Fomenko–Mischenko Theory, Hessenberg Varieties, and Polarizations

The symmetric algebra S(g) over a Lie algebra g has the structure of a Poisson algebra. Assume g is complex semisimple. Then results of Fomenko–Mischenko (translation of invariants) and A. Tarasev construct a polynomial subalgebra H = C[q1, . . . , qb] of S(g) which is maximally Poisson commutative. Here b is the dimension of a Borel subalgebra of g. Let G be the adjoint group of g and let l = rank g. Using the Killing form, identify g with its dual so that any G-orbit O in g has the structure (KKS) of a symplectic manifold and S(g) can be identified with the affine algebra of g. An element x ∈ g will be called strongly regular if {(dqi)x}, i = 1, . . . , b, are linearly independent. Then the set g of all strongly regular elements is Zariski open and dense in g and also g ⊂ g where g is the set of all regular elements in g. A Hessenberg variety is the b-dimensional affine plane in g, obtained by translating a Borel subalgebra by a suitable principal nilpotent element. Such a variety was introduced in [K2]. Defining Hess to be a particular Hessenberg variety, Tarasev has shown that Hess ⊂ g. Let R be the set of all regular G-orbits in g. Thus if O ∈ R, then O is a symplectic manifold of dimension 2n where n = b − l. For any O ∈ R let O = g ∩ O. One shows that O is Zariski open and dense in O so that O is again a symplectic manifold of dimension 2n. For any O ∈ R let Hess(O) = Hess ∩ O. One proves that Hess(O) is a Lagrangian submanifold of O and that Hess = ⊔O∈RHess(O). The main result of this paper is to show that there exists simultaneously over all O ∈ R, an explicit polarization (i.e., a “fibration” by Lagrangian submanifolds) of O which makesO simulate, in some sense, the cotangent bundle of Hess(O).