Transformation Groups, Vol. 6, No. 2, 2001, pp QBirkh~user Boston (2001) INVARIANT LINEAR CONNECTIONS ON HOMOGENEOUS SYMPLECTIC VARIETIES

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1 Transformation Groups, Vol. 6, No. 2, 2001, pp QBirkh~user Boston (2001) INVARIANT LINEAR CONNECTIONS ON HOMOGENEOUS SYMPLECTIC VARIETIES S. V. PIKULIN E. A. TEVELEV* Department of Mathematics Chair of Algebra Moscow State University Moscow, , Russia Department of Mathematics Chair of Algebra Moscow State University Moscow, , Russia Abstract. We find all homogeneous symplectic varieties of connected semisimple Lie groups that admit an invariant linear connection. Introduction Let G be a connected Lie group. One can consider the following problem: to describe all homogeneous G-spaces X that admit an invariant linear connection (see [2], [3], [5]). Not much is known in general and the main attention was focused on the description of invariant linear connections on some particularly nice homogeneous spaces, mainly on reductive homogeneous spaces (see [1], [2]). The aim of this paper is to give a complete solution of this problem in the following situation: G is a connected semisimpte Lie group over field K = ~ or C and X TM G/H is a symplectic G-variety, that is, X is either an adjoint orbit of G or its covering. Our answer is given by the following theorem: Theorem 1. Let X cover the adjoint orbit Ad(G)x, x C g. Let x = x~ + Xn be the Jordan decomposition in 9, 3(x~) = ~ m the decomposition into a sum of the 1 m center ~ and simple ideals 9k of the centralizer of xs, and one has xn = x,~ x~, k where x n E 9k, k = 1,...,m. Then X admits an invariant linear connection if and only if, for k = 1,..., m, xk ~ 0 ~ 9k ~- 5p(2nk, ]K), nk C N and Xk is a highest root vector in 9~. (1) It is known [3] that semisimple adjoint orbits always admit an invariant linear connection and the situation of nilpotent orbit is one of the most important. In w we deduce Theorem 1 from the following particular cases. Theorem 2. Let G be a connected simple complex Lie group, and let X nilpotent adjoint G-orbit. The following assertions are equivalent: (a) there exists an invariant linear connection on X; (b) there exists an invariant linear connection on some covering X of X; (c) 9 ~ zp~(c) and X = (Ad G)e, where e is a highest root vector. be a nonzero *The research was supported by the grant INTAS-OPEN of the INTAS Foundation. Received February 23, Accepted December 26, 2000.

2 194 $. V. PIKULIN AND E. A. TEVELEV Theorem 3. Let G be a connected simple real Lie group, and let X be some covering of a nonzero nilpotent adjoint orbit Ad(G)e, e E t- Then X admits an invariant linear connection if and only if t ~ ~Pn(~) and e is the highest root vector in t c = Spn(C). A connection is called symplectic if the corresponding covariant derivative of the basic symplectic form ~ vanishes on X. It is observed in [7, w that for any G-invariant linear connection V on X without torsion there exists a canonically defined symplectic G-invariant linear connection V also without torsion. Namely, V differs from V by a symmetric tensor T(~, 7) = V~(~) - V~(~) of type (1, 2) defined by w(t(~, ~), ~) = ~((V~ca)(~/, ~) - (Vnw)(~, ~)), (2) for any vector fields ~, 7, C on X. Our result can be applied to studying *-products on X and formal deformations of the Poisson Lie algebra on X (see. [6]-[9]). In particular, it is known ([9]) that X admits a G-invariant.