals for semisimple Lie algebras, LieAlgebras and Related Topics, CMS Conf.

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France117 (1989), 451–477.Department of Mathematics, Pennsylvania State University, University Park,Pennsylvania 16802Department of Mathematics, Massachusetts Institute of Technology, Cambridge,Massachusetts 02139

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