Fomenko-Mischenko Theory, Hessenberg Varieties, and Polarizations
Abstract
The symmetric algebra g (denoted S()) over a Lie algebra (frak g) has the structure of a Poisson algebra. Assume is complex semi-simple. Then results of Fomenko- Mischenko (translation of invariants) and A.Tarasev construct a polynomial subalgebra H = C[q1,...,qb] of S() which is maximally Poisson commutative. Here b is the dimension of a Borel subalgebra of . Let G be the adjoint group of and let = rank . Identify with its dual so that any G-orbit O in has the structure (KKS) of a symplectic manifold and S() can be identified with the affine algebra of . An element x ∈ is strongly regular if \(dqi)x\, i=1,...,b, are linearly independent. Then the set sreg of all strongly regular elements is Zariski open and dense in , and also sreg ⊂ reg where reg is the set of all regular elements in . A Hessenberg variety is the b-dimensional affine plane in , obtained by translating a Borel subalgebra by a suitable principal nilpotent element. This variety was introduced in [K2]. Defining Hess to be a particular Hessenberg variety, Tarasev has shown that Hess ⊂ sreg. Let R be the set of all regular G-orbits in . Thus if O ∈ R, then O is a symplectic manifold of dim 2n where n= b-. For any O∈ R let Osreg = sreg O. We show that Osreg is Zariski open and dense in O so that Osreg is again a symplectic manifold of dim 2n. For any O ∈ R let Hess (O) = Hess O. We prove that Hess(O) is a Lagrangian submanifold of Osreg and Hess =O ∈ R Hess(O). The main result here shows that there exists, simultaneously over all O ∈ R, an explicit polarization (i.e., a "fibration" by Lagrangian submanifolds) of Osreg which makes Osreg simulate, in some sense, the cotangent bundle of Hess(O).