We show that every Lie algebroid A over a manifold P has a natural representation on the line bundle Q A = ∧ top A ⊗ ∧ top T * P . The line bundle Q A may be viewed as the Lie algebroid analog of the orientation bundle in topology, and sections of Q A may be viewed as transverse measures to A. As a consequence, there is a well-defined class in the first Lie algebroid cohomology H 1 (A) called the modular class of the Lie algebroid A. This is the same as the one introduced earlier by Weinstein using the Poisson structure on A * . We show that there is a natural pairing between the Lie algebroid cohomology spaces of A with trivial coefficients and with coefficients in Q A . This generalizes the pairing used in the Poincare duality of finite-dimensional Lie algebra cohomology. The case of holomorphic Lie algebroids is also discussed, where the existence of the modular class is connected with the Chern class of the line bundle Q A .

We study Lagrangian subalgebras of a semisimple Lie algebra with respect to the imaginary part of the Killing form. We show that the variety L of Lagrangian subalgebras carries a natural Poisson structure Π. We determine the irreducible components of L, and we show that each irreducible component is a smooth fiber bundle over a generalized flag variety, and that the fiber is the product of the real points of a De Concini-Procesi compactification and a compact homogeneous space. We study some properties of the Poisson structure Π and show that it contains many interesting Poisson submanifolds.

Abstract.Let X be the flag variety of a compact Lie group and let h* be a complex-oriented generalized cohomology theory. We introduce operators on h*(X) which generalize operators introduced by Bernstein, Gel fand, and Gel fand for rational cohomology and by Demazure for if-theory. Using the Becker-Gottlieb transfer, we give a formula for these operators, which enables us to prove that they satisfy braid relations only for the two classical cases, thereby giving a topological interpretation of a theorem proved by the authors and extended by Gutkin.One of the central issues in Lie theory is the geometry of the flag variety associated to a compact Lie group G. An important problem concerning the flag variety is the Schubert calculus, which studies the ring structure of the cohomology of the flag variety. Work initiated by Borel, Bott and Kostant, which culminated in a paper by Bernstein, Gel'fand and Gel'fand [BGG], gave a complete solution to the problem. Demazure studied the same problem for ./sT-theory. Moreover, these techniques have been generalized to the Kac-Moody situation by Kac-Peterson, Kostant-Kumar, and others. This work has focussed on algebro-geometric properties of the flag variety.Here, on the other hand we study the flag variety from the point of view of algebraic topology. As a consequence, not only do we recover the classical results described above, but we extend these results to a certain class of cohomology theories-those which are complex-oriented. Examples of complex-oriented theories include ordinary cohomology, A^-theory, complex cobordism, and elliptic cohomology. Since the context we have chosen in very general, the proofs are universal and are often simpler than the classical arguments.In the work of BGG, a crucial role is played by operators Ai associated to each simple reflection s¡ of the Weyl group of G (defined by Demazure in if-theory). These operators Ai satisfy the braid relations, which are the relations between pairs of simple reflections. In this paper, we generalize the Ai to give operators Di acting on h*(G/T) for any complex-oriented theory h*. We prove that braid relations are satisfied only for cohomology theories with the formal group law of rational cohomology or of AMheory (Theorem

Motivated by Drinfeld's theorem on Poisson homogeneous spaces, we study the variety L of Lagrangian subalgebras of g ⊕ g for a complex semi-simple Lie algebra g. Let G be the adjoint group of g. We show that the (G × G)-orbit closures in L are smooth spherical varieties. We also classify the irreducible components of L and show that they are smooth. Using some methods of M. Yakimov, we give a new description and proof of Karolinsky's classification of the diagonal G-orbits in L, which, as a special case, recovers the Belavin-Drinfeld classification of quasi-triangular r-matrices on g. Furthermore, L has a canonical Poisson structure, and we compute its rank at each point and describe its symplectic leaf decomposition in terms of intersections of orbits of two subgroups of G × G.