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Fomenko-Mischenko Theory, Hessenberg Varieties, and Polarizations

The symmetric algebra g (denoted S(\g)) over a Lie algebra \g (frak g) has the structure of a Poisson algebra. Assume \g is complex semi-simple. Then results of Fomenko- Mischenko (translation of invariants) and A.Tarasev construct a polynomial subalgebra \cal H = \bf C[q_1,...,q_b] 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 \ell = rank \g. 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 \in \g is strongly regular if \{(dq_i)_x\}, i=1,...,b, are linearly independent. Then the set \g^{sreg} of all strongly regular elements is Zariski open and dense in \g, and also \g^{sreg \subset \g^{reg} where \g^{reg} 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. This variety was introduced in [K2]. Defining Hess to be a particular Hessenberg variety, Tarasev has shown that Hess \subset \g^sreg. Let R be the set of all regular G-orbits in \g. Thus if O \in R, then O is a symplectic manifold of dim 2n where n= b-\ell. For any O\in R let O^{sreg} = \g^{sreg}\cap O. We show that O^{sreg} is Zariski open and dense in O so that O^{sreg} is again a symplectic manifold of dim 2n. For any O \in R let Hess (O) = Hess \cap O. We prove that Hess(O) is a Lagrangian submanifold of O^{sreg} and Hess =\sqcup_{O \in R} Hess(O). The main result here shows that there exists, simultaneously over all O \in R, an explicit polarization (i.e., a "fibration" by Lagrangian submanifolds) of O^{sreg} which makes O^{sreg} simulate, in some sense, the cotangent bundle of Hess(O).