-product if and only if there exists a symplectic G-invariant linear connection on X, i.e., only in the cases described in Theorem 1. The paper is organized as follows. In w we recall some basics on invariant linear connections, and deduce Theorems 1 and 3 from Theorem 2. In w we prove the Basic Lemma, which says that if there exists an invariant linear connection on a nilpotent orbit X, then X is locally G-equivariantly isomorphic to an open orbit in some G-module. The Basic Lemma is proved a priori using an s[2-trick. In w we prove Theorem 2 using the well-known list of all G-modules having an open orbit. The authors are grateful to E. B. Vinberg for useful discussions and the simplification of some proofs. w Basics on invariant linear connections Let G be a connected Lie group over ]I{ or C with Lie algebra 1t. Let X be a homogeneous G-variety. Denote by H = Go C G the stabilizer of some point o C X, and let D be the Lie algebra of H. The tangent space ToX is canonically identified with t/~. For any~ E t we denote the vector~+~ C t/~ byt. IrA :~ ~ gis alinear operator such that Ab C 0, then A is the induced operator on g/b, A(~) = A~. Any C g defines the vector field ~* on X. The corresponding map t --+ Vect(X) is a Lie algebra homomorphism. Since X is homogeneous, the values of the vector fields ~* at any point of X span the tangent space at that point. In particular, we have {* (o) = 7. It is known (see [3, Theorem 2] or [5]) that there exists a 1-1 correspondence between the set of invariant linear connections on X and the set of linear maps F : t --+ End g/0 that satisfy the following conditions for any ~ C t, a C O, x E H: F(a) = ad(a) P(Ad(x)~) = Ad(x)r(~)Ad(x) (3) (4) If H is connected, then (4) is equivalent to = (5)

3 INVARIANT LINEAR CONNECTIONS ON SYMPLECTIC VARIETIES 195 In fact, the conditions (3), (5) are equivalent to the existence of an invariant linear connection on the universal covering of G/H. The covariant derivative of the vector field ~* in the direction ~* (o) = ~ is given by v,c (o) = (6) A connection is called locally fiat if it has zero curvature and torsion. The curvature p and the torsion ~- of the connection V at the point o are given by = v]) - r(v)], v) = - r(v) - v]. (7) Let us deduce Theorem 3 from Theorem 2. Proof of Theorem 3. If X admits an invariant linear connection, then the system of linear equations (3) and (5) has a solution. It follows from Theorem 2 that the system is consistent only if gc ~ sp~(c) and 0c is the centralizer of a highest root vector. On the other hand, 0 is the centralizer of some element e E g. Hence, O c is the centralizer of e in gc and so e is a highest root vector in gc But spn(i~) is the unique real form of 5p~(C) containing such vector. Conversely, let ~ ~ sp~(i~) and e is a highest root vector in tic. Then X covers the orbit Ad(G)e ~ (N n \ {0})/{tl} that admits a natural G-invariant linear connection. [] Since the space X = G/H is symplectic, one has the moment map # : X -+ 9 (we identify f~* and 9 by means of the Cartan inner product) and the covering map X -+ Ad(G)x, x = #(o) C g, where o = ell. Let x = xs + x, be the Jordan decomposition of x in g, where xs is semisimple, xn is nilpotent and [xs,x~] = 0. Since G is reduetive, the centralizer G1 = ZG(xs) C G of xs in G is a connected reductive subgroup. One has xn E gl and [~ = 91N3~(Xn)- Lemma 1. The space Ad(G)x (resp. its covering X) admits a G-invariant linear connection if and only if there exists a Gl-invariant linear connection on Ad(G1)xn (resp. on its covering X1 = G1/H). Proof. The assertion follows from the decomposition ~ = gl m, where Ad(G1)m 9 m, b C ~1 and Theorem 4 of [3]. [] Proof of Theorem 1. Due to Lemma 1 we may assume that x = x, is nilpotent. Suppose that X admits an invariant linear connection. The corresponding map F satisfies conditions (3) and (5). Denote bi = b N g~. Consider the map F~ : gi -+ End(~Ji/bi) defined by Fi(~)~ = pr~f(~)~ for ~ E g~, ~ E 9i/Oi where pri : g/[l ~ ~i/[li is the projecting map along Ojr t?j/bj. It is easy to see that F~ gives rise to an invariant linear connection on the universal covering of the space Xi = Ad(G)xi. Actually, Xi is a nilpotent adjoint orbit of the simple Lie group Gi C Ad(G) with the tangent algebra gi- Now (1) follows from Theorems 2 and 3. Let (1) hold. Then by Theorems 2 and 3 there exist invariant linear connections on Ad(G~)xi. The direct sum of the corresponding maps F satisfies conditions (3), (4) and hence gives rise to an invariant linear connection on Ad(G)x. Since X covers Ad(G)x, it also admits an invariant linear connection. []

