Symplectic geometry of semisimple orbits (original) (raw)

2008, Indagationes Mathematicae

Let G be a complex semisimple group, T ⊂ G a maximal torus and B a Borel subgroup of G containing T. Let Ω be the Kostant-Kirillov holomorphic symplectic structure on the adjoint orbit O = Ad(G)c G/Z(c), where c ∈ Lie(T), and Z(c) is the centralizer of c in G. We prove that the real symplectic form Re Ω (respectively, Im Ω) on O is exact if and only if all the eigenvalues ad (c) are real (respectively, purely imaginary). Furthermore, each of these real symplectic manifolds is symplectomorphic to the cotangent bundle of the partial flag manifold G/Z(c)B, equipped with the Liouville symplectic form. The latter result is generalized to hyperbolic adjoint orbits in a real semisimple Lie algebra. H. AZAD, E. P. VAN DEN BAN, AND I. BISWAS Unless specified otherwise, we will use B to identify g with g *. In particular, by pull-back under B of the canonical Kostant-Liouville holomorphic symplectic form on any coadjoint orbit O ⊂ g * may be viewed as a holomorphic symplectic form on the associated adjoint orbit B −1 (O). If c ∈ g, then by ad (c) we denote the endomorphism Y → [c, Y ] of g. The element c is called semisimple if and only if ad (c) diagonalizes. Equivalently, this means that c is contained in the Lie algebra of a maximal torus (or Cartan subgroup) T of G. The centralizer of c in G is denoted by Z(c). If c is semisimple, then Z(c) is known to be the Levi component of a parabolic subgroup P of G. In fact, one may take P = Z(c)B, where B is a Borel subgroup containing a maximal torus which contains c. We will prove the following generalization of Arnold's result. Theorem 1.2. Let G be a connected complex semisimple group, and let c be a semisimple element of its Lie algebra g. Let Ω be the Kostant-Kirillov holomorphic symplectic form on the orbit O = Ad(G)c G/Z(c). Then the real and imaginary parts Re Ω and Im Ω are real symplectic forms on O. Moreover, the following hold. (a) The form Re Ω (respectively, Im Ω) on O is exact if and only if all eigenvalues of ad (c) are real (respectively, purely imaginary). (b) In either case, these symplectic manifolds with exact real symplectic forms are symplectomorphic to the total space of the cotangent bundle of G/P, equipped with the Liouville symplectic form, where P is any parabolic subgroup of G with Levi component Z(c). In fact, we will prove a refinement of assertion (b) in the more general context of a real hyperbolic adjoint orbit of a real semisimple Lie group; see Theorems 2.11 and 6.1. Here are a few words about our interpretation of the above mentioned result of Arnold. Set G = SL(n + 1, C), and let T ⊂ G be the subgroup of diagonal matrices. For any c ∈ Lie(T) with distinct eigenvalues we have Z(c) = T, so that the adjoint orbit of c can be identified with G/T. Let c i denote the i-th diagonal entry of c. The eigenvalues of ad (c) are all the numbers of the form c i − c j , with 1 ≤ i, j ≤ n. As j c j = 0, it follows that the eigenvalues of c are all real if and only if those of ad (c) are.