Gelfand-Zeitlin theory from the perspective of classical mechanics II

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In this paper, Part II, of a two part paper we apply the results of [KW], Part I, to establish, with an explicit dual coordinate system, a commutative analogue of the Gelfand-Kirillov theorem for M(n), the algebra of n× n complex matrices. The function field F(n) of M(n) has a natural Poisson structure and an exact analogue would be to show that F(n) is isomorphic to the function field of a n(n-1)-dimensional phase space over a Poisson central rational function field in n variables. Instead we show that this the case for a Galois extension, F(n, e), of F(n). The techniques use a maximal Poisson commutative algebra of functions arising from Gelfand-Zeitlin theory, the algebraic action of a n(n-1)/2--dimensional torus on F(n, e), and the structure of a Zariski open subset of M(n) as a n(n-1)/2--dimensional torus bundle over a n(n+1)/2--dimensional base space of Hessenberg matrices.