4 196 S. V. PIKULIN AND E. A. TEVELEV From now on we assume that G is a complex Lie group. It is important that we can average connections in the following sense: Lemma 2. Suppose that K C G is a reductive subgroup that normalizes H and X admits a G-invariant linear connection V. Then X admits an invariant linear connection V such that the corresponding map F satisfies the following additional condition: r(ad(x)~) = Ad(x)C(~)Ad(x) for any x C K, ~ e g. Pro@ We note that conditions (3) and (4) define an affine subspace M in the linear space L = Horn(9, End 9/ )- The group K acts on M in a natural way. If a reductive group K acts linearly on a vector space L and preserves an affine subspace M, then M must contain a K-fixed vector that represents the desired connection. [] The lemma's assertion means that V is invariant under the G-equivariant action of K on G/H by right shifts. w Invariant linear connections on nilpotent orbits Let e E ~ be a nilpotent element, X = Ad(G)e, and let -Y be the universal covering of X. By the Jacobson-Morozov Theorem the nilpotent element e E g can be included into the zl2-triptet (f, h, e) C g. Basic Lemma. Suppose that there exists an invariant linear connection on X. Then there exists the unique invariant linear connection on X such that the corresponding map F satisfies the following additional condition: This connection is locally fiat and symplectic. r([h, ~1) =[ad h, P(r for any r E g. (8) Proof. Let T be the one-parameter subgroup whose Lie algebra is spanned by h. Clearly T normalizes H = Gc and hence its identity component H ~ By Lemma 2 we obtain (8) for some connection on X. Lemma 3. There exists an s[2-triplet (~,~,g) C Endg/b, such that g = ade and ~/= ad h + Ida/0. Proof. Let Y C g be an irreducible ~[2-submodule in g with highest weight it and highest weight vector y. Then ~ - Y ~ 0 = (y). The operator adely induces the unique structure of a simple s[2-module on Y - Y/O, the highest weight of which is it - 1 and the highest vector is If, y]. We assume by definition that the desired z[2-triplet acts on Y in accordance with described structure. We notice that if [h, ~] = k~ for ~ C Y, then HIvE = (k + 1)~; therefore ~/= ad h + Idg/0. [] It follows from Lemma 3, equation (8), and (5) that for any ~ E 9 we have C([e,~]) = [g, C(~)], C([h,~]) = [7/,F(~)] for any ~ C g. (9)

5 INVARIANT LINEAR CONNECTIONS ON SYMPLECTIC VARIETIES 197 It is well known that for any reductive group S and any S-modules U and V the linear map U --+ V is S-equivariant if and only if it is equivariant with respect to a Borel subgroup of S. It follows that the equations (9) imply that we also have r([f, ~]) = [f, r(~)] for any ~ e ~. (10) Substituting e and h instead of ~ in (10) we get 7/= F(h) and F = F(f). Therefore, for any ~ E r([/,~]) = [F(f),r(~)], r([h,~]) = [r(h),r(~)], r([e,~]) = [r(e),r(~)]. (11) The subalgebras O C ~ and (f, h, e) C ~ together generate ~. Therefore, (5) and (11) imply that F([~, ~]) = [F(~), F(~)] for any ~, ~ e O. (12) So F is a uniquely defined representation of ~ in ~/O and the curvature of the corresponding invariant linear connection V vanishes by (7). We are to check now that V is torsion-free. Let us consider the symmetrization of ~7, which is an invariant linear connection without torsion. Obviously, it satisfies (8) and consequently it coincides with V, so V is locally flat. Furthermore, consider