m at h . SG ] 2 4 A ug 2 00 4 Gelfand – Zeitlin theory from the perspective of classical mechanics I

Let M(n) be the algebra (both Lie and associative) of n×n matrices over C. Then M(n) inherits a Poisson structure from its dual using the bilinear form (x, y) = −tr xy. The Gl(n) adjoint orbits are the symplectic leaves and the algebra, P (n), of polynomial functions on M(n) is a Poisson algebra. In particular if f ∈ P (n) then there is a corresponding vector field ξf on M(n). If m ≤ n then M(m) embeds as a Lie subalgebra of M(n) (upper left hand block) and P (m) embeds as a Poisson subalgebra of P (n). Then, as an analogue of the Gelfand–Zeitlin algebra in the enveloping algebra of M(n), let J(n) be the subalgebra of P (n) generated by P (m) for m = 1, . . . , n. One observes that J(n) ∼= P (1) ⊗ · · · ⊗ P (n) We prove that J(n) is a maximal Poisson commutative subalgebra of P (n) and that for any p ∈ J(n) the holomorphic vector field ξf is integrable and generates a global one parameter group σp(z) of holomorphic transformations of M(n). If d(n) = n(n+1)/2 then J(n) is a polynomial ring C[p1, . . . , pd(n)] and the vector fields ξpi , i ∈ Id(n), span a commutative Lie algebra of dimension d(n − 1). Let A be a corresponding simply-connected Lie group so that A ∼= C. Then A operates on M(n) by an action σ so that if a ∈ A then σ(a) = σp1(z1) · · ·σpd(n)(zd(n)) where a is the product of exp zi ξpi for i = 1, . . . , d(n). We prove that the orbits of A are independent of the choice of the generators pi. In addition we prove the following results about this rather remarkable group action. (The latter is a very extensive enlargement of an abelian group action introduced in [GS]). * Research supported in part by NSF grant DMS-0209473 and in part by the KG&G Foundation. ** Research supported in part by NSF grant MTH 0200305. 2000 Mathematics Subject Classification: Primary14L30, 14R20, 33C45, 53D17