the symplectic connection V defined by (2). It follows from (6) that ~7 satisfies (8). Hence V = V is a symplectic connection. The Basic Lemma is proved. [] w Proof of Theorem 2 Proof. The implication (a) ~ (b) is trivial since any invariant linear connection on X can be lifted to an invariant linear connection on )~. Let us prove that (c) ~ (a). Let G ~ Sp~(C). The adjoint representation of G is isomorphic to the symmetric square of the standard representation of G. This isomorphism takes the orbit of the highest root vector to the variety of nonzero "perfect squares" {u ~ E S 2 C n [u E cn\{0}}. The open orbit in the standard representation of G in C n covers this orbit. Therefore, X is obtained from C n \ {0} with a natural action of G by identifying opposite points. Obviously, this variety admits an invariant linear connection. Finally let us check that (b) ~ (c). It follows from (b) that there exists an invariant linear connection on the universal covering )(. Moreover, we can assume that this connection satisfies the assertions of the Basic Lemma. It follows from [3, w that the invariant linear connection ~J given by the linear map F : fl -+ End g/~ is locally flat if and only if the map P is a locally transitive representation of g in ~t/~, that is, if there exists a vector v E g/b such that F(g)v = g/o- Furthermore, it follows from [3, w that all locally transitive representations of simple algebras 0 are given in the following table: 1 At, 1>1 2 A~I, l _> 2 3 A2t, l > 2 4 A2z, l > 2 5 C~, l _> 2 6 D5 F kr(~l), k=l,2,...,1 R( 2) R( I) R( 4)

6 198 S. V. PIKULIN AND E. A. TEVELEV Here 7rk is the k-th fundamental weight of t~ (in the numbering of [4]). R(A) is an irreducible representation of g with the highest weight A, and R(,~)* is its dual. R1 is the direct sum of R1 and R~, kr is the direct sum of k copies of R. Note that since the connection V is sympleetic by the Basic Lemma, the corresponding representation F is also sympleetic. So, case 5 is the only possible one. The adjoint representation of a simple group has a unique nonzero orbit of minimal dimension, namely, the orbit of a highest root vector. Therefore in the case 5 the only possible variant was considered above. The theorem is proved. [] References [1] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, vol. 1, 1963, vol. 2, 1969~ Interscience Publishers, New York, London, Sydney. Russian transl.: HI. IdoSazc~, t/. HoM~3y, Ocnoeb* Our,eoJuempuu, TT. I H II, M~p, M., [2] K. Nomizu, Lie Groups and Differential Geometry, Math. Society of Japan~ Russian transl.: K. HOM~3y~ Fpynnb~ flu u Ou~epen~tuanbna~ zeomempu~, BHS~. c5. "MaTe- MaT~Ka", FIJI, M., [3] E. B. B~HSepr, Ifnoapuanmn~e nune~n~e co~r~ocmu e oonopoono~u npocmpancmee, Tpyg~i Mock. MaT. O-Ba, 9 (1960), [4] E. 13. B~HSepr, A. JI. O~I4~, Ce~uunap no ~pynnam flu u an~espaunecroum ~pynna,u, Hay~a, M., English transl.: A. L. Onishchik, E. B. Vinberg, Lie Groups and Algebraic Groups, Springer, Berlin, [5] H.-C. Wang, On invariant connections over a principal fibre bundle, Nagoya Math., 13 (1958), [6] M. de Wilde, P. Lecomte, Formal deformations of the Poisson Lie algebra of a symplectic manifold and star-products. Existence, equivalence~ derivations, in: Deformation Theory of Algebras and Structures and Applications, Kluwer Academic Publishers, 1988, [7] A. Lichnerowicz, D~formations d'alg~bres assocides d une varidtd symplectique (los.,- produits)~ Ann. Inst. Fourier 32 (1982), no. 1, [8] O. M. Neroslavsky, A. T. Vlassov, Sur los ddformations de FaIg~bres des fonctions d'une varidtd sympleetique, C. R. Acad. Sc. Paris 292 (1981), [9] D. Melotte, Invariant deformations of the Poisson Lie algebra of a symplectic manifold and star-products, in: Deformation Theory of Algebras and Structures and Applieations~ Kluwer Academic Publ., 